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==[[Schedule of Project Presentations]] ==
 
==[[Schedule of Project Presentations]] ==
 
==[[Proposal Fall 2010]] ==
 
==[[Proposal Fall 2010]] ==
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==[[Mark your contribution here]]==
  
 
==[[statf10841Scribe|Editor sign up]] ==
 
==[[statf10841Scribe|Editor sign up]] ==
 
{{Cleanup|date=October 8 2010|reason=Provide a summary for each topic here.}}
 
{{Cleanup|date=October 8 2010|reason=Provide a summary for each topic here.}}
== Summary ==
 
=== Classification ===
 
'''Statistical classification''', or simply known as classification, is an area of [http://en.wikipedia.org/wiki/Supervised_learning supervised learning] that addresses the problem of how to systematically assign unlabeled (classes unknown) novel data to their labels (classes or groups or types) by using knowledge of their features (characteristics or attributes) that are obtained from observation and/or measurement. A [http://en.wikipedia.org/wiki/Classifier_%28mathematics%29 classifier] is a specific technique or method for performing classification.
 
To classify new data, a classifier first uses labeled (classes are known) [http://en.wikipedia.org/wiki/Training_set training data] to [http://en.wikipedia.org/wiki/Mathematical_model#Training train] a model, and then it uses a function known as its [http://en.wikipedia.org/wiki/Decision_rule classification rule] to assign a label to each new data input after feeding the input's known feature values into the model to determine how much the input belongs to each class.
 
 
===LDA x QDA===
 
 
Linear discriminant analysis[http://en.wikipedia.org/wiki/Linear_discriminant_analysis] is a statistical method used to find the ''linear combination'' of features which best separate two or more classes of objects or events. It is widely applied in classifying diseases, positioning, product management, and marketing research. LDA assumes that the different classes have the same covariance matrix <math>\, \Sigma</math>.
 
 
Quadratic Discriminant Analysis[http://en.wikipedia.org/wiki/Quadratic_classifier], on the other hand, aims to find the ''quadratic combination'' of features. It is more general than linear discriminant analysis. Unlike LDA, QDA does not make the assumption that the different classes have the same covariance matrix <math>\, \Sigma</math>. Instead, QDA makes the assumption that each class <math>\, k</math> has its own covariance matrix <math>\, \Sigma_k</math>.
 
=== Principle Component Analysis ===
 
Principal component analysis (PCA) is a dimensionality-reduction method invented by [http://en.wikipedia.org/wiki/Karl_Pearson Karl Pearson] in 1901 [http://stat.smmu.edu.cn/history/pearson1901.pdf]. Depending on where this methodology is applied, other common names of PCA include the [http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem Karhunen–Loève transform (KLT)] , the [http://en.wikipedia.org/wiki/Harold_Hotelling Hotelling transform], and the proper orthogonal decomposition (POD). PCA is the simplist [http://en.wikipedia.org/wiki/Eigenvector eigenvector]-based [http://en.wikipedia.org/wiki/Multivariate_analysis multivariate analysis]. It reduces the dimensionality of the data by revealing the internal structure of the data in a way that best explains the variance in the data. To this end, PCA works by using a user-defined number of the most important directions of variation (dimensions or '''principal components''') of the data to project the data onto these directions so as to produce a lower-dimensional representation of the original data. The resulting lower-dimensional representation of our data is usually much easier to visualize and it also exhibits the most informative aspects (dimensions) of our data whilst capturing as much of the variation exhibited by our data as it possibly could.
 
 
 
==[[f10_Stat841_digest |Digest ]] ==
 
==[[f10_Stat841_digest |Digest ]] ==
  
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As another example, suppose we wish to classify newly-given fruits into apples and oranges by considering three features of a fruit that comprise its color, its diameter, and its weight. After selecting a classifier and constructing a model using training data <math>\,\{(X_{color, 1}, X_{diameter, 1}, X_{weight, 1}, Y_{1}), \dots , (X_{color, n}, X_{diameter, n}, X_{weight, n}, Y_{n})\}</math>, we could then use the classifier's classification rule <math>\,h</math> to assign any newly-given fruit having known feature values <math>\,x = (\,x_{color}, x_{diameter} , x_{weight})</math> the label <math>\, \hat{Y}=h(x) \in \mathcal{Y}= \{apple,orange\}</math>.
 
As another example, suppose we wish to classify newly-given fruits into apples and oranges by considering three features of a fruit that comprise its color, its diameter, and its weight. After selecting a classifier and constructing a model using training data <math>\,\{(X_{color, 1}, X_{diameter, 1}, X_{weight, 1}, Y_{1}), \dots , (X_{color, n}, X_{diameter, n}, X_{weight, n}, Y_{n})\}</math>, we could then use the classifier's classification rule <math>\,h</math> to assign any newly-given fruit having known feature values <math>\,x = (\,x_{color}, x_{diameter} , x_{weight})</math> the label <math>\, \hat{Y}=h(x) \in \mathcal{Y}= \{apple,orange\}</math>.
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=== Examples of Classification ===
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• Email spam filtering (spam vs not spam).
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• Detecting credit card fraud (fraudulent or legitimate).
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• Face detection in images (face or background).
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• Web page classification (sports vs politics vs entertainment etc).
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• Steering an autonomous car across the US (turn left, right, or go straight).
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• Medical diagnosis (classification of disease based on observed symptoms).
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=== Independent and Identically Distributed (iid) Data Assumption ===
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Suppose that we have training data X which contains N data points.  The Independent and Identically Distributed (IID) assumption declares that the datapoints are drawn independently from identical distributions.  This assumption implies that ordering of the data points does not matter, and the assumption is used in many classification problems.  For an example of data that is not IID, consider daily temperature: today's temperature is not independent of the yesterday's temperature -- rather, there is a strong correlation between the temperatures of the two days.
  
 
=== Error rate ===
 
=== Error rate ===
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In practice, the empirical error rate is obtained to estimate the true error rate, whose value is impossible to be known because the parameter values of the underlying process cannot be known but can only be estimated using available data. The empirical error rate, in practice, estimates the true error rate quite well in that, as mentioned [http://www.liebertonline.com/doi/pdf/10.1089/106652703321825928 here], it is an unbiased estimator of the true error rate.
 
In practice, the empirical error rate is obtained to estimate the true error rate, whose value is impossible to be known because the parameter values of the underlying process cannot be known but can only be estimated using available data. The empirical error rate, in practice, estimates the true error rate quite well in that, as mentioned [http://www.liebertonline.com/doi/pdf/10.1089/106652703321825928 here], it is an unbiased estimator of the true error rate.
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An Error Rate Comparison of Classification Methods [http://pdfserve.informaworld.com/311525_770885140_713826662.pdf]
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=== Decision Theory ===
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we can identify three distinct approaches to solving decision problems, all of which have been used in practical applications. These are given, in decreasing order of complexity, by:
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a. First solve the inference problem of determining the class-conditional densities <math>\ p(x|C_k)</math> for each class <math>\ C_k</math> individually. Also separately infer the prior class probabilities <math>\ p(C_k)</math>. Then use Bayes’ theorem in the form
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<math>\begin{align}p(C_k|x)=\frac{p(x|C_k)p(C_k)}{p(x)} \end{align}</math>
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to find the posterior class probabilities <math>\ p(C_k|x)</math>. As usual, the denominator in Bayes’ theorem can be found in terms of the quantities appearing in the numerator, because
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<math>\begin{align}p(x)=\sum_{k} p(x|C_k)p(C_k) \end{align}</math>
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Equivalently, we can model the joint distribution <math>\ p(x, C_k)</math> directly and then normalize to obtain the posterior probabilities. Having found the posterior probabilities, we use decision theory to determine class membership for each new input <math>\ x</math>. Approaches that explicitly or implicitly model the distribution of inputs as well as outputs are known as "generative models", because by sampling from them it is possible to generate synthetic data points in the input space.
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b. First solve the inference problem of determining the posterior class probabilities <math>\ p(C_k|x)</math>, and then subsequently use decision theory to assign each new <math>\ x</math> to one of the classes. Approaches that model the posterior probabilities directly
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are called "discriminative models".
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c. Find a function <math>\ f(x)</math>, called a discriminant function, which maps each input <math>\ x</math> directly onto a class label. For instance, in the case of two-class problems, <math>\ f(.)</math> might be binary valued and such that <math>\ f = 0</math> represents class <math>\ C_1</math> and <math>\ f = 1</math> represents class <math>\ C_2</math>. In this case, probabilities play no role.
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This topic has brought to you from Pattern Recognition and Machine Learning by Christopher M. Bishop (Chapter 1)
  
 
=== Bayes Classifier ===
 
=== Bayes Classifier ===
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:[[File:裁剪.jpg]]
 
:[[File:裁剪.jpg]]
{{Cleanup|date=September 2010|reason=this graph is not complete, the reason is that it should be in consistent with the computation below.}}
 
  
 
For each student, his/her feature values is <math>\, x = \{G, M, H\} </math> and his or her class is <math>\, y \in \{0, 1\} </math>.
 
For each student, his/her feature values is <math>\, x = \{G, M, H\} </math> and his or her class is <math>\, y \in \{0, 1\} </math>.
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<br />
 
<br />
<math>\, \hat r(x) = P(Y=1|X =(0,1,0))=\frac{P(X=(0,1,0)|Y=1)P(Y=1)}{P(X=(0,1,0)|Y=0)P(Y=0)+P(X=(0,1,0)|Y=1)P(Y=1)}=\frac{0.05*0.5}{0.05*0.5+0.2*0.5}=\frac{0.025}{0.075}=\frac{1}{3}<\frac{1}{2}.</math><br />
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<math>\, \hat r(x) = P(Y=1|X =(0,1,0))=\frac{P(X=(0,1,0)|Y=1)P(Y=1)}{P(X=(0,1,0)|Y=0)P(Y=0)+P(X=(0,1,0)|Y=1)P(Y=1)}=\frac{0.05*0.5}{0.05*0.5+0.2*0.5}=\frac{0.025}{0.125}=\frac{1}{5}<\frac{1}{2}.</math><br />
  
 
The Bayes classifier assigns the new student into the class <math>\, h^*(x)=0 </math>. Therefore, we predict that the new student would fail the course.
 
The Bayes classifier assigns the new student into the class <math>\, h^*(x)=0 </math>. Therefore, we predict that the new student would fail the course.
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More information regarding the Bayesian and the frequentist schools of thought are available [http://www.statisticalengineering.com/frequentists_and_bayesians.htm here]. Furthermore, an interesting and informative youtube video that explains the Bayesian and frequentist views of probability is available [http://www.youtube.com/watch?v=hLKOKdAircA here].
 
More information regarding the Bayesian and the frequentist schools of thought are available [http://www.statisticalengineering.com/frequentists_and_bayesians.htm here]. Furthermore, an interesting and informative youtube video that explains the Bayesian and frequentist views of probability is available [http://www.youtube.com/watch?v=hLKOKdAircA here].
  
== '''Linear and Quadratic Discriminant Analysis'''  ==
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There is useful information about Machine Learning, Neural and Statistical Classification in this link [http://www.amsta.leeds.ac.uk/~charles/statlog/] Machine Learning, Neural and Statistical Classification; there is some description of Classification in chapter 2 Classical Statistical Methods in chapter 3 and  Modern Statistical Techniques in chapter 4.
First, we shall limit ourselves to the case where there are two classes, i.e. <math>\, \mathcal{Y}=\{0, 1\}</math>. In the above discussion, we introduced the Bayes classifier's ''decision boundary'' <math>\,D(h^*)=\{x: P(Y=1|X=x)=P(Y=0|X=x)\}</math>, which represents a [http://en.wikipedia.org/wiki/Hyperplane hyperplane] that determines the class of any new data input depending on which side of the hyperplane the input lies in. Now, we shall look at how to derive the Bayes classifier's decision boundary under certain assumptions of the data. [http://en.wikipedia.org/wiki/Linear_discriminant_analysis Linear discriminant analysis (LDA)] and [http://en.wikipedia.org/wiki/Quadratic_classifier#Quadratic_discriminant_analysis quadratic discriminant analysis (QDA)] are two of the most well-known ways for deriving the Bayes classifier's decision boundary, and we shall look at each of them in turn.
+
 
 +
=== Extension: Statistical Classification Framework ===
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In statistical classification, each object is represented by 'd' (a set of features) a measurement vector, and the goal of classifier becomes finding compact and disjoint regions for classes in a d-dimensional feature space. Such decision regions are defined by decision rules that are known or can be trained. The simplest configuration of a classification consists of a decision rule and multiple membership functions; each membership function represents a class. The following figures illustrate this general framework.
  
Let us denote the likelihood <math>\ P(X=x|Y=y) </math> as <math>\ f_y(x) </math> and the prior probability <math>\ P(Y=y) </math> as <math>\ \pi_y </math>.
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[[File:cs1.png]]
  
First, we shall examine LDA. As explained above, the Bayes classifier is optimal. However, in practice, the prior and conditional densities are not known. Under LDA, one gets around this problem by making the assumptions that both of the two classes have [http://en.wikipedia.org/wiki/Multivariate_normal_distribution multivariate normal (Gaussian) distributions] and the two classes have the same covariance matrix <math>\, \Sigma</math>. Under the assumptions of LDA, we have: <math>\ P(X=x|Y=y) = f_y(x) = \frac{1}{ (2\pi)^{d/2}|\Sigma|^{1/2} }\exp\left( -\frac{1}{2} (x - \mu_k)^\top \Sigma^{-1} (x - \mu_k) \right)</math>. Now, to derive the Bayes classifier's decision boundary using LDA, we equate <math>\, P(Y=1|X=x) </math> to <math>\, P(Y=0|X=x) </math> and proceed from there. The derivation of <math>\,D(h^*)</math> is as follows:
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Simple Conceptual Classifier.
  
:<math>\,Pr(Y=1|X=x)=Pr(Y=0|X=x)</math>
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[[File:cs2.png]]
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 +
[http://www.orfeo-toolbox.org/SoftwareGuide/SoftwareGuidech17.html#x44-2480011 Statistical Classification Framework]
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 +
 
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The classification process can be described as follows:
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 +
A measurement vector is input to each membership function.
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Membership functions feed the membership scores to the decision rule.
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A decision rule compares the membership scores and returns a class label.
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 +
== '''Linear and Quadratic Discriminant Analysis'''  ==
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 +
===Introduction===
 +
'''Linear discriminant analysis''' ([http://en.wikipedia.org/wiki/Linear_discriminant_analysis LDA]) and the related '''Fisher's linear discriminant''' are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterize or separate two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification.
 +
 
 +
LDA is also closely related to principal component analysis ([http://en.wikipedia.org/wiki/Principal_component_analysis PCA]) and [http://en.wikipedia.org/wiki/Factor_analysis factor analysis] in that both look for linear combinations of variables which best explain the data. LDA explicitly attempts to model the difference between the classes of data. PCA on the other hand does not take into account any difference in class, and factor analysis builds the feature combinations based on differences rather than similarities. Discriminant analysis is also different from factor analysis in that it is not an interdependence technique: a distinction between independent variables and dependent variables (also called criterion variables) must be made.
 +
 
 +
LDA works when the measurements made on independent variables for each observation are continuous quantities. When dealing with categorical independent variables, the equivalent technique is '''discriminant correspondence analysis'''.
 +
 
 +
=== Content ===
 +
First, we shall limit ourselves to the case where there are two classes, i.e. <math>\, \mathcal{Y}=\{0, 1\}</math>. In the above discussion, we introduced the Bayes classifier's ''decision boundary'' <math>\,D(h^*)=\{x: P(Y=1|X=x)=P(Y=0|X=x)\}</math>, which represents a [http://en.wikipedia.org/wiki/Hyperplane hyperplane] that determines the class of any new data input depending on which side of the hyperplane the input lies in. Now, we shall look at how to derive the Bayes classifier's decision boundary under certain assumptions of the data. [http://en.wikipedia.org/wiki/Linear_discriminant_analysis Linear discriminant analysis (LDA)] and [http://en.wikipedia.org/wiki/Quadratic_classifier#Quadratic_discriminant_analysis quadratic discriminant analysis (QDA)] are two of the most well-known ways for deriving the Bayes classifier's decision boundary, and we shall look at each of them in turn.
 +
 
 +
Let us denote the likelihood <math>\ P(X=x|Y=y) </math> as <math>\ f_y(x) </math> and the prior probability <math>\ P(Y=y) </math> as <math>\ \pi_y </math>.
 +
 
 +
First, we shall examine LDA. As explained above, the Bayes classifier is optimal. However, in practice, the prior and conditional densities are not known. Under LDA, one gets around this problem by making the assumptions that both of the two classes have [http://en.wikipedia.org/wiki/Multivariate_normal_distribution multivariate normal (Gaussian) distributions] and the two classes have the same covariance matrix <math>\, \Sigma</math>. Under the assumptions of LDA, we have: <math>\ P(X=x|Y=y) = f_y(x) = \frac{1}{ (2\pi)^{d/2}|\Sigma|^{1/2} }\exp\left( -\frac{1}{2} (x - \mu_k)^\top \Sigma^{-1} (x - \mu_k) \right)</math>. Now, to derive the Bayes classifier's decision boundary using LDA, we equate <math>\, P(Y=1|X=x) </math> to <math>\, P(Y=0|X=x) </math> and proceed from there. The derivation of <math>\,D(h^*)</math> is as follows:
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 +
:<math>\,Pr(Y=1|X=x)=Pr(Y=0|X=x)</math>
 
:<math>\,\Rightarrow \frac{Pr(X=x|Y=1)Pr(Y=1)}{Pr(X=x)}=\frac{Pr(X=x|Y=0)Pr(Y=0)}{Pr(X=x)}</math> (using Bayes' Theorem)
 
:<math>\,\Rightarrow \frac{Pr(X=x|Y=1)Pr(Y=1)}{Pr(X=x)}=\frac{Pr(X=x|Y=0)Pr(Y=0)}{Pr(X=x)}</math> (using Bayes' Theorem)
 
:<math>\,\Rightarrow Pr(X=x|Y=1)Pr(Y=1)=Pr(X=x|Y=0)Pr(Y=0)</math> (canceling the denominators)
 
:<math>\,\Rightarrow Pr(X=x|Y=1)Pr(Y=1)=Pr(X=x|Y=0)Pr(Y=0)</math> (canceling the denominators)
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* LDA implicitly assumes that the mean rather than the variance is the discriminating factor.
 
* LDA implicitly assumes that the mean rather than the variance is the discriminating factor.
 
* LDA may over-fit the training data.
 
* LDA may over-fit the training data.
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The following link provides a comparison of discriminant analysis and artificial neural networks [http://www.jstor.org/stable/2584434?seq=4]
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====Different Approaches to LDA====
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Data sets can be transformed and test vectors can be classified in the transformed space by two
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different approaches.
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*Class-dependent transformation: This type of approach involves maximizing the ratio of between
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class variance to within class variance. The main objective is to maximize this ratio so that adequate
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class separability is obtained. The class-specific type approach involves using two optimizing criteria
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for transforming the data sets independently.
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*Class-independent transformation: This approach involves maximizing the ratio of overall variance
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to within class variance. This approach uses only one optimizing criterion to transform the data sets
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and hence all data points irrespective of their class identity are transformed using this transform. In
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this type of LDA, each class is considered as a separate class against all other classes.
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== Further reading  ==
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The following are some applications that use LDA and QDA:
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1- Linear discriminant analysis for improved large vocabulary continuous speech recognition [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=225984 here]
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2- 2D-LDA: A statistical linear discriminant analysis for image matrix  [http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V15-4DK6B5P-4-1&_cdi=5665&_user=1067412&_pii=S0167865504002272&_origin=search&_coverDate=04%2F01%2F2005&_sk=999739994&view=c&wchp=dGLzVlz-zSkzV&md5=60ea1cf7ff045f76421f5bde64bf855a&ie=/sdarticle.pdf here]
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3- Regularization studies of linear discriminant analysis in small sample size scenarios with application to face recognition  [http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V15-4DTJVF4-2-9&_cdi=5665&_user=1067412&_pii=S0167865504002260&_origin=search&_coverDate=01%2F15%2F2005&_sk=999739997&view=c&wchp=dGLzVtb-zSkzk&md5=1bba55e357b1c79579987638dcbf6828&ie=/sdarticle.pdf here]
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4- The sparse discriminant vectors are useful for supervised dimension reduction for high dimensional data.
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Naive application of classical Fisher’s LDA to high dimensional, low sample size settings suffers from the data piling problem.  In [http://www.iaeng.org/IJAM/issues_v39/issue_1/IJAM_39_1_06.pdf] they have use sparse LDA method which selects important variables for discriminant analysis and thereby
 +
yield improved classification. Introducing sparsity in the discriminant vectors is very effective in eliminating data piling and the associated overfitting
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problem.
  
 
== '''Linear and Quadratic Discriminant Analysis cont'd - September 23, 2010''' ==
 
== '''Linear and Quadratic Discriminant Analysis cont'd - September 23, 2010''' ==
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In the case where we need a common covariance matrix, we get the estimate using the following equation:
 
In the case where we need a common covariance matrix, we get the estimate using the following equation:
  
<math>\,\Sigma=\frac{\sum_{r=1}^{k}(n_r\Sigma_r)}{n-k} </math>
+
<math>\,\Sigma=\frac{\sum_{r=1}^{k}(n_r\Sigma_r)}{\sum_{l=1}^{k}(n_l)} </math>
  
 
Where: <math>\,n_r</math> is the number of data points in class r, <math>\,\Sigma_r</math> is the covariance of class r and <math>\,n</math> is the total number of data points,
 
Where: <math>\,n_r</math> is the number of data points in class r, <math>\,\Sigma_r</math> is the covariance of class r and <math>\,n</math> is the total number of data points,
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A similar transformation of all the centers can be done from <math>\,\mu_k</math> to <math>\,\mu_k^*</math> where <math> \, \mu_k^* \leftarrow S_k^{-\frac{1}{2}}U_k^\top \mu_k </math>.
 
A similar transformation of all the centers can be done from <math>\,\mu_k</math> to <math>\,\mu_k^*</math> where <math> \, \mu_k^* \leftarrow S_k^{-\frac{1}{2}}U_k^\top \mu_k </math>.
  
It is now possible to do classification with <math>\,x^*</math> and <math>\,\mu_k^*</math>, treating them as in Case 1 above.
+
It is now possible to do classification with <math>\,x^*</math> and <math>\,\mu_k^*</math>, treating them as in Case 1 above. This strategy is correct because by transforming <math>\, x</math> to <math>\,x^*</math> where <math> \, x^* \leftarrow S_k^{-\frac{1}{2}}U_k^\top x </math>, the new variable variance is <math>I</math>
 
 
{{Cleanup|date=October 18 2010|reason=The sentence above may cause some misleading. In general case, <math>\,\Sigma_k </math> may not be the same . So you can't treat them completely the same as in Case 1 above. You need to compute <math>\, log{|\Sigma_k |} </math> differently. Here is a detailed discussion below:}}
 
{{Cleanup|date=October 18 2010|reason=The sentence above is right since by transforming<math>\,x</math> to <math>\,x^*</math> where <math> \, x^* \leftarrow S_k^{-\frac{1}{2}}U_k^\top x </math>, the new variable variance is <math>I</math>}}
 
 
 
  
 
Note that when we have multiple classes, we also need to compute <math>\, log{|\Sigma_k|}</math> respectively. Then we compute <math> \,\delta_k </math> for QDA .
 
Note that when we have multiple classes, we also need to compute <math>\, log{|\Sigma_k|}</math> respectively. Then we compute <math> \,\delta_k </math> for QDA .
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If the classes have different shapes, in another word, have different covariance <math>\,\Sigma_k</math>, can we use the same method to transform all data points <math>\,x</math> to <math>\,x^*</math>?
 
If the classes have different shapes, in another word, have different covariance <math>\,\Sigma_k</math>, can we use the same method to transform all data points <math>\,x</math> to <math>\,x^*</math>?
 
The answer is NO. Consider that you have two classes with different shapes, then consider transforming them to the same shape. Given a data point, justify which class this point belongs to. The question is, which transformation can you use? For example, if you use the transformation of class A, then you have assumed that this data point belongs to class A.
 
 
{{Cleanup|date=October 18 2010|reason=The statement above may not be true, because in assignment 1, we did do the QDA computation using this approach although the corresponding three covarience matrices are different, the reason why the answer is Yes is as below }}
 
  
 
The answer is Yes. Consider that you have two classes with different shapes. Given a data point, justify which class this point belongs to. You just do the transformations corresponding to the 2 classes respectively, then you get <math>\,\delta_1 ,\delta_2 </math> ,then you determine which class the data point belongs to by comparing <math> \,\delta_1 </math>  and <math> \,\delta_2 </math> .
 
The answer is Yes. Consider that you have two classes with different shapes. Given a data point, justify which class this point belongs to. You just do the transformations corresponding to the 2 classes respectively, then you get <math>\,\delta_1 ,\delta_2 </math> ,then you determine which class the data point belongs to by comparing <math> \,\delta_1 </math>  and <math> \,\delta_2 </math> .
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:: Step 1: For each class <math>\,k</math>, apply singular value decomposition on <math>\,X_k</math> to obtain <math>\,S_k</math> and <math>\,U_k</math>.
 
:: Step 1: For each class <math>\,k</math>, apply singular value decomposition on <math>\,X_k</math> to obtain <math>\,S_k</math> and <math>\,U_k</math>.
  
:: Step 2: For each class <math>\,k</math>, transform each <math>\,x</math> belonging to that class to <math>\,x^* = S_k^{-\frac{1}{2}}U_k^\top x</math>, and transform its center <math>\,\mu_k</math> to <math>\,\mu_k^* = S_k^{-\frac{1}{2}}U_k^\top \mu_k</math>.
+
:: Step 2: For each class <math>\,k</math>, transform each <math>\,x</math> belonging to that class to <math>\,x_k^* = S_k^{-\frac{1}{2}}U_k^\top x</math>, and transform its center <math>\,\mu_k</math> to <math>\,\mu_k^* = S_k^{-\frac{1}{2}}U_k^\top \mu_k</math>.
  
:: Step 3: For each data point <math>\,x \in X</math>, find the squared Euclidean distance between the transformed data point <math>\,x^*</math> and the transformed center <math>\,\mu^*</math> of each class, and assign <math>\,x</math> to the class such that the squared Euclidean distance between <math>\,x^*</math> and <math>\,\mu^*</math> is the least over all of the classes.
+
:: Step 3: For each data point <math>\,x \in X</math>, find the squared Euclidean distance between the transformed data point <math>\,x_k^*</math> and the transformed center <math>\,\mu_k^*</math> of each class <math>\,k</math>, and assign <math>\,x</math> to class <math>\,k</math> such that the squared Euclidean distance between <math>\,x_k^*</math> and <math>\,\mu_k^*</math> is the least for all possible <math>\,k</math>'s.
  
  
Line 410: Line 490:
  
 
[[File:Lda-qda-parameters.png|frame|center|A plot of the number of parameters that must be estimated, in terms of (K-1). The x-axis represents the number of dimensions in the data. As is easy to see, QDA is far less robust than LDA for high-dimensional data sets.]]
 
[[File:Lda-qda-parameters.png|frame|center|A plot of the number of parameters that must be estimated, in terms of (K-1). The x-axis represents the number of dimensions in the data. As is easy to see, QDA is far less robust than LDA for high-dimensional data sets.]]
 +
 +
===More information on Regularized Discriminant Analysis (RDA)===
 +
Discriminant analysis (DA) is widely used in classification problems. Except LDA and QDA, there is also an intermediate method between LDA and QDA, a regularized version of discriminant analysis (RDA) proposed by Friedman [1989], and it has been shown to be more flexible in dealing with various class distributions. RDA applies the regularization techniques by using two regularization parameters, which are selected to jointly maximize the classification performance. The optimal pair of parameters is commonly estimated via cross-validation from a set of candidate pairs. More detail about this method can be found in the book by Hastie et al. [2001]. On the other hand, the time of computing last long for high dimensional data, especially when the candidate set is large, which limits the applications of RDA to low dimensional data. In 2006, Ye Jieping and Wang Tie develop a novel algorithm for RDA for high dimensional data. It can estimate the optimal regularization parameters from a large set of parameter candidates efficiently. Experiments on a variety of datasets confirm the claimed theoretical estimate of the efficiency, and also show that, for a properly chosen pair of regularization parameters, RDA performs favourably in classification, in comparison with other existing classification methods. For more details, see Ye, Jieping; Wang, Tie
 +
Regularized discriminant analysis for high dimensional, low sample size data Conference on Knowledge Discovery in Data: Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining; 20-23 Aug. 2006
 +
 +
===Further Reading for Regularized Discriminant Analysis (RDA)===
 +
 +
1. Regularized Discriminant Analysis and Reduced-Rank LDA
 +
[http://www.stat.psu.edu/~jiali/course/stat597e/notes2/lda2.pdf]
 +
 +
2. Regularized discriminant analysis for the small sample size in face recognition
 +
[http://www.google.ca/url?sa=t&source=web&cd=2&sqi=2&ved=0CCQQFjAB&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.84.6960%26rep%3Drep1%26type%3Dpdf&rct=j&q=Regularized%20Discriminant%20Analysis&ei=IPr2TJ_2MKWV4gaP5eH-Bg&usg=AFQjCNHB3fk6eVe5fSjlQCMfK44kU1-lug&sig2=5EJv_AV3W_ngSVFIa1nfRg&cad=rja.pdf]
 +
 +
3. Regularized Discriminant Analysis and Its Application in Microarrays
 +
[http://www-stat.stanford.edu/~hastie/Papers/RDA-6.pdf]
  
 
== Trick: Using LDA to do QDA  - September 28, 2010==
 
== Trick: Using LDA to do QDA  - September 28, 2010==
Line 426: Line 521:
 
Suppose we can estimate some vector <math>\underline{w}^T</math> such that
 
Suppose we can estimate some vector <math>\underline{w}^T</math> such that
  
<math>y = \underline{w}^Tx</math>
+
<math>y = \underline{w}^T\underline{x}</math>
  
where <math>\underline{w}</math> is a d-dimensional column vector, and <math style="vertical-align:0%;">x\ \epsilon\ \mathbb{R}^d</math> (vector in d dimensions).
+
where <math>\underline{w}</math> is a d-dimensional column vector, and <math style="vertical-align:0%;">\underline{x}\ \epsilon\ \mathbb{R}^d</math> (vector in d dimensions).
  
We also have a non-linear function <math>g(x) = y = x^Tvx + \underline{w}^Tx</math> that we cannot estimate.
+
We also have a non-linear function <math>g(x) = y = \underline{x}^Tv\underline{x} + \underline{w}^T\underline{x}</math> that we cannot estimate.
  
Using our trick, we create two new vectors, <math>\,\underline{w}^*</math> and <math>\,x^*</math> such that:
+
Using our trick, we create two new vectors, <math>\,\underline{w}^*</math> and <math>\,\underline{x}^*</math> such that:
  
 
<math>\underline{w}^{*T} = [w_1,w_2,...,w_d,v_1,v_2,...,v_d]</math>
 
<math>\underline{w}^{*T} = [w_1,w_2,...,w_d,v_1,v_2,...,v_d]</math>
Line 438: Line 533:
 
and
 
and
  
<math>x^{*T} = [x_1,x_2,...,x_d,{x_1}^2,{x_2}^2,...,{x_d}^2]</math>
+
<math>\underline{x}^{*T} = [x_1,x_2,...,x_d,{x_1}^2,{x_2}^2,...,{x_d}^2]</math>
  
We can then estimate a new function, <math>g^*(x,x^2) = y^* = \underline{w}^{*T}x^*</math>.
+
We can then estimate a new function, <math>g^*(\underline{x},\underline{x}^2) = y^* = \underline{w}^{*T}\underline{x}^*</math>.
  
Note that we can do this for any <math>x</math> and in any dimension; we could extend a <math>D \times n</math> matrix to a quadratic dimension by appending another <math>D \times n</math> matrix with the original matrix squared, to a cubic dimension with the original matrix cubed, or even with a different function altogether, such as a <math>\,sin(x)</math> dimension. Pay attention, We don't do QDA with LDA. If we try QDA directly on this problem the resulting decision boundary will be different. Here we try to find a nonlinear boundary for a better possible boundary but it is different with general QDA method. We can call it nonlinear LDA.
+
Note that we can do this for any <math>\, x</math> and in any dimension; we could extend a <math>D \times n</math> matrix to a quadratic dimension by appending another <math>D \times n</math> matrix with the original matrix squared, to a cubic dimension with the original matrix cubed, or even with a different function altogether, such as a <math>\,sin(x)</math> dimension. Pay attention, We don't do QDA with LDA. If we try QDA directly on this problem the resulting decision boundary will be different. Here we try to find a nonlinear boundary for a better possible boundary but it is different with general QDA method. We can call it nonlinear LDA.
  
 
=== By Example ===
 
=== By Example ===
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:Not only does LDA give us a better result than it did previously, it actually beats QDA, which only correctly classified 371 data points for this data set. Continuing this procedure by adding another two dimensions with <math>x^4</math> (i.e. we set <code>X_star(i,j+2) = X_star(i,j)^4</code>) we can correctly classify 376 points.
 
:Not only does LDA give us a better result than it did previously, it actually beats QDA, which only correctly classified 371 data points for this data set. Continuing this procedure by adding another two dimensions with <math>x^4</math> (i.e. we set <code>X_star(i,j+2) = X_star(i,j)^4</code>) we can correctly classify 376 points.
  
=== LDA and QDA in Matlab ===
+
===Working Example - Diabetes Data Set===
 +
 
 +
Let's take a look at specific data set. This is a [http://archive.ics.uci.edu/ml/datasets/Diabetes diabetes data set] from the UC Irvine Machine Learning Repository. It is a fairly small data set by today's standards. The original data had eight variable dimensions. What I did here was to obtain the two prominent principal components from these eight variables. Instead of using the original eight dimensions we will just use these two principal components for this example.
 +
 
 +
The Diabetes data set has two types of samples in it. One sample type are healthy individuals the other are individuals with a higher risk of diabetes. Here are the prior probabilities estimated for both of the sample types, first for the healthy individuals and second for those individuals at risk:
 +
 
 +
[[File:eq1.png]]
 +
 
 +
The first type has a prior probability estimated at 0.651. This means that among the data set, (250 to 300 data points), about 65% of these belong to class one and the other 35% belong to class two. Next, we computed the mean vector for the two classes separately:[[File:eq2.png]]
 +
 
 +
Then we computed [[File:eq3.jpg]] using the formulas discussed earlier.
 +
 
 +
Once we have done all of this, we compute the linear discriminant function and found the classification rule. Classification rule:[[File:eq4.jpg]]
 +
 
 +
In this example, if you give me an <math>\, x</math>, I then plug this value into the above linear function. If the result is greater than or equal to zero, I claim that it is in class one. Otherwise, it is in class two.
 +
Below is a scatter plot of the dominant principle components. The two classes are represented. The first, without diabetes, is shown with red stars (class 1), and the second class, with diabetes, is shown with blue circles (class 2). The solid line represents the classification boundary obtained by LDA. It appears the two classes are not that well separated. The dashed or dotted line is the boundary obtained by linear regression of indicator matrix. In this case, the results of the two different linear boundaries are very close.
 +
 
 +
[[File:eq5.jpg]]
 +
 
 +
It is always good practice to visualize the scheme to check for any obvious mistakes.
 +
 
 +
• Within training data classification error rate: 28.26%.
 +
• Sensitivity: 45.90%.
 +
• Specificity: 85.60%.
 +
 
 +
Below is the contour plot for the density of the diabetes data (the marginal density for <math>\, x</math> is a mixture of two Gaussians, 2 classes). It looks like a single Gaussian distribution. The reason for this is that the two classes are so close together that they merge into a single mode.
 +
 
 +
[[File:eq6.jpg]]
 +
 
 +
=== LDA and QDA in Matlab ===
  
 
We have examined the theory behind Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA) above; how do we use these algorithms in practice? Matlab offers us a function called [http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/classify.html <code>classify</code>] that allows us to perform LDA and QDA quickly and easily.
 
We have examined the theory behind Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA) above; how do we use these algorithms in practice? Matlab offers us a function called [http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/classify.html <code>classify</code>] that allows us to perform LDA and QDA quickly and easily.
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'''Recall: An analysis of the function of <code>princomp</code> in matlab.'''
 
'''Recall: An analysis of the function of <code>princomp</code> in matlab.'''
<br />In our assignment 1, we have learnt that how to perform Principal Component Analysis using SVD method. In fact, the matlab offers us a function called [http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/princomp.html&http://www.google.cn/search?hl=zh-CN&q=mathwork+princomp&btnG=Google+%E6%90%9C%E7%B4%A2&aq=f&oq= <code>princomp</code>] which can perform PCA conveniently. From the matlab help file on <code>princomp</code>, you can find the details about this function. But here we will analyze the code of the function of <code>princomp()</code> in matlab to find something different when comparing with SVD method. The following is the code of princomp and explanations to some emphasized steps.
+
<br />In our assignment 1, we learned how to perform Principal Component Analysis using the SVD (Singular Value Decomposition) method. In fact, matlab offers a built-in function called [http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/princomp.html&http://www.google.cn/search?hl=zh-CN&q=mathwork+princomp&btnG=Google+%E6%90%9C%E7%B4%A2&aq=f&oq= <code>princomp</code>] which performs PCA. From the matlab help file on <code>princomp</code>, you can find the details about this function. Here we will analyze Matlab's <code>princomp()</code> code. We find something different than the SVD method we used on our first assignment. The following is Matlab's code for princomp followed by some explanations to emphasize some key steps.
  
 
     function [pc, score, latent, tsquare] = princomp(x);
 
     function [pc, score, latent, tsquare] = princomp(x);
Line 585: Line 709:
 
     tsquare = sum(tmp.*tmp)';
 
     tsquare = sum(tmp.*tmp)';
  
From the above code, we should pay attention to the following aspects when comparing with SVD method:
+
We should compare the following aspects of the above code with the SVD method:
  
 
First, Rows of <math>\,X</math> correspond to observations, columns to variables. When using princomp on 2_3 data in assignment 1, note that we take the transpose of <math>\,X</math>.
 
First, Rows of <math>\,X</math> correspond to observations, columns to variables. When using princomp on 2_3 data in assignment 1, note that we take the transpose of <math>\,X</math>.
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The third, when <math>\,X=UdV'</math>, princomp uses <math>\,V</math> as coefficients for principal components, rather than <math>\,U</math>.
 
The third, when <math>\,X=UdV'</math>, princomp uses <math>\,V</math> as coefficients for principal components, rather than <math>\,U</math>.
  
The following is an example to perform PCA using princomp and SVD respectively to get the same results.
+
The following is an example to perform PCA using princomp and SVD respectively to get the same result.
 
:SVD method
 
:SVD method
 
   >> load 2_3
 
   >> load 2_3
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Then we can see that y=score, v=U.
 
Then we can see that y=score, v=U.
  
'''useful resouces:'''
+
'''useful resources:'''
 
LDA and QDA in Matlab[http://www.mathworks.com/products/statistics/demos.html?file=/products/demos/shipping/stats/classdemo.html],[http://www.mathworks.com/matlabcentral/fileexchange/189],[http://seed.ucsd.edu/~cse190/media07/MatlabClassificationDemo.pdf]
 
LDA and QDA in Matlab[http://www.mathworks.com/products/statistics/demos.html?file=/products/demos/shipping/stats/classdemo.html],[http://www.mathworks.com/matlabcentral/fileexchange/189],[http://seed.ucsd.edu/~cse190/media07/MatlabClassificationDemo.pdf]
  
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[http://www.uni-leipzig.de/~strimmer/lab/courses/ss06/seminar/slides/daniela-2x4.pdf LDA & QDA]
 
[http://www.uni-leipzig.de/~strimmer/lab/courses/ss06/seminar/slides/daniela-2x4.pdf LDA & QDA]
  
 +
Using discriminant analysis for multi-class classification: an experimental investigation [http://www.springerlink.com/content/6851416084227k8p/fulltext.pdf]
 +
 +
===Reference articles on solving a small sample size problem when LDA is applied===
 +
( Based on Li-Fen Chen, Hong-Yuan Mark Liao, Ming-Tat Ko, Ja-Chen Lin, Gwo-Jong Yu  A new LDA-based face recognition system which can solve the small sample size problem Pattern Recognition 33 (2000) 1713-1726 )
 +
 +
Small sample size indicates that the number of samples is smaller than the dimension of each sample. In this case, the within-class covariance we stated in class could be a singular matrix and naturally we cannot find its inverse matrix for further analysis.However, many researchers tried to solve it by different techniques:<br />
 +
1.Goudail et al. proposed a technique which calculated 25 local autocorrelation coefficients from each sample image to achieve dimensionality reduction. (Referenced by F. Goudail, E. Lange, T. Iwamoto, K. Kyuma, N. Otsu, Face recognition system using local autocorrelations and multiscale integration, IEEE Trans. Pattern Anal. Mach. Intell. 18 (10) (1996) 1024-1028.)<br />
 +
2.Swets and Weng applied the PCA approach to accomplish reduction of image dimensionality. (Referenced by D. Swets, J. Weng, Using discriminant eigen features for image retrieval, IEEE Trans. Pattern Anal. Mach. Intell.18 (8) (1996) 831-836.)<br />
 +
3.Fukunaga proposed a more efficient algorithm and calculated eigenvalues and eigenvectors from an m*m matrix, where n is the dimensionality of the samples and m is the rank of the within-class scatter matrix Sw. (Referenced by K. Fukunaga, Introduction to Statistical Pattern Recognition, Academic Press, New York, 1990.)<br />
 +
4.Tian et al. used a positive pseudoinverse matrix instead of calculating the inverse matrix Sw. (Referenced by Q. Tian, M. Barbero, Z.H. Gu, S.H. Lee, Image classification by the Foley-Sammon transform, Opt. Eng. 25 (7) (1986) 834-840.)<br />
 +
5.Hong and Yang tried to add the singular value perturbation in Sw and made Sw a nonsingular matrix. (Referenced by Zi-Quan Hong, Jing-Yu Yang, Optimal discriminant plane for a small number of samples and design method of classifier on the plane, Pattern Recognition 24 (4) (1991) 317-324)<br />
 +
6.Cheng et al. proposed another method based on the principle of rank decomposition of matrices. The above three methods are all based on the conventional Fisher's criterion function. (Referenced by Y.Q. Cheng, Y.M. Zhuang, J.Y. Yang, Optimal fisher discriminant analysis using the rank decomposition, Pattern Recognition 25 (1) (1992) 101-111.)<br />
 +
7.Liu et al. modified the conventional Fisher's criterion function and conducted a number of researches based on the new criterion function. They used the total scatter matrix as the divisor of the original Fisher's function instead of merely using the within-class scatter matrix. (Referenced by K. Liu, Y. Cheng, J. Yang, A generalized optimal set of discriminant vectors, Pattern Recognition 25 (7) (1992) 731-739.)
  
 
==Principal Component Analysis  - September 30, 2010==
 
==Principal Component Analysis  - September 30, 2010==
 +
 +
===Brief introduction on dimension reduction method===
 +
 +
Dimension reduction is a process to reduce the number of variables of the data by some techniques. [http://en.wikipedia.org/wiki/Principal_component_analysis Principal components analysis] (PCA) and factor analysis are two primary classical methods on dimension reduction. PCA is a method to create some new variables by a linear combination of the variables in the data and the number of new variables depends on what proportion of the variance the new ones contribute. On the contrary, factor analysis method tries to express the old variables by the linear combination of new variables. So before creating the expressions, a certain number of factors should be determined firstly by analysis on the features of old variables. In general, the idea of both PCA and factor analysis is to use as less as possible mixed variables to reflect as more as possible information.
 +
 
===Rough definition===
 
===Rough definition===
  
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Furthermore, if one considers the lower dimensional representation produced by PCA as a least squares fit of our original data, then it can also be easily shown that this representation is the one that minimizes the reconstruction error of our data. It should be noted however, that one usually does not have control over which dimensions PCA deems to be the most informative for a given set of data, and thus one usually does not know which dimensions PCA selects to be the most informative dimensions in order to create the lower-dimensional representation.  
+
Furthermore, if one considers the lower dimensional representation produced by PCA as a least square fit of our original data, then it can also be easily shown that this representation is the one that minimizes the reconstruction error of our data. It should be noted, however, that one usually does not have control over which dimensions PCA deems to be the most informative for a given set of data, and thus one usually does not know which dimensions PCA should be selected to be the most informative dimensions in order to create the lower-dimensional representation.  
  
  
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PCA takes a sample of ''d'' - dimensional vectors and produces an orthogonal(zero covariance) set of ''d'' 'Principal Components'. The first Principal Component is the direction of greatest variance in the sample. The second principal component is the direction of second greatest variance (orthogonal to the first component), etc.
+
PCA takes a sample of <math>\, d</math> - dimensional vectors and produces an orthogonal(zero covariance) set of <math>\, d</math> 'Principal Components'. The first Principal Component is the direction of greatest variance in the sample. The second principal component is the direction of second greatest variance (orthogonal to the first component), etc.
  
Then we can preserve most of the variance in the sample in a lower dimension by choosing the first ''k'' Principle Components and approximating the data in ''k'' - dimensional space, which is easier to analyze and plot.
+
Then we can preserve most of the variance in the sample in a lower dimension by choosing the first <math>\, k</math> Principle Components and approximating the data in <math>\, k</math> - dimensional space, which is easier to analyze and plot.
  
 
===Principal Components of handwritten digits===
 
===Principal Components of handwritten digits===
 
Suppose that we have a set of 130 images (28 by 23 pixels) of handwritten threes.  
 
Suppose that we have a set of 130 images (28 by 23 pixels) of handwritten threes.  
{{Cleanup|date=September 6 2010|reason=If anyone can tell me where I can find the 2-3 data set, I would create the new image. In the mean time, I found a non-copyrighted image of different looking 3s online, but as you can see, it is not as nice as one we could make.}}
 
{{Cleanup|date=September 6 2010|reason=I think you can find it on your UW-ACE account for this course.}}
 
 
[[File:Handwritten 3s.gif]]
 
  
  
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===Derivation of the first Principle Component===
 
===Derivation of the first Principle Component===
{{Cleanup|date=October 2010|reason=I think English of this section must be improved}}
+
We want to find the direction of maximum variation. Let <math>\begin{align}\textbf{w}\end{align}</math>  be an arbitrary direction, <math>\begin{align}\textbf{x}\end{align}</math> a data point and <math>\begin{align}\displaystyle u\end{align}</math> the length of the projection of <math>\begin{align}\textbf{x}\end{align}</math> in direction <math>\begin{align}\textbf{w}\end{align}</math>.
+
For finding the direction of maximum variation, Let <math>\begin{align}\textbf{w}\end{align}</math>  be an arbitrary direction, <math>\begin{align}\textbf{x}\end{align}</math> a data point, and <math>\begin{align}\displaystyle u\end{align}</math> be the length of the projection of <math>\begin{align}\textbf{x}\end{align}</math> in the direction <math>\begin{align}\textbf{w}\end{align}</math>.
 
<br /><br />
 
<br /><br />
 
<math>\begin{align}
 
<math>\begin{align}
Line 704: Line 842:
 
</math>
 
</math>
 
<br /><br />
 
<br /><br />
The direction <math>\begin{align}\textbf{w}\end{align}</math> is the same as <math>\begin{align}c\textbf{w}\end{align}</math>, for any scalar <math>c</math>, so without loss of generality, we assume that: <br>
+
The direction <math>\begin{align}\textbf{w}\end{align}</math> is the same as <math>\begin{align}c\textbf{w}\end{align}</math>, for any scalar <math>c</math>, so without loss of generality we assume that: <br>
 
<br />
 
<br />
 
<math>
 
<math>
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</math>
 
</math>
 
<br /><br />
 
<br /><br />
Let <math>x_1, \ldots, x_D</math> be random variables, then our goal is to maximize the variance of <math>u</math>,
+
Let <math>x_1, \ldots, x_D</math> be random variables, then we set our goal as to maximize the variance of <math>u</math>,
 
<br /><br />
 
<br /><br />
 
<math>
 
<math>
Line 719: Line 857:
 
</math>
 
</math>
 
<br /><br />
 
<br /><br />
For a finite data set we replace the covariance matrix <math>\Sigma</math> by <math>s</math>, the sample covariance matrix  
+
For a finite data set we replace the covariance matrix <math>\Sigma</math> by <math>s</math>. The sample covariance matrix  
 
<br /><br />
 
<br /><br />
 
<math>\textrm{var}(u) = \textbf{w}^T s\textbf{w} .</math>
 
<math>\textrm{var}(u) = \textbf{w}^T s\textbf{w} .</math>
 
<br /><br />
 
<br /><br />
The above is the variance of <math>\begin{align}\displaystyle u \end{align}</math> formed by the weight vector <math>\begin{align}\textbf{w} \end{align}</math>. The first principal component is the vector <math>\begin{align}\textbf{w} \end{align}</math> that maximizes the variance,
+
The above mentioned variable is the variance of <math>\begin{align}\displaystyle u \end{align}</math> formed by the weight vector <math>\begin{align}\textbf{w} \end{align}</math>. The first principal component is the vector <math>\begin{align}\textbf{w} \end{align}</math> that maximizes the variance,
 
<br /><br />
 
<br /><br />
 
<math>
 
<math>
Line 747: Line 885:
 
====Lagrange Multiplier====
 
====Lagrange Multiplier====
  
Before we can proceed, we must review Lagrange Multipliers.
+
Before we can proceed, we must review Lagrange multipliers.
  
 
[[Image:LagrangeMultipliers2D.svg.png|right|thumb|200px|"The red line shows the constraint g(x,y) = c. The blue lines are contours of f(x,y). The point where the red line tangentially touches a blue contour is our solution." [Lagrange Multipliers, Wikipedia]]]
 
[[Image:LagrangeMultipliers2D.svg.png|right|thumb|200px|"The red line shows the constraint g(x,y) = c. The blue lines are contours of f(x,y). The point where the red line tangentially touches a blue contour is our solution." [Lagrange Multipliers, Wikipedia]]]
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====Example====
 
====Example====
Suppose we wish to maximise the function <math>\displaystyle f(x,y)=x-y</math> subject to the constraint <math>\displaystyle x^{2}+y^{2}=1</math>. We can apply the Lagrange multiplier method on this example; the lagrangian is:
+
Suppose we wish to maximize the function <math>\displaystyle f(x,y)=x-y</math> subject to the constraint <math>\displaystyle x^{2}+y^{2}=1</math>. We can apply the Lagrange multiplier method on this example; the lagrangian is:
  
 
<math>\displaystyle L(x,y,\lambda) = x-y - \lambda (x^{2}+y^{2}-1)</math>
 
<math>\displaystyle L(x,y,\lambda) = x-y - \lambda (x^{2}+y^{2}-1)</math>
Line 802: Line 940:
 
<math>\displaystyle S\textbf{w}^* = \lambda^*\textbf{w}^* </math>
 
<math>\displaystyle S\textbf{w}^* = \lambda^*\textbf{w}^* </math>
 
<br><br />
 
<br><br />
{{Cleanup|date=October 2010|reason=It is good discussion, what will happen if we don't have distinct eigenvalues and eigenvectors? What does this situation mean? }}
 
{{Cleanup|date=October 2010|reason=If the eigenvalues are not distinct, I suppose we could still take the leftmost eigenvector by default. Not sure if this is the correct approach, so can anyone please explain further? Thanks }}
 
{{Cleanup|date=October 2010|reason= As U is the eigenvector of a symetric matrix, is it possible that we have 2 similar eigen vector? }}
 
  
 
From the eigenvalue equation <math>\, \textbf{w}^*</math> is an eigenvector of '''S''' and <math>\, \lambda^*</math> is the corresponding eigenvalue of '''S'''. If we substitute <math>\displaystyle\textbf{w}^*</math> in <math>\displaystyle \textbf{w}^T S\textbf{w}</math> we obtain, <br /><br />
 
From the eigenvalue equation <math>\, \textbf{w}^*</math> is an eigenvector of '''S''' and <math>\, \lambda^*</math> is the corresponding eigenvalue of '''S'''. If we substitute <math>\displaystyle\textbf{w}^*</math> in <math>\displaystyle \textbf{w}^T S\textbf{w}</math> we obtain, <br /><br />
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<br><br />
 
<br><br />
 
In order to maximize the objective function we choose the eigenvector corresponding to the largest eigenvalue. We choose the first PC, '''u<sub>1</sub>''' to have the maximum variance<br /> (i.e. capturing as much variability in <math>\displaystyle x_1, x_2,...,x_D </math> as possible.) Subsequent principal components will take up successively smaller parts of the total variability.
 
In order to maximize the objective function we choose the eigenvector corresponding to the largest eigenvalue. We choose the first PC, '''u<sub>1</sub>''' to have the maximum variance<br /> (i.e. capturing as much variability in <math>\displaystyle x_1, x_2,...,x_D </math> as possible.) Subsequent principal components will take up successively smaller parts of the total variability.
 
  
 
D dimensional data will have D eigenvectors
 
D dimensional data will have D eigenvectors
Line 820: Line 954:
 
<math>Var(u_1) \geq Var(u_2) \geq ... \geq Var(u_D)</math>
 
<math>Var(u_1) \geq Var(u_2) \geq ... \geq Var(u_D)</math>
  
 +
If two eigenvalues happen to be equal, then the data has the same amount of variation in each of the two directions that they correspond to with. If only one of the two equal eigenvalues are to be chosen for dimensionality reduction, then either will do. Note that if ALL of the eigenvalues are the same then this means that the data is on the surface of a d-dimensional sphere (all directions have the same amount of variation).
  
 
Note that the Principal Components decompose the total variance in the data:
 
Note that the Principal Components decompose the total variance in the data:
Line 864: Line 999:
  
 
====Application of PCA - Feature Extraction ====
 
====Application of PCA - Feature Extraction ====
 +
PCA, depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT), the Hotelling transform or proper orthogonal decomposition (POD).
 
One of the applications of PCA is to group similar data (e.g. images).  There are generally two methods to do this.  We can classify the data (i.e. give each data a label and compare different types of data) or cluster (i.e. do not label the data and compare output for classes).
 
One of the applications of PCA is to group similar data (e.g. images).  There are generally two methods to do this.  We can classify the data (i.e. give each data a label and compare different types of data) or cluster (i.e. do not label the data and compare output for classes).
  
Line 890: Line 1,026:
 
::#We can compare the reconstructed test sample to the reconstructed training sample to classify the new data.
 
::#We can compare the reconstructed test sample to the reconstructed training sample to classify the new data.
  
== Fisher's (Linear) Discriminant Analysis (FDA) - Two Class Problem  - October 5, 2010 ==
 
  
===Sir Ronald A. Fisher===
+
==== Feature Extraction Uses and Discussion ====
Fisher's Discriminant Analysis (FDA), also known as Fisher's Linear Discriminant Analysis (LDA) in some sources, is a classical feature extraction technique. It was originally described in 1936 by Sir [http://en.wikipedia.org/wiki/Ronald_A._Fisher Ronald Aylmer Fisher], an English statistician and eugenicist who has been described as one of the founders of modern statistical science. His original paper describing FDA can be found [http://digital.library.adelaide.edu.au/dspace/handle/2440/15227 here]; a Wikipedia article summarizing the algorithm can be found [http://en.wikipedia.org/wiki/Linear_discriminant_analysis#Fisher.27s_linear_discriminant  here].
 
In this paper Fisher used for the first time the term DISCRIMINANT FUNCTION. The term DISCRIMINANT ANALYSIS was introduced later by Fisher himself in a subsequent paper which can be found [http://digital.library.adelaide.edu.au/coll/special//fisher/155.pdf here].
 
  
=== Contrasting FDA with PCA ===
+
PCA, as well as other feature extraction methods not within the scope of the course [http://en.wikipedia.org/wiki/Feature_extraction] are used as a first step to classification in enhancing generalization capability: one of the classification aspects that will be discussed later in the course is model complexity. As a classification model becomes more complex over its training set, classification error over test data tends to increase. By performing feature extraction prior to attempting classification, we restrict model inputs to only the most important variables, thus decreasing complexity and potentially improving test results.
As in PCA, the goal of FDA is to project the data in a lower dimension. You might ask, why was FDA invented when PCA already existed? There is a simple explanation for this that can be found [http://www.ics.uci.edu/~welling/classnotes/papers_class/Fisher-LDA.pdf here]. PCA is an unsupervised method for classification, so it does not take into account the labels in the data. Suppose we have two clusters that have very different or even opposite labels from each other but are nevertheless positioned in a way such that they are very much parallel to each other and also very near to each other. In this case, most of the total variation of the data is in the direction of these two clusters. If we use PCA in cases like this, then both clusters would be projected onto the direction of greatest variation of the data to become sort of like a single cluster after projection. PCA would therefore mix up these two clusters that, in fact, have very different labels. What we need to do instead, in this cases like this, is to project the data onto a direction that is orthogonal to the direction of greatest variation of the data. This direction is in the least variation of the data. On the 1-dimensional space resulting from such a projection, we would then be able to effectively classify the data, because these two clusters would be perfectly or nearly perfectly separated from each other taking into account of their labels. This is exactly the idea behind FDA.  
 
  
The main difference between FDA and PCA is that, in FDA, in contrast to PCA, we are not interested in retaining as much of the variance of our original data as possible.  Rather, in FDA, our goal is to find a direction that is useful for classifying the data (i.e. in this case, we are looking for a direction that is most representative of a particular characteristic e.g. glasses vs. no-glasses). 
+
Feature ''selection'' methods, that are used to select subsets of relevant features for building robust learning models, differ from extraction methods, where features are transformed. Feature selection has the added benefit of improving model interpretability.
Suppose we have 2-dimensional data, then FDA would attempt to project the data of each class onto a point in such a way that the resulting two points would be as far apart from each other as possible. Intuitively, this basic idea behind FDA is the optimal way for separating each pair of classes along a certain direction.  
 
  
{{Cleanup|date=October 2010|reason= Just a thought: how relevant is "Dimensionality reduction techniques" to the concept of "subspace clustering"? As in subspace clustering, the goal is to find a set of features (relevant features, the concept is referred to as local feature relevance in the literature) in the high dimensional space, where potential subspaces accommodating different classes of data points can be defined. This means; the data points are dense when they are considered in a subset of dimensions (features).}}
 
{{Cleanup|date=October 2010|reason=If I'm not mistaken, classification techniques like FDA use labeled training data whereas clustering techniques use unlabeled training data instead. Any other input regarding this would be much appreciated. Thanks}}
 
{{Cleanup|date=October 2010|reason=An extension of clustering is subspace clustering in which different subspace are searched through to find the relavant and appropriate dimentions. High dimentional data sets are roughly equiedistant from each other, so feature selection methods are used to remove the irrelavant dimentions. These techniques do not keep the relative distance so PCA is not useful for these applications. It should be noted that subspace clustering localize their search unlike feature selection algorithms.for more information click here[http://portal.acm.org/citation.cfm?id=1007731]}}
 
  
The number of dimensions that we want to reduce the data to depends on the number of classes:
+
 
<br>
+
=== Independent Component Analysis ===
For a 2-classes problem, we want to reduce the data to one dimension (a line), <math>\displaystyle Z \in \mathbb{R}^{1}</math>   
+
As we have already seen, the Principal Component Analysis (PCA) performed by the Karhunen-Lokve transform produces features <math>\ y ( i ) ; i = 0, 1, . . . , N - 1</math>, that are mutually uncorrelated. The obtained by the KL transform solution is optimal when dimensionality reduction is the goal and one wishes to minimize the approximation mean square error. However, for certain applications, the obtained solution falls short of the expectations. In contrast, the more recently developed Independent Component Analysis (ICA) theory, tries to achieve much more than simple decorrelation of the data. The ICA task is casted as follows: Given the set of input samples <math>\ x</math>, determine an <math>\ N \times N</math> invertible matrix <math>\ W</math> such that the entries <math>\ y(i), i = 0, 1, . . . , N - 1</math>, of the transformed vector
<br>
+
 
Generally, for a k-classes problem, we want to reduce the data to k-1 dimensions, <math>\displaystyle Z \in \mathbb{R}^{k-1}</math>
+
<math>\ y = W.x</math>
 +
 
 +
are mutually independent. The goal of statistical independence is a stronger condition than the uncorrelatedness required by the PCA. The two conditions are equivalent only for Gaussian random variables. Searching for independent rather than uncorrelated features gives us the means of exploiting a lot more of information, hidden in the higher order statistics of the data.
 +
 
 +
This topic has brought to you from Pattern Recognition by Sergios Theodoridis and Konstantinos Koutroumbas. (Chapter 6) For further details on the ICA and its varieties, refer to this book.
 +
 
 +
=== References ===
 +
1. Probabilistic Principal Component Analysis
 +
[http://onlinelibrary.wiley.com/doi/10.1111/1467-9868.00196/abstract]
 +
 
 +
2. Nonlinear Component Analysis as a Kernel Eigenvalue Problem
 +
[http://www.mitpressjournals.org/doi/abs/10.1162/089976698300017467]
 +
 
 +
3. Kernel principal component analysis
 +
[http://www.springerlink.com/content/w0t1756772h41872/]
 +
 
 +
4. Principal Component Analysis
 +
[http://onlinelibrary.wiley.com/doi/10.1002/0470013192.bsa501/full] and [http://support.sas.com/publishing/pubcat/chaps/55129.pdf]
 +
 
 +
=== Further Readings ===
 +
1. I. T. Jolliffe "Principal component analysis" Available [http://books.google.ca/books?id=_olByCrhjwIC&printsec=frontcover&dq=principal+component+analysis&hl=en&ei=TooCTaesN42YnweR843lDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC4Q6AEwAA#v=onepage&q&f=false here].
 +
 
 +
2. James V. Stone "Independent component analysis: a tutorial introduction" Available [http://books.google.ca/books?id=P0rROE-WFCwC&pg=PA129&dq=principal+component+analysis&hl=en&ei=TooCTaesN42YnweR843lDQ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEYQ6AEwBg#v=onepage&q=principal%20component%20analysis&f=false here].
 +
 
 +
3. Aapo Hyvärinen, Juha Karhunen, Erkki Oja "Independent component analysis" Available [http://books.google.ca/books?id=96D0ypDwAkkC&printsec=frontcover&dq=independent+component+analysis&hl=en&ei=F4wCTZqjJY2RnAew6pnlDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCoQ6AEwAA#v=onepage&q&f=false here].
 +
 
 +
== Fisher's (Linear) Discriminant Analysis (FDA) - Two Class Problem  - October 5, 2010 ==
 +
 
 +
===Sir Ronald A. Fisher===
 +
Fisher's Discriminant Analysis (FDA), also known as Fisher's Linear Discriminant Analysis ([http://en.wikipedia.org/wiki/Linear_discriminant_analysis LDA]) in some sources, is a classical [http://en.wikipedia.org/wiki/Feature_extraction feature extraction] technique. It was originally described in 1936 by Sir [http://en.wikipedia.org/wiki/Ronald_A._Fisher Ronald Aylmer Fisher], an English statistician and eugenicist who has been described as one of the founders of modern statistical science. His original paper describing FDA can be found [http://digital.library.adelaide.edu.au/dspace/handle/2440/15227 here]; a Wikipedia article summarizing the algorithm can be found [http://en.wikipedia.org/wiki/Linear_discriminant_analysis#Fisher.27s_linear_discriminant  here].
 +
In this paper Fisher used for the first time the term DISCRIMINANT FUNCTION. The term DISCRIMINANT ANALYSIS was introduced later by Fisher himself in a subsequent paper which can be found [http://digital.library.adelaide.edu.au/coll/special//fisher/155.pdf here].
 +
 
 +
===Introduction===
 +
'''Linear discriminant analysis''' ([http://en.wikipedia.org/wiki/Linear_discriminant_analysis LDA]) and the related '''Fisher's linear discriminant''' are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterize or separate two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification.
 +
 
 +
LDA is also closely related to principal component analysis ([http://en.wikipedia.org/wiki/Principal_component_analysis PCA]) and [http://en.wikipedia.org/wiki/Factor_analysis factor analysis] in that both look for linear combinations of variables which best explain the data. LDA explicitly attempts to model the difference between the classes of data. PCA on the other hand does not take into account any difference in class, and factor analysis builds the feature combinations based on differences rather than similarities. Discriminant analysis is also different from factor analysis in that it is not an interdependence technique: a distinction between independent variables and dependent variables (also called criterion variables) must be made.
 +
 
 +
LDA works when the measurements made on independent variables for each observation are continuous quantities. When dealing with categorical independent variables, the equivalent technique is '''discriminant correspondence analysis'''.
 +
 
 +
=== Contrasting FDA with PCA ===
 +
As in PCA, the goal of FDA is to project the data in a lower dimension. You might ask, why was FDA invented when PCA already existed? There is a simple explanation for this that can be found [http://www.ics.uci.edu/~welling/classnotes/papers_class/Fisher-LDA.pdf here]. PCA is an unsupervised method for classification, so it does not take into account the labels in the data. Suppose we have two clusters that have very different or even opposite labels from each other but are nevertheless positioned in a way such that they are very much parallel to each other and also very near to each other. In this case, most of the total variation of the data is in the direction of these two clusters. If we use PCA in cases like this, then both clusters would be projected onto the direction of greatest variation of the data to become sort of like a single cluster after projection. PCA would therefore mix up these two clusters that, in fact, have very different labels. What we need to do instead, in this cases like this, is to project the data onto a direction that is orthogonal to the direction of greatest variation of the data. This direction is in the least variation of the data. On the 1-dimensional space resulting from such a projection, we would then be able to effectively classify the data, because these two clusters would be perfectly or nearly perfectly separated from each other taking into account of their labels. This is exactly the idea behind FDA.
 +
 
 +
The main difference between FDA and PCA is that, in FDA, in contrast to PCA, we are not interested in retaining as much of the variance of our original data as possible.  Rather, in FDA, our goal is to find a direction that is useful for classifying the data (i.e. in this case, we are looking for a direction that is most representative of a particular characteristic e.g. glasses vs. no-glasses). 
 +
Suppose we have 2-dimensional data, then FDA would attempt to project the data of each class onto a point in such a way that the resulting two points would be as far apart from each other as possible. Intuitively, this basic idea behind FDA is the optimal way for separating each pair of classes along a certain direction.
 +
 
 +
Please note dimention reduction in PCA is different from subspace cluster , see the details about the subspace cluser [http://en.wikipedia.org/wiki/Clustering_high-dimensional_data]
 +
{{Cleanup|date=October 2010|reason= Just a thought: how relevant is "Dimensionality reduction techniques" to the concept of "subspace clustering"? As in subspace clustering, the goal is to find a set of features (relevant features, the concept is referred to as local feature relevance in the literature) in the high dimensional space, where potential subspaces accommodating different classes of data points can be defined. This means; the data points are dense when they are considered in a subset of dimensions (features).}}
 +
{{Cleanup|date=October 2010|reason=If I'm not mistaken, classification techniques like FDA use labeled training data whereas clustering techniques use unlabeled training data instead. Any other input regarding this would be much appreciated. Thanks}}
 +
{{Cleanup|date=October 2010|reason=An extension of clustering is subspace clustering in which different subspace are searched through to find the relavant and appropriate dimentions. High dimentional data sets are roughly equiedistant from each other, so feature selection methods are used to remove the irrelavant dimentions. These techniques do not keep the relative distance so PCA is not useful for these applications. It should be noted that subspace clustering localize their search unlike feature selection algorithms.for more information click here[http://portal.acm.org/citation.cfm?id=1007731]}}
 +
 
 +
The number of dimensions that we want to reduce the data to depends on the number of classes:
 +
<br>
 +
For a 2-classes problem, we want to reduce the data to one dimension (a line), <math>\displaystyle Z \in \mathbb{R}^{1}</math>   
 +
<br>
 +
Generally, for a k-classes problem, we want to reduce the data to k-1 dimensions, <math>\displaystyle Z \in \mathbb{R}^{k-1}</math>
  
 
As we will see from our objective function, we want to maximize the separation of the classes and to minimize the within-variance of each class.  That is, our ideal situation is that the individual classes are as far away from each other as possible, and at the same time the data within each class are as close to each other as possible (collapsed to a single point in the most extreme case).
 
As we will see from our objective function, we want to maximize the separation of the classes and to minimize the within-variance of each class.  That is, our ideal situation is that the individual classes are as far away from each other as possible, and at the same time the data within each class are as close to each other as possible (collapsed to a single point in the most extreme case).
Line 973: Line 1,155:
  
 
=== Two-class problem  ===
 
=== Two-class problem  ===
In the two-class problem, we have the pre-knowledge that data points belong to two classes. Intuitively speaking points of each class form a cloud around the mean of the class, with each class having possibly different size. To be able to separate the two classes we must determine the class whose mean is closest to a given point while also accounting for the different size of each class, which is represented by the covariance of each class.
+
In the two-class problem, we have prior knowledge that the data points belong to two classes. Conceptually, points of each class form a cloud around the class mean, and each class has an distinct size. To divide points among the two classes, we must determine the class whose mean is closest to each point, and we must also account for the different size of each class given by the covariance of each class.
  
 
Assume <math>\underline{\mu_{1}}=\frac{1}{n_{1}}\displaystyle\sum_{i:y_{i}=1}\underline{x_{i}}</math> and <math>\displaystyle\Sigma_{1}</math>,
 
Assume <math>\underline{\mu_{1}}=\frac{1}{n_{1}}\displaystyle\sum_{i:y_{i}=1}\underline{x_{i}}</math> and <math>\displaystyle\Sigma_{1}</math>,
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{{Cleanup|date=October 2010|reason=In 2. above, I wonder if the computation would be much more complex if we instead find a weighted sum of the covariances of the two classes where the weights are the sizes of the two classes?}}
 
{{Cleanup|date=October 2010|reason=In 2. above, I wonder if the computation would be much more complex if we instead find a weighted sum of the covariances of the two classes where the weights are the sizes of the two classes?}}
  
 +
{{Cleanup|date=December 2010|reason= If using the weighted sum of two covariances, you will need to use the shared mean of the two classes, and the weighted sum will be the shared covariance. Doing this will result in collapsing the two classes into one point, which contradicts the purpose of using FDA}}
  
 
As is demonstrated below, both of these goals can be accomplished simultaneously.
 
As is demonstrated below, both of these goals can be accomplished simultaneously.
Line 1,116: Line 1,299:
  
 
===Some of FDA applications===
 
===Some of FDA applications===
There are many applications for FDA in many domains some of them are stated below:
+
There are many applications for FDA in many domains; a few examples are stated below:
  
* SPEECH/MUSIC/NOISE CLASSIFICATION IN HEARING AIDS
+
* Speech/Music/Noise Classification in Hearing Aids
FDA can be used to enhance listening comprehension when the user goes from a sound
+
FDA can be used to enhance listening comprehension when the user goes from one sound environment to another different one. In practice, many people who require hearing aids do not wear them due in part to the nusiance of having to adjust the settings each time a user changes noise environments (for example, from a quiet walk in the to park to a crowded cafe). If the hearing aid itself could distinguish between the type of sound environment and automatically adjust its settings itself, many more people may be willing to wear and use the hearing aids. The paper referenced below examines the difference in using a classifier based on one level and three classes ("speech", "noisy" or "music" environments) and a classifier based on two levels with two classes each ("speech" versus "non-speech" and then for the "non-speech" group, between "noisy" and "music") and also includes a discussion about the feasibility of implementing these classifiers in the hearing aids. For more information review this paper by Alexandre et al. [http://www.eurasip.org/Proceedings/Eusipco/Eusipco2008/papers/1569101740.pdf here].
environment to another different one. For more information review this paper by Alexandre et al.[http://www.eurasip.org/Proceedings/Eusipco/Eusipco2008/papers/1569101740.pdf here]
 
  
 
* Application to Face Recognition
 
* Application to Face Recognition
FDA can be used in face recognition at different situation. Using FDA Kong et al. proposes an Application to Face
+
FDA can be used in face recognition for different situations. Instead of using the one-dimensional LDA where the data is transformed into long column vectors with less-than-full-rank covariance matrices for the within-class and between-class covariance matrices, several other approaches of using FDA are suggested here including a two-dimensional approach where the data is stored as a matrix rather than a column vector. In this case, the covariance matrices are full-rank. Details can be found in the paper by Kong et al. [http://person.hst.aau.dk/pimuller/2D_FDA_Face_CVPR05fish.pdf here].
Recognition with Small Number of Training Samples [http://person.hst.aau.dk/pimuller/2D_FDA_Face_CVPR05fish.pdf here].
 
  
 
* Palmprint Recognition
 
* Palmprint Recognition
FDA is used in biometrics, to implement an automated palmprint recognition system. See An Automated Palmprint Recognition System by Tee et al. [http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V09-4FJ5XPN-1-1&_cdi=5641&_user=1067412&_pii=S0262885605000089&_origin=search&_coverDate=05%2F01%2F2005&_sk=999769994&view=c&wchp=dGLbVzz-zSkWb&md5=a064b67c9bdaaba7e06d800b6c9b209b&ie=/sdarticle.pdf here].
+
FDA is used in biometrics to implement an automated palmprint recognition system. In Tee et al. [http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V09-4FJ5XPN-1-1&_cdi=5641&_user=1067412&_pii=S0262885605000089&_origin=search&_coverDate=05%2F01%2F2005&_sk=999769994&view=c&wchp=dGLbVzz-zSkWb&md5=a064b67c9bdaaba7e06d800b6c9b209b&ie=/sdarticle.pdf here] An Automated Palmprint Recognition System was proposed and FDA was used to match images in a compressed subspace where these subspaces best discriminate among classes. It is different from PCA in the aspect that it deals directly with class separation while PCA treats images in its entirety without considering the underlying class structure.
  
{{Cleanup|date=October 2010|reason=I think briefing about the other applications would be easier than browsing through all of these applications}}
+
* Other Applications
  
other applications could found in references 4,5,6,7,8 and so on.
+
Other applications can be seen in [4] where FDA was used to authenticate different olive oil types, or classify multiple fault classes [5]. As well as, applications on face recognition [6] and shape deformations to localize epilepsy [8].
  
 
=== '''References'''===
 
=== '''References'''===
Line 1,155: Line 1,336:
  
 
8. Kodipaka, S.; Vemuri, B.C.; Rangarajan, A.; Leonard, C.M.; Schmallfuss, I.; Eisenschenk, S.; "Kernel Fisher discriminant for shape-based classification in epilepsy" Hournal Medical Image Analysis, 2007. [http://www.sciencedirect.com/science/article/B6W6Y-4MH8BS0-1/2/055fb314828d785a5c3ca3a6bf3c24e9 8]
 
8. Kodipaka, S.; Vemuri, B.C.; Rangarajan, A.; Leonard, C.M.; Schmallfuss, I.; Eisenschenk, S.; "Kernel Fisher discriminant for shape-based classification in epilepsy" Hournal Medical Image Analysis, 2007. [http://www.sciencedirect.com/science/article/B6W6Y-4MH8BS0-1/2/055fb314828d785a5c3ca3a6bf3c24e9 8]
 +
 +
9. Fisher LDA and Kernel Fisher LDA [http://www.ics.uci.edu/~welling/classnotes/papers_class/Fisher-LDA.pdf]
  
 
==Fisher's (Linear) Discriminant Analysis (FDA) - Multi-Class Problem  - October 7, 2010==
 
==Fisher's (Linear) Discriminant Analysis (FDA) - Multi-Class Problem  - October 7, 2010==
  
====Obtaining Covariance Matrices====
+
===Obtaining Covariance Matrices===
  
  
Line 1,326: Line 1,509:
 
eigenvalues with respect to
 
eigenvalues with respect to
  
{{Cleanup|date=What if we encounter complex eigenvalues? Then concept of being large does not dense. What is the solution in that case? }}
+
{{Cleanup|reason=What if we encounter complex eigenvalues? Then concept of being large does not dense. What is the solution in that case? }}
 +
{{Cleanup|date=December 2010|reason=Covariance matrices are positive semi-definite. The inverse of a positive semi-definite matrix is positive semi-definite.  The product of positive semi-definite matrices is positive semi-definite. The eigenvalues of a positive semi-definite matrix are all real, non-negative values. As a result, the eigenvalues of \mathbf{S}_{W}^{-1}\mathbf{S}_{B} will always be real, non-negative values.}}
  
 
:<math>
 
:<math>
Line 1,428: Line 1,612:
 
{{Cleanup|date=October 2010|reason=Would you please show how could we reconstruct our original data from the data that its dimentionality is reduced by FDA.}}
 
{{Cleanup|date=October 2010|reason=Would you please show how could we reconstruct our original data from the data that its dimentionality is reduced by FDA.}}
 
{{Cleanup|date=October 2010|reason= When you reduce the dimensionality of data in most general form you lose some features of the data and you cannot reconstruct the data from redacted space unless the data have special features that help you in reconstruction like sparsity. In FDA it seems that we cannot reconstruct data in general form using reducted version of data  }}
 
{{Cleanup|date=October 2010|reason= When you reduce the dimensionality of data in most general form you lose some features of the data and you cannot reconstruct the data from redacted space unless the data have special features that help you in reconstruction like sparsity. In FDA it seems that we cannot reconstruct data in general form using reducted version of data  }}
 +
 +
====Advantages of FDA compared with PCA====
 +
 +
-PCA find components which are useful for representing data.
 +
 +
-While there is no reason to assume that components are useful to discriminate data between classes.
 +
 +
-In FDA , we try to use labels to find the components which are useful for discriminating data.
  
 
===Generalization of Fisher's Linear Discriminant Analysis ===
 
===Generalization of Fisher's Linear Discriminant Analysis ===
Line 1,435: Line 1,627:
  
 
Also notice that LDA can be seen as a dimensionality reduction technique. In general k-class problems, we have k means which lie on a linear subspace with dimension k-1. Given a data point, we are looking for the closest class mean to this point. In LDA, we project the data point to the linear subspace and calculate distances within that subspace. If the dimensionality of the data, d, is much larger than the number of classes, k, then we have a considerable drop in dimensionality from d dimensions to k - 1 dimensions.
 
Also notice that LDA can be seen as a dimensionality reduction technique. In general k-class problems, we have k means which lie on a linear subspace with dimension k-1. Given a data point, we are looking for the closest class mean to this point. In LDA, we project the data point to the linear subspace and calculate distances within that subspace. If the dimensionality of the data, d, is much larger than the number of classes, k, then we have a considerable drop in dimensionality from d dimensions to k - 1 dimensions.
 +
 +
===Multiple Discriminant Analysis===
 +
 +
(MDA) is also termed Discriminant Factor Analysis and Canonical Discriminant Analysis. It adopts a similar perspective to PCA: the rows of the data matrix to be examined constitute points in a multidimensional space, as also do the group mean vectors. Discriminating axes are determined in this space, in such a way that optimal separation of the predefined groups is attained. As with PCA, the problem becomes mathematically the eigenreduction of a real, symmetric matrix. The eigenvalues represent the discriminating power of the associated eigenvectors. The nYgroups lie in a space of dimension at most <math>n_{y-1}</math>. This will be the number of discriminant axes or factors obtainable in the most common practical case when n > m > nY (where n is the number of rows, and m the number of columns of the input data matrix.
 +
 +
===Matlab Example: Multiple Discriminant Analysis for Face Recognition===
 +
 +
% The following MATLAB code is an example of using MDA in face recognition. The used dataset can be    % found be found [http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html here]. IT contains % a set of face images taken between April 1992 and April 1994 at the lab. The database was used in the % context of a face recognition project carried out in collaboration with the Speech, Vision and      % Robotics Group of the Cambridge University Engineering Department.
 +
 +
load orl_faces_112x92.mat
 +
u=(mean(faces'))';
 +
stfaces=faces-u*ones(1,400);
 +
S=stfaces'*stfaces;
 +
[V,E] = eig(S);
 +
U=zeros(length(stfaces),150);%%%%%%
 +
for i=400:-1:251
 +
    U(:,401-i)=stfaces*V(:,i)/sqrt(E(i,i));
 +
end
 +
 +
defaces=U'*stfaces;
 +
for i=1:40
 +
    for j=1:5
 +
        lsamfaces(:,j+5*i-5)=defaces(:,j+10*i-10);
 +
        ltesfaces(:,j+5*i-5)=defaces(:,j+10*i-5);
 +
    end
 +
end
 +
stlsamfaces=lsamfaces-lsamfaces*wdiag(ones(5,5),40)/5;
 +
Sw=stlsamfaces*stlsamfaces';
 +
zstlsamfaces=lsamfaces-(mean(lsamfaces'))'*ones(1,200);
 +
St=zstlsamfaces*zstlsamfaces';
 +
Sb=St-Sw;
 +
[V D]=eig(Sw\Sb);
 +
U=V(:,1:39);
 +
desamfaces=U'*lsamfaces;
 +
detesfaces=U'*ltesfaces;
 +
rightnum=0;
 +
for i=1:200
 +
    mindis=10^10;minplace=1;
 +
    for j=1:200
 +
        distan=norm(desamfaces(:,i)-detesfaces(:,j));
 +
        if mindis>distan
 +
            mindis=distan;
 +
            minplace=j;
 +
        end
 +
    end
 +
    if floor(minplace/5-0.2)==floor(i/5-0.2)
 +
        rightnum=rightnum+1;
 +
    end
 +
end
 +
rightrate=rightnum/200
 +
 +
===K-NNs Discriminant Analysis===
 +
 +
Non-parametric (distribution-free) methods dispense with the need for assumptions regarding the probability density function. They have become very popular especially in the image processing area. The K-NNs method assigns an object of unknown affiliation to the group to which the majority of its K nearest neighbours belongs.
 +
 +
There is no best discrimination method. A few remarks concerning the advantages and disadvantages of the methods studied are as follows.
 +
 +
:1.Analytical simplicity or computational reasons may lead to initial consideration of linear discriminant analysis or the NN-rule.
 +
:2.Linear discrimination is the most widely used in practice. Often the 2-group method is used repeatedly for the analysis of pairs of multigroup data (yielding <math>\frac{k(k-1)}{2}</math>decision surfaces for k groups).
 +
:3.To estimate the parameters required in quadratic discrimination more computation and data is required than in the case of linear discrimination. If there is not a great difference in the group covariance matrices, then the latter will perform as well as quadratic discrimination.
 +
:4.The k-NN rule is simply defined and implemented, especially if there is insufficient data to adequately define sample means and covariance matrices.
 +
:5.MDA is most appropriately used for feature selection. As in  the case of PCA, we may want to focus on the variables used in order to investigate the differences between groups; to create synthetic variables which improve the grouping ability of the data; to arrive at a similar objective by discarding irrelevant variables; or to determine the most parsimonious variables for graphical representational purposes.
 +
 +
=== Fisher Score ===
 +
Fisher Discriminant Analysis should be distinguished from Fisher Score. Feature score is a means, by which we can evaluate the importance of each of the features in a binary classification task. Here is the Fisher score, or in brief <math>\ FS</math>.
 +
 +
<math>FS_i=\frac{(\mu_i^1-\mu_i)^2+(\mu_i^2-\mu_i)^2}{var_i^1+var_i^2}</math>
 +
 +
Where <math>\ \mu_i^1</math>, and <math>\ \mu_i^2</math> are the average of the feature <math>\ i</math> for the class 1 and 2 respectively and <math>\ \mu_i</math> is the average of the feature <math>\ i</math> over both of the classes. And <math>\ var_i^1</math>, and <math>\ var_i^2</math> are the variances of the feature <math>\ i</math> in the two classes of 1 and 2 respectively.
 +
 +
We can estimate the FS over all of the features and then select those features with the highest FS. We want features to discriminate as much as possible between two classes and describe each of the classes as dense as possible; this is exactly the criterion that has been taken into consideration for defining the Fisher Score.
 +
 +
 +
===References===
 +
 +
1. Optimal Fisher discriminant analysis using the rank decomposition
 +
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V14-48MPMK5-14R&_user=10&_coverDate=01%2F31%2F1992&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1550315473&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=b8b00da9ab59b76a40eca456f5aa99b6&searchtype=a]
 +
 +
2. Face recognition using Kernel-based Fisher Discriminant Analysis
 +
[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1004157]
 +
 +
3. Fisher discriminant analysis with kernels
 +
[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=788121]
 +
 +
4. Fisher LDA and Kernel Fisher LDA [http://www.ics.uci.edu/~welling/classnotes/papers_class/Fisher-LDA.pdf]
 +
 +
5. Previous STAT 841 notes. [http://www.math.uwaterloo.ca/~aghodsib/courses/f07stat841/notes/lecture7.pdf]
 +
 +
6. Another useful pdf introducing FDA [http://www.cedar.buffalo.edu/~srihari/CSE555/Chap3.Part6.pdf]
 +
 +
==Random Projection==
 +
Random Project (RP) is an approach of projecting a point from a high dimensional space to a lower dimensional space. In general, a target subspace, presented as a uniform random orthogonal matrix, should be determined firstly and the projected vector can be described as v=c.p.u, where u is a d-dimension vector, p is the uniform random orthogonal matrix with d’ rows and d columns, v is the projected vector with d’-dimension and c is scaling factor such that the expected squared length of v is equal to the squared length of u. For the projected vectors by RP, they have two main properties:
 +
1. The distance between any two of the original vectors is approximately equal to the distance of their corresponding projected vectors by RP.
 +
2. If each of entries in the uniform random orthogonal matrix is randomly selected followed by distribution N(0,1), then the expected squared length of v is equal to the squared length of u.
 +
For more details of RP, please see The Random Projection Method by Santosh S. Vempala.
 +
  
 
==Linear and Logistic Regression - October 12, 2010==
 
==Linear and Logistic Regression - October 12, 2010==
Line 1,540: Line 1,828:
 
\begin{align}
 
\begin{align}
 
\mathbf{\hat y} = \mathbf{X}^{T}\hat{\beta} =  
 
\mathbf{\hat y} = \mathbf{X}^{T}\hat{\beta} =  
\mathbf{X}^{T}(\mathbf{X}\mathbf{X}^{T})^{-1}\mathbf{X}\mathbf{y}
+
\mathbf{X}^{T}(\mathbf{X}\mathbf{X}^{T})^{-1}\mathbf{X}\mathbf{y} =
 +
\mathbf{H}\mathbf{y}
 
\end{align}
 
\end{align}
 
</math>
 
</math>
Line 1,546: Line 1,835:
 
where <math>\mathbf{H} = \mathbf{X}^{T}(\mathbf{X}\mathbf{X}^{T})^{-1}\mathbf{X} </math> is called the [http://en.wikipedia.org/wiki/Hat_matrix hat matrix].
 
where <math>\mathbf{H} = \mathbf{X}^{T}(\mathbf{X}\mathbf{X}^{T})^{-1}\mathbf{X} </math> is called the [http://en.wikipedia.org/wiki/Hat_matrix hat matrix].
  
<br/>
+
A more efficient way to do this is by [http://en.wikipedia.org/wiki/QR_decomposition QR Factorization]
*'''Note'''  For classification purposes, this is not a correct model.  Recall the following application of Bayes classifier:<br/>
+
 
<math>r(x)= P( Y=k | X=x )= \frac{f_{k}(x)\pi_{k}}{\Sigma_{l}f_{l}(x)\pi_{l}}</math><br/>
+
<math>
It is clear that to make sense mathematically, <math>\displaystyle r(x)</math> must be a value between 0 and 1 and must also sum up to 1.  If this is estimated with the  
+
X^T = QR </math> where Q is an orthonormal matrix and R is an upper triangular matrix
regression function <math>\displaystyle r(x)=E(Y|X=x)</math> and <math>\mathbf{\hat\beta} </math> is learned as above, then there is nothing that would restrict <math>\displaystyle r(x)</math> to meet these two criteria. This is more direct approach to classification since it do not need to estimate <math>\ f_k(x) </math> and <math>\ \pi_k </math>.
+
 
<math>\ 1 \times P(Y=1|X=x)+0 \times P(Y=0|X=x)=E(Y|X) </math>.  
+
<math>
This model does not classify Y between 0 and 1, so it is not good but at times it can lead to a decent classifier. <math>\ y_i=\frac{1}{n_1} </math>  <math>\ \frac{-1}{n_2} </math>
+
\begin{align}
[[File:Example.jpg]]
+
\hat{\beta} &=& ((QR){^T}(QR))^{-1}(QR)^{T}y \\
 +
  &=& ((R^{T}Q^{T}QR))^{-1}(QR)^{T}y \\
 +
  &=& (R^{T}R)^{-1}R^{T}Qy \\
 +
  &=& R^{-1}(R^{-T}R^{T})Qy \\
 +
  &=& R^{-1}Qy
 +
\end{align}
 +
</math>
 +
 
 +
Therefore <math>\hat{\beta}</math> can be solved for by solving <math> R\hat{\beta} = Qy</math>
 +
 
 +
<br/>
 +
*'''Note'''  For classification purposes, this is not a correct model.  Recall the following application of Bayes classifier:<br/>
 +
<math>r(x)= P( Y=k | X=x )= \frac{f_{k}(x)\pi_{k}}{\Sigma_{l}f_{l}(x)\pi_{l}}</math><br/>
 +
It is clear that to make sense mathematically, <math>\displaystyle r(x)</math> must be a value between 0 and 1 and must also sum up to 1.  If this is estimated with the  
 +
regression function <math>\displaystyle r(x)=E(Y|X=x)</math> and <math>\mathbf{\hat\beta} </math> is learned as above, then there is nothing that would restrict <math>\displaystyle r(x)</math> to meet these two criteria. This is more direct approach to classification since it do not need to estimate <math>\ f_k(x) </math> and <math>\ \pi_k </math>.
 +
<math>\ 1 \times P(Y=1|X=x)+0 \times P(Y=0|X=x)=E(Y|X) </math>.  
 +
This model does not classify Y between 0 and 1, so it is not good but at times it can lead to a decent classifier. <math>\ y_i=\frac{1}{n_1} </math>  <math>\ \frac{-1}{n_2} </math>
 +
[[File:Example.jpg]]
 +
 
 +
==== Recursive Linear Regression ====
 +
In some applications, we need to estimate the weights of the linear regression in an online scheme. For real-time applications, efficiency of computations comes to be very important. In cases like this, we have a batch data and more data samples are still being observed and according to the whole observed data points, we need for example to predict the class label of the upcoming samples. To be able to do that in real-time we should better take advantage of the computations that we have done up to any given sample point and estimate the new weights -having seen the new sample point- using the previous weights -before observing the new sample point. So we want to update the weights, like this:
 +
 
 +
<math>\ W_{new}=h(W_{old},x_{new},y_{new})</math>
 +
 
 +
In which <math>\ W_{new}</math> and <math>\ W_{old}</math> are the linear regression weights after and before observation of the new sample pair, <math>\ (x_{new},y_{new})</math>. The function <math>\ h</math> could be obtained using the following procedure.
 +
 
 +
<math>\begin{align}
 +
W_{old}&=(XX^T-x_{new}x_{new}^T)^{-1}(XY-x_{new}y_{new}) \\
 +
\rightarrow (XX^T-x_{new}x_{new}^T)W_{old}&=XY-x_{new}y_{new}  \\
 +
\rightarrow XX^TW_{old}&=XY-x_{new}y_{new}+x_{new}x_{new}^TW_{old}  \\
 +
\rightarrow W_{old}&=(XX^T)^{-1}(XY-x_{new}y_{new}+x_{new}x_{new}^TW_{old})  \\
 +
\rightarrow W_{old}&=W_{new}+(XX^T)^{-1}(-x_{new}y_{new}+x_{new}x_{new}^TW_{old}) \\
 +
\rightarrow W_{new}&=W_{old}-(XX^T)^{-1}(-x_{new}y_{new}+x_{new}x_{new}^TW_{old})
 +
\end{align}</math>
 +
 
 +
Where <math>\ X</math>, and <math>\ Y</math> represent the whole set of sample points pairs, including the recently seen sample pair, <math>\ (x_{new},y_{new})</math>.
 +
 
 +
====Comments about Linear regression model====
 +
 
 +
Linear regression model is almost the easiest and most popular way to analyze the relationship of different data sets. However, it has some disadvantages as well as its advantages. We should be clear about them before we apply the model.
 +
 
 +
''Advantages'': Linear least squares regression has earned its place as the primary tool for process modeling because of its effectiveness and completeness. Though there are types of data that are better described by functions that are nonlinear in the parameters, many processes in science and engineering are well-described by linear models. This is because either the processes are inherently linear or because, over short ranges, any process can be well-approximated by a linear model. The estimates of the unknown parameters obtained from linear least squares regression are the optimal estimates from a broad class of possible parameter estimates under the usual assumptions used for process modeling. Practically speaking, linear least squares regression makes very efficient use of the data. Good results can be obtained with relatively small data sets. Finally, the theory associated with linear regression is well-understood and allows for construction of different types of easily-interpretable statistical intervals for predictions, calibrations, and optimizations. These statistical intervals can then be used to give clear answers to scientific and engineering questions.
 +
 
 +
''Disadvantages'': The main disadvantages of linear least squares are limitations in the shapes that linear models can assume over long ranges, possibly poor extrapolation properties, and sensitivity to outliers. Linear models with nonlinear terms in the predictor variables curve relatively slowly, so for inherently nonlinear processes it becomes increasingly difficult to find a linear model that fits the data well as the range of the data increases. As the explanatory variables become extreme, the output of the linear model will also always more extreme. This means that linear models may not be effective for extrapolating the results of a process for which data cannot be collected in the region of interest. Of course extrapolation is potentially dangerous regardless of the model type. Finally, while the method of least squares often gives optimal estimates of the unknown parameters, it is very sensitive to the presence of unusual data points in the data used to fit a model. One or two outliers can sometimes seriously skew the results of a least squares analysis. This makes model validation, especially with respect to outliers, critical to obtaining sound answers to the questions motivating the construction of the model.
 +
 
 +
=====Inverse-Computation Trick for Matrices that are Nearly-Singular=====
 +
 
 +
The calculation of <math>\, \underline{\beta}</math> in linear regression and in logistic regression (described in the next lecture) requires the calculation of a matrix inverse.  For linear regression, <math>\, (\mathbf{X}\mathbf{X}^T)^{-1} </math> must be calculated.  Likewise, <math>\, (XWX^T)^{-1}</math> must be produced during the iterative method used for logistic regression. When the matrix <math>\, \mathbf{X}\mathbf{X}^T </math> or <math>\, XWX^T</math> is nearly singular, error resulting from numerical roundoff can be very large.  In the case of logistic regression, it may not be possible to determine a solution because the iterative method relies on convergence; with such large error in calculation of the inverse, the solution for entries of <math>\, \underline{\beta}</math> may grow without bound.  To improve the condition of the nearly-singular matrix prior to calculating its inverse, one trick is to add to it a very small identity matrix like <math>\, (10^{-10})I</math>.  This modification has very little effect on the exact result for the inverse matrix, but it improves the numerical calculation considerably.  Now, the inverses to be calculated are <math>\, (\mathbf{X}\mathbf{X}^T + (10^{-10})I)^{-1} </math> and <math>\, (XWX^T + (10^{-10})I)^{-1}</math>
 +
 
 +
====Multiple Linear Regression Analysis====
 +
Multiple linear regression is a statistical analysis which is similar to Linear Regression with the exception that there can be more than one predictor variable. The assumptions of outliers, linearity and constant variance need to be met. One additional assumption that needs to be examined is multicollinearity. Multicollinearity is the extent to which the predictor variables are related to each other. Multicollinearity can be assessed by asking SPSS for the Variance Inflation Factor (VIF). While different researchers have different criteria for what constitutes too high a VIF number, VIF of 10 or greater is certainly reason for pause. If the VIF is 10 or greater, consider collapsing the variables.
  
 
===Logistic Regression===
 
===Logistic Regression===
Line 1,567: Line 1,906:
 
1. <math>y = \frac{1}{1+e^{-x}}</math>
 
1. <math>y = \frac{1}{1+e^{-x}}</math>
  
2. <math>\frac{dy}{dx} = y(1-y)=\frac{e^{x}}{(1+e^{x})^{2}}</math>
+
2. <math>\frac{dy}{dx} = y(1-y)=\frac{-e^{-x}}{(1+e^{-x})^{2}}</math>
  
 
3. <math>y(0) = \frac{1}{2}</math>
 
3. <math>y(0) = \frac{1}{2}</math>
Line 1,611: Line 1,950:
  
 
====Fitting a Logistic Regression====
 
====Fitting a Logistic Regression====
Logistic regression tries to fit a distribution.  The common practice in statistics is to fit density function, posterior density of each class (Pr(Y|X), to data using [http://en.wikipedia.org/wiki/Maximum_likelihood maximum likelihood]. The maximum likelihood estimate of <math>\underline\beta</math>, denoted <math>\hat \beta_{ML}</math>, maximizes the probability of observing the training data <math>\{y_i,\, x_{i1}, \ldots, x_{id}\}, i=1, \ldots, n</math> from the known distribution.  Combining <math>\displaystyle P(Y=1 | X=x)</math> and <math>\displaystyle P(Y=0 | X=x)</math> as follows, we can consider the two classes at the same time:
+
Logistic regression tries to fit a distribution to the data.  The common practice in statistics is to fit a density function to data using [http://en.wikipedia.org/wiki/Maximum_likelihood maximum likelihood]. The maximum likelihood estimate of <math>\underline\beta</math>, denoted <math>\hat \beta_{ML}</math>, maximizes the probability of observing the training data <math>\{y_i,\, x_{i1}, \ldots, x_{id}\}, i=1, \ldots, n</math> from the maximum likelihood distribution.  Combining <math>\displaystyle P(Y=1 | X=x)</math> and <math>\displaystyle P(Y=0 | X=x)</math> as follows, we can consider the two classes at the same time (this is a useful trick, since <math> y_i \in \{0, 1\}</math>):
  
 
:<math>p(\underline{x_{i}};\underline{\beta}) = \left(\frac{\exp(\underline{\beta}^T \underline{x_i})}{1+\exp(\underline{\beta}^T \underline{x_i})}\right)^{y_i} \left(\frac{1}{1+\exp(\underline{\beta}^T \underline{x_i})}\right)^{1-y_i}</math>
 
:<math>p(\underline{x_{i}};\underline{\beta}) = \left(\frac{\exp(\underline{\beta}^T \underline{x_i})}{1+\exp(\underline{\beta}^T \underline{x_i})}\right)^{y_i} \left(\frac{1}{1+\exp(\underline{\beta}^T \underline{x_i})}\right)^{1-y_i}</math>
Line 1,620: Line 1,959:
 
\begin{align}
 
\begin{align}
 
\mathcal{L}(\theta)&=p({x_{1},...,x_{n}};\theta)\\
 
\mathcal{L}(\theta)&=p({x_{1},...,x_{n}};\theta)\\
&=\displaystyle p(x_{1};\theta) p(x_{2};\theta)... p(x_{n};\theta)  \quad   \mbox{(by independence and identical distribution)}\\
+
&=\displaystyle p(x_{1};\theta) p(x_{2};\theta)... p(x_{n};\theta)  \quad
&= \prod_{i=1}^n p(x_{i};\theta)
+
\end{align}
 +
</math>  (by independence and identical distribution)
 +
:::<math>
 +
\begin{align}
 +
    = \prod_{i=1}^n p(x_{i};\theta)
 
\end{align}
 
\end{align}
 
</math>
 
</math>
Line 1,652: Line 1,995:
 
To solve this equation, the [http://numericalmethods.eng.usf.edu/topics/newton_raphson.html Newton-Raphson algorithm] is used which requires the second derivative of the log-likelihood <math>\,l(\beta)</math> with respect to <math>\,\beta</math> in addition to the first derivative of <math>\,l(\beta)</math> with respect to <math>\,\beta</math>. This is demonstrated in the next section.
 
To solve this equation, the [http://numericalmethods.eng.usf.edu/topics/newton_raphson.html Newton-Raphson algorithm] is used which requires the second derivative of the log-likelihood <math>\,l(\beta)</math> with respect to <math>\,\beta</math> in addition to the first derivative of <math>\,l(\beta)</math> with respect to <math>\,\beta</math>. This is demonstrated in the next section.
  
====Extension====
+
=== Example: logistic Regression in MATLAB ===
 +
 
 +
% function x = logistic(a, y, w, ridge, param)
 +
%
 +
% Logistic regression.  Design matrix A, targets Y, optional instance
 +
% weights W, optional ridge term RIDGE, optional parameters object PARAM.
 +
%
 +
% W is a vector with length equal to the number of training examples; RIDGE
 +
% can be either a vector with length equal to the number of regressors, or
 +
% a scalar (the latter being synonymous to a vector with all entries the
 +
% same).
 +
%
 +
% PARAM has fields PARAM.MAXITER (an iteration limit), PARAM.VERBOSE
 +
% (whether to print diagnostic information), PARAM.EPSILON (used to test
 +
% convergence), and PARAM.MAXPRINT (how many regression coefficients to
 +
% print if VERBOSE==1).
 +
%
 +
% Model is
 +
%
 +
%  E(Y) = 1 ./ (1+exp(-A*X))
 +
%
 +
% Outputs are regression coefficients X.
  
* When we are dealing with a problem with more than two classes, we need to generalize our logistic regression to a [http://en.wikipedia.org/wiki/Multinomial_logit Multinomial Logit model].
+
function x = logistic(a, y, w, ridge, param)
  
* Limitations of Logistic Regression:
+
% process parameters
 +
 
 +
[n, m] = size(a);
 +
if ((nargin < 3) || (isempty(w)))
 +
  w = ones(n, 1);
 +
end
 +
if ((nargin < 4) || (isempty(ridge)))
 +
  ridge = 1e-5;
 +
end
 +
if (nargin < 5)
 +
  param = [];
 +
end
 +
if (length(ridge) == 1)
 +
    ridgemat = speye(m) * ridge;
 +
elseif (length(ridge(:)) == m)
 +
    ridgemat = spdiags(ridge(:), 0, m, m);
 +
else
 +
    error('ridge weight vector should be length 1 or %d', m);
 +
end
 +
if (~isfield(param, 'maxiter'))
 +
  param.maxiter = 200;
 +
end
 +
if (~isfield(param, 'verbose'))
 +
  param.verbose = 0;
 +
end
 +
if (~isfield(param, 'epsilon'))
 +
  param.epsilon = 1e-10;
 +
end
 +
if (~isfield(param, 'maxprint'))
 +
  param.maxprint = 5;
 +
end
 +
 
 +
% do the regression
 +
x = zeros(m,1);
 +
oldexpy = -ones(size(y));
 +
for iter = 1:param.maxiter
 +
  adjy = a * x;
 +
  expy = 1 ./ (1 + exp(-adjy));
 +
  deriv = expy .* (1-expy);
 +
  wadjy = w .* (deriv .* adjy + (y-expy));
 +
  weights = spdiags(deriv .* w, 0, n, n);
 +
  x = inv(a' * weights * a + ridgemat) * a' * wadjy;
 +
  if (param.verbose)
 +
    len = min(param.maxprint, length(x));
 +
    fprintf('%3d: [',iter);
 +
    fprintf(' %g', x(1:len));
 +
    if (len < length(x))
 +
      fprintf(' ... ');
 +
    end
 +
    fprintf(' ]\n');
 +
  end
 +
  if (sum(abs(expy-oldexpy)) < n*param.epsilon)
 +
    if (param.verbose)
 +
      fprintf('Converged.\n');
 +
    end
 +
    return;
 +
  end
 +
    oldexpy = expy;
 +
end
 +
warning('logistic:notconverged', 'Failed to converge');
 +
 
 +
 
 +
====Extension====
 +
 
 +
* When we are dealing with a problem with more than two classes, we need to generalize our logistic regression to a [http://en.wikipedia.org/wiki/Multinomial_logit Multinomial Logit model].
 +
*An extension of the logistic model to sets of interdependent variables is the [http://en.wikipedia.org/wiki/Conditional_random_field Conditional random field].
 +
 
 +
* Advantages and Limitations of Linear Regression Model:
 +
:1. Linear regression implements a statistical model that, when relationships between the independent variables and the dependent variable are almost linear, shows optimal results.
 +
:2. Linear regression is often inappropriately used to model non-linear relationships.
 +
:3. Linear regression is limited to predicting numeric output.
 +
:4. A lack of explanation about what has been learned can be a problem.
 +
 
 +
* Limitations of Logistic Regression:
 
:1. We know that there is no assumptions made about the distributions of the features of the data (i.e. the explanatory variables). However, the features should not be highly correlated with one another because this could cause problems with estimation.
 
:1. We know that there is no assumptions made about the distributions of the features of the data (i.e. the explanatory variables). However, the features should not be highly correlated with one another because this could cause problems with estimation.
 
:2. Large number of data points (i.e.the sample sizes) are required for logistic regression to provide sufficient estimates of the paramters in both classes. The more number of features/dimensions of the data, the larger the sample size required.
 
:2. Large number of data points (i.e.the sample sizes) are required for logistic regression to provide sufficient estimates of the paramters in both classes. The more number of features/dimensions of the data, the larger the sample size required.
 
:3. According to [http://www.google.ca/url?sa=t&source=web&cd=3&ved=0CC0QFjAC&url=http%3A%2F%2Fwww.csun.edu%2F~ata20315%2Fpsy524%2Fdocs%2FPsy524%2520lecture%252018%2520logistic.ppt&rct=j&q=logistic%20regression%20limitations&ei=mN7RTOC5HcWOnwfP0eho&usg=AFQjCNFBQ8BNxnc7xVArBgVgVWJOnDLMlw&sig2=_6j0mR3r92_xVGtzEJl7oA&cad=rja this source] however, the only real limitation of logistic regression as compared to other types of regression such as linear regression is that the response variable <math>\,y</math> can only take discrete values.
 
:3. According to [http://www.google.ca/url?sa=t&source=web&cd=3&ved=0CC0QFjAC&url=http%3A%2F%2Fwww.csun.edu%2F~ata20315%2Fpsy524%2Fdocs%2FPsy524%2520lecture%252018%2520logistic.ppt&rct=j&q=logistic%20regression%20limitations&ei=mN7RTOC5HcWOnwfP0eho&usg=AFQjCNFBQ8BNxnc7xVArBgVgVWJOnDLMlw&sig2=_6j0mR3r92_xVGtzEJl7oA&cad=rja this source] however, the only real limitation of logistic regression as compared to other types of regression such as linear regression is that the response variable <math>\,y</math> can only take discrete values.
 +
 +
====Further reading ====
 +
Some supplemental readings on linear and logistic regression:
 +
 +
1- A simple method of sample size calculation for linear and logistic regression [http://onlinelibrary.wiley.com/doi/10.1002/%28SICI%291097-0258%2819980730%2917:14%3C1623::AID-SIM871%3E3.0.CO;2-S/pdf here]
 +
 +
2- Choosing Between Logistic Regression and Discriminant Analysis [http://www.jstor.org/stable/pdfplus/2286261.pdf?acceptTC=true here]
 +
 +
3- On the existence of maximum likelihood estimates in logistic regression models [http://biomet.oxfordjournals.org/content/71/1/1.full.pdf+html here]
  
 
==Lecture summary==
 
==Lecture summary==
Line 1,669: Line 2,115:
 
===Logistic Regression Model===
 
===Logistic Regression Model===
  
Recall that in the last lecture, we learned the logistic regression model.
+
In statistics, '''logistic regression''' (sometimes called the '''logistic model''' or '''logit model''') is used for prediction of the probability of occurrence of an event by fitting data to a logit function logistic curve. It is a generalized linear model used for binomial regression. Like many forms of regression analysis, it makes use of several predictor variables that may be either numerical or categorical. For example, the probability that a person has a heart attack within a specified time period might be predicted from knowledge of the person's age, sex and body mass index. Logistic regression is used extensively in the medical and social sciences fields, as well as marketing applications such as prediction of a customer's propensity to purchase a product or cease a subscription.
 +
 
 +
 
 +
Recall that in the last lecture, we learned the logistic regression model:
  
 
* <math>P(Y=1 | X=x)=P(\underline{x};\underline{\beta})=\frac{exp(\underline{\beta}^T \underline{x})}{1+exp(\underline{\beta}^T \underline{x})}</math>
 
* <math>P(Y=1 | X=x)=P(\underline{x};\underline{\beta})=\frac{exp(\underline{\beta}^T \underline{x})}{1+exp(\underline{\beta}^T \underline{x})}</math>
Line 1,685: Line 2,134:
 
'''Newton-Raphson Algorithm:'''<br />
 
'''Newton-Raphson Algorithm:'''<br />
  
If we want to find <math>\ x^* </math> such that <math>\ f(x^*)=0</math>
+
If we want to find <math>\ x^* </math> such that <math>\ f(x^*)=0</math>, we proceed by first arbitrarily picking a starting point <math>\,x^* = x^{old}</math> and we iterate the following two steps until convergence, i.e. when <math>\, x^{new}</math> is sufficiently close to <math>\, x^{old}</math> using an arbitrary criterion of closeness:
 
+
<br \>
We first pick a starting point <math>\,x^* = x^{old}</math> and and we solve:
+
Step 1:
 +
<math>\, x^{new} \leftarrow x^{old}-\frac {f(x^{old})}{f'(x^{old})} </math><br />
 
<br \>
 
<br \>
 +
Step 2:
 +
<math>\, x^{old} \leftarrow x^{new}</math> <br />
  
<math>\ x^{new} \leftarrow x^{old}-\frac {f(x^{old})}{f'(x^{old})} </math> <br />
 
<math> \ x^{old} \leftarrow x^{new}</math>
 
<br />
 
This is repeated until convergence.
 
  
If <math>\ f'(x)=0</math> , we can solve:
+
If <math>\ f'(x)=0</math> , then we can replace the two steps above by the following two steps:
 
<br \>
 
<br \>
 +
Step 1: <math>\ x^{new} \leftarrow x^{old}-\frac {f'(x^{old})}{f''(x^{old})} </math> <br />
 +
<br \>
 +
Step 2:
 +
<math> \ x^{old} \leftarrow x^{new}</math> <br />
  
<math>\ x^{new} \leftarrow x^{old}-\frac {f'(x^{old})}{f''(x^{old})} </math> <br />
+
If we want to maximize or minimize <math>\ f(x) </math>, then we solve for the value of <math>\,x</math> at which <math>\ f'(x)=0 </math> using the following iterative updating rule that generates <math>\ x^{new}</math> from <math>\ x^{old}</math>:
<math> \ x^{old} \leftarrow x^{new}</math>
+
<br \><math>\ x^{new} \leftarrow x^{old}-\frac {f'(x^{old})}{f''(x^{old})} </math><br />
<br />
 
  
If we want to maximize or minimize <math>\ f(x) </math>, then solve for  <math>\ f'(x)=0 </math>
+
Using vector notation, the above rule can be written as <br />
 
 
<math>\ x^{new} \leftarrow x^{old}-\frac {f'(x^{old})}{f''(x^{old})} </math>
 
 
 
<br />
 
 
 
In vector notation the above can be written as <br />
 
  
 
<math>
 
<math>
X^{new} \leftarrow X^{old} - H^{-1}\nabla
+
X^{new} \leftarrow X^{old} - H^{-1}(f)(X^{old})\nabla f(X^{old})
 
</math>
 
</math>
 
<br />
 
<br />
H is the [http://en.wikipedia.org/wiki/Hessian_matrix Hessian matrix] or second derivative matrix and <math>\,\nabla</math> is the gradient both evaluated at <math>\,X^{old}</math>
+
where <math>\,H</math> is the [http://en.wikipedia.org/wiki/Hessian_matrix Hessian matrix] or second derivative matrix and <math>\,\nabla</math> is the [http://en.wikipedia.org/wiki/Gradient gradient] or first derivative vector.
 
<br />
 
<br />
  
Line 1,734: Line 2,179:
 
::<math>=\sum_{i=1}^n \frac{(-\underline{x}_i exp(\underline{\beta}^T\underline{x}_i) \underline{x}_i^T)}{(1+exp(\underline{\beta}^T \underline{x}))(1+exp(\underline{\beta}^T \underline{x}))}</math> (by cancellation)
 
::<math>=\sum_{i=1}^n \frac{(-\underline{x}_i exp(\underline{\beta}^T\underline{x}_i) \underline{x}_i^T)}{(1+exp(\underline{\beta}^T \underline{x}))(1+exp(\underline{\beta}^T \underline{x}))}</math> (by cancellation)
  
::<math>=\sum_{i=1}^n - \underline{x}_i \underline{x}_i^T P(\underline{x}_i;\underline{\beta}))[1-P(\underline{x}_i;\underline{\beta})])</math>(since <math>P(\underline{x}_i;\underline{\beta})=\frac{exp(\underline{\beta}^T \underline{x})}{1+exp(\underline{\beta}^T \underline{x})}</math> and <math>1-P(\underline{x}_i;\underline{\beta})=\frac{1}{1+exp(\underline{\beta}^T \underline{x})}</math>)
+
::<math>=\sum_{i=1}^n - \underline{x}_i \underline{x}_i^T P(\underline{x}_i;\underline{\beta}))[1-P(\underline{x}_i;\underline{\beta})])</math>(since <math>P(\underline{x}_i;\underline{\beta})=\frac{exp(\underline{\beta}^T \underline{x}_i)}{1+exp(\underline{\beta}^T \underline{x}_i)}</math> and <math>1-P(\underline{x}_i;\underline{\beta})=\frac{1}{1+exp(\underline{\beta}^T \underline{x}_i)}</math>)
  
 
The same second derivative can be achieved if we reduce the occurrences of beta to 1 by the identity<math>\frac{a}{1+a}=1-\frac{1}{1+a}</math>
 
The same second derivative can be achieved if we reduce the occurrences of beta to 1 by the identity<math>\frac{a}{1+a}=1-\frac{1}{1+a}</math>
  
And solving <math>\frac{\partial}{\partial \underline{\beta}^T}\sum_{i=1}^n \left[{y_i} \underline{x}_i-\left[1-\frac{1}{1+exp(\underline{\beta}^T \underline{x}_i)}\right]\underline{x}_i\right] </math>
+
and then solving <math>\frac{\partial}{\partial \underline{\beta}^T}\sum_{i=1}^n \left[{y_i} \underline{x}_i-\left[1-\frac{1}{1+exp(\underline{\beta}^T \underline{x}_i)}\right]\underline{x}_i\right] </math>
  
  
Starting with <math>\,\underline{\beta}^{old}</math>, the Newton-Raphson update is
+
In each of the iterative steps, starting with the existing <math>\,\underline{\beta}^{old}</math> which is initialized with an arbitrarily chosen value, the [http://en.wikipedia.org/wiki/Newton-Raphson Newton-Raphson] updating rule for obtaining <math>\,\underline{\beta}^{new}</math> is
  
 
<math>\,\underline{\beta}^{new}\leftarrow \,\underline{\beta}^{old}- (\frac{\partial ^2 l}{\partial \underline{\beta}\partial \underline{\beta}^T})^{-1}(\frac{\partial l}{\partial \underline{\beta}})</math> where the derivatives are evaluated at <math>\,\underline{\beta}^{old}</math>
 
<math>\,\underline{\beta}^{new}\leftarrow \,\underline{\beta}^{old}- (\frac{\partial ^2 l}{\partial \underline{\beta}\partial \underline{\beta}^T})^{-1}(\frac{\partial l}{\partial \underline{\beta}})</math> where the derivatives are evaluated at <math>\,\underline{\beta}^{old}</math>
  
The iteration will terminate when <math>\underline{\beta}^{new}</math> is very close to <math>\underline{\beta}^{old}</math>.
+
The iterations terminate when <math>\underline{\beta}^{new}</math> is very close to <math>\underline{\beta}^{old}</math> according to an arbitrarily defined criterion.
  
The iteration can be described in matrix form.
+
Each iteration can be described in matrix form.
  
 
* Let <math>\,\underline{Y}</math> be the column vector of <math>\,y_i</math>.  (<math>n\times1</math>)
 
* Let <math>\,\underline{Y}</math> be the column vector of <math>\,y_i</math>.  (<math>n\times1</math>)
Line 1,760: Line 2,205:
 
<math>\frac{\partial ^2 l}{\partial \underline{\beta}\partial \underline{\beta}^T} = -XWX^T</math>
 
<math>\frac{\partial ^2 l}{\partial \underline{\beta}\partial \underline{\beta}^T} = -XWX^T</math>
  
The Newton-Raphson step is
+
The [http://en.wikipedia.org/wiki/Newton-Raphson Newton-Raphson] step is
  
 
<math>\underline{\beta}^{new} \leftarrow \underline{\beta}^{old}+(XWX^T)^{-1}X(\underline{Y}-\underline{P})</math>
 
<math>\underline{\beta}^{new} \leftarrow \underline{\beta}^{old}+(XWX^T)^{-1}X(\underline{Y}-\underline{P})</math>
Line 1,768: Line 2,213:
 
<math>
 
<math>
 
\begin{align}
 
\begin{align}
\underline{\beta}^{new} &= (XWX^T)^{-1}(XWX^T)\underline{\beta}^{old}+(XWX^T)^{-1}XWW^{-1}(\underline{Y}-\underline{P})\\
+
\underline{\beta}^{new} &= \,\underline{\beta}^{old}- (\frac{\partial ^2 l}{\partial \underline{\beta}\partial \underline{\beta}^T})^{-1}(\frac{\partial l}{\partial \underline{\beta}})\\
 +
&= \,\underline{\beta}^{old}- (-XWX^T)^{-1}X(\underline{Y}-\underline{P})\\
 +
&= \,(XWX^T)^{-1}(XWX^T)\underline{\beta}^{old}- (XWX^T)^{-1}(XWX^T)(-XWX^T)^{-1}X(\underline{Y}-\underline{P})\\
 +
&= (XWX^T)^{-1}(XWX^T)\underline{\beta}^{old}+(XWX^T)^{-1}XWW^{-1}(\underline{Y}-\underline{P})\\
 
&=(XWX^T)^{-1}XW[X^T\underline{\beta}^{old}+W^{-1}(\underline{Y}-\underline{P})]\\
 
&=(XWX^T)^{-1}XW[X^T\underline{\beta}^{old}+W^{-1}(\underline{Y}-\underline{P})]\\
 
&=(XWX^T)^{-1}XWZ
 
&=(XWX^T)^{-1}XWZ
Line 1,775: Line 2,223:
 
where <math>Z=X^{T}\underline{\beta}^{old}+W^{-1}(\underline{Y}-\underline{P})</math>
 
where <math>Z=X^{T}\underline{\beta}^{old}+W^{-1}(\underline{Y}-\underline{P})</math>
  
This is an adjusted response and it is solved repeatedly when <math>\ p </math>, <math>\ W </math>,  and <math>\ z </math> changes. This algorithm is called [http://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares iteratively reweighted least squares] because it solves the weighted least squares problem repeatedly.
+
This is an adjusted response and it is solved repeatedly as <math>\, P </math>, <math>\, W </math>,  and <math>\, Z </math> are iteratively updated during the steps until convergence is achieved. This algorithm is called [http://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares iteratively reweighted least squares] because it solves the weighted least squares problem iteratively.
  
Recall that linear regression by least squares finds the following minimum: <math>\min_{\underline{\beta}}(\underline{y}-\underline{\beta}^T X)^T(\underline{y}-\underline{\beta}^TX)</math>
+
Recall that linear regression by least squares finds the following minimum: <math>\min_{\underline{\beta}}(\underline{y}-X^T \underline{\beta})^T(\underline{y}-X^T \underline{\beta})</math>
  
 
we have <math>\underline\hat{\beta}=(XX^T)^{-1}X\underline{y}</math>
 
we have <math>\underline\hat{\beta}=(XX^T)^{-1}X\underline{y}</math>
Line 1,783: Line 2,231:
 
Similarly, we can say that <math>\underline{\beta}^{new}</math> is the solution of a weighted least square problem:
 
Similarly, we can say that <math>\underline{\beta}^{new}</math> is the solution of a weighted least square problem:
  
<math>\underline{\beta}^{new} \leftarrow arg \min_{\underline{\beta}}(Z-X\underline{\beta}^T)W(Z-X\underline{\beta}^T)</math>
+
<math>\underline{\beta}^{new} \leftarrow arg \min_{\underline{\beta}}(Z-X^T \underline{\beta})W(Z-X^T \underline{\beta})</math>
  
 
====Pseudo Code====
 
====Pseudo Code====
#<math>\underline{\beta} \leftarrow 0</math>
+
First, initialize <math>\,\underline{\beta}^{old} \leftarrow 0</math> and set <math>\,\underline{Y}</math>, the labels associated with the observations <math>\,i=1...n</math>.
#Set <math>\,\underline{Y}</math>, the label associated with each observation <math>\,i=1...n</math>.
+
Then, in each iterative step, perform the following:
 
#Compute <math>\,\underline{P}</math> according to the equation <math>P(\underline{x}_i;\underline{\beta})=\frac{exp(\underline{\beta}^T \underline{x}_i)}{1+exp(\underline{\beta}^T \underline{x}_i)}</math> for all <math>\,i=1...n</math>.
 
#Compute <math>\,\underline{P}</math> according to the equation <math>P(\underline{x}_i;\underline{\beta})=\frac{exp(\underline{\beta}^T \underline{x}_i)}{1+exp(\underline{\beta}^T \underline{x}_i)}</math> for all <math>\,i=1...n</math>.
#Compute the diagonal matrix <math>\,W</math> by setting <math>\,w_{i,i}</math> to <math>P(\underline{x}_i;\underline{\beta}))[1-P(\underline{x}_i;\underline{\beta})]</math> for all <math>\,i=1...n</math>.
+
#Compute the diagonal matrix <math>\,W</math> by setting <math>\,W_{i,i}</math> to <math>P(\underline{x}_i;\underline{\beta}))[1-P(\underline{x}_i;\underline{\beta})]</math> for all <math>\,i=1...n</math>.
#<math>Z \leftarrow X^T\underline{\beta}+W^{-1}(\underline{Y}-\underline{P})</math>.
+
#Compute <math>Z \leftarrow X^T\underline{\beta}+W^{-1}(\underline{Y}-\underline{P})</math>.
#<math>\underline{\beta} \leftarrow (XWX^T)^{-1}XWZ</math>.
+
#<math>\underline{\beta}^{new} \leftarrow (XWX^T)^{-1}XWZ</math>.
#If the new <math>\underline{\beta}</math> value is sufficiently close to the old value, stop; otherwise go back to step 3.
+
#If <math>\underline{\beta}^{new}</math> is sufficiently close to <math>\underline{\beta}^{old}</math> according to an arbitrarily defined criterion, then stop; otherwise, set <math>\,\underline{\beta}^{old} \leftarrow \underline{\beta}^{new}</math> and another iterative step is made towards convergence between <math>\underline{\beta}^{new}</math> and <math>\underline{\beta}^{old}</math>.
 +
 
 +
The following Matlab code implements the method above:
 +
 
 +
  Error = 0.01;
 +
 
 +
  %Initialize logistic variables
 +
  B_old=0.1*ones(m,1); %beta
 +
  W=0.5*ones(n,n); %weights
 +
  P=zeros(n,1);
 +
  Norm=1;
 +
 
 +
  while Norm>Error %while the change in Beta (represented by the norm between B_new and B_old) is higher than the threshold, iterate
 +
      for i=1:n
 +
          P(i,1)=exp(B_old'*Xnew(:,i))/(1+exp(B_old'*Xnew(:,i)));
 +
          W(i,i)=P(i,1)*(1-P(i,1));
 +
      end
 +
      z = Xnew'*B_old + pinv(W)*(ytrain-P);
 +
      B_new = pinv(Xnew*W*Xnew')*Xnew*W*z;
 +
      Norm=sqrt((B_new-B_old)'*(B_new-B_old));
 +
      B_old = B_new;
 +
  end
 +
 
 +
====Classification====
 +
To implement classification, we should compute <math> \underline{\beta}^{T} x</math>. If <math> \underline{\beta}^{T} x <0 </math>, then <math>\, x </math> belongs to class 0 , otherwise it belongs to class 1 .
  
 
===Comparison with Linear Regression===
 
===Comparison with Linear Regression===
Line 1,800: Line 2,272:
 
:'''note:'''For linear regression, we assume the model is linear. The boundary is <math>P(Y=k|X=x)=\underline{\beta}^T\underline{x}_i+\underline{\beta}_0=0.5</math> (linear)
 
:'''note:'''For linear regression, we assume the model is linear. The boundary is <math>P(Y=k|X=x)=\underline{\beta}^T\underline{x}_i+\underline{\beta}_0=0.5</math> (linear)
  
::For logistic regression, the boundary is <math>P(Y=k|X=x)=\frac{exp(\underline{\beta}^T \underline{x})}{1+exp(\underline{\beta}^T \underline{x})}=0.5 \Rightarrow exp(\underline{\beta}^T \underline{x})=1\Rightarrow \underline{\beta}^T \underline{x}=0</math> (linear)  
+
::For logistic regression, the boundary is <math>P(Y=k|X=x)=\frac{exp(\underline{\beta}^T \underline{x})}{1+exp(\underline{\beta}^T \underline{x})}=0.5 \Rightarrow exp(\underline{\beta}^T \underline{x})=1\Rightarrow \underline{\beta}^T \underline{x}=0</math> (nonlinear)  
  
 
*'''Differences'''
 
*'''Differences'''
 +
  
 
#Linear regression: <math>\,P(Y=k|X=x)</math> is linear function of <math>\,x</math>, <math>\,P(Y=k|X=x)</math> is not guaranteed to fall between 0 and 1 and to sum up to 1. There exists a closed form solution for least squares.
 
#Linear regression: <math>\,P(Y=k|X=x)</math> is linear function of <math>\,x</math>, <math>\,P(Y=k|X=x)</math> is not guaranteed to fall between 0 and 1 and to sum up to 1. There exists a closed form solution for least squares.
#Logistic regression: <math>\,P(Y=k|X=x)</math> is a nonlinear function of <math>\,x</math>, and it is guaranteed to range from 0 to 1 and to sum  up to 1. No closed form solution exists
+
#Logistic regression: <math>\,P(Y=k|X=x)</math> is a nonlinear function of <math>\,x</math>, and it is guaranteed to range from 0 to 1 and to sum  up to 1. No closed form solution exists, so the Newton-Raphson algorithm is typically used to arrive at an estimate for the parameters.
  
 
===Comparison with LDA===
 
===Comparison with LDA===
Line 1,843: Line 2,316:
 
   >>[B,dev,stats] = mnrfit(sample,group);
 
   >>[B,dev,stats] = mnrfit(sample,group);
 
   >>x=[ones(1,400); sample'];
 
   >>x=[ones(1,400); sample'];
:Now we use [http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/mnrfit.html&http://www.google.cn/search?hl=zh-CN&q=mnrfit+matlab&btnG=Google+%E6%90%9C%E7%B4%A2&aq=f&oq= mnrfit] to use logistic regression to classfy the data. This function can return B which is a <math>\,(d+1)</math><math>\,\times</math><math>\,(k-1)</math> matrix of estimates, where each column corresponds to the estimated intercept term and predictor coefficients. In this case, B is a <math>3\times{1}</math> matrix.
+
:Now we use [http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/mnrfit.html&http://www.google.cn/search?hl=zh-CN&q=mnrfit+matlab&btnG=Google+%E6%90%9C%E7%B4%A2&aq=f&oq= mnrfit] to use logistic regression to classfy the data. This function can return <math>\underline{\beta}</math> which is a <math>\,(d+1)</math><math>\,\times</math><math>\,(k-1)</math> matrix of estimates, where each column corresponds to the estimated intercept term and predictor coefficients. In this case, <math>\underline{\beta}</math> is a <math>3\times{1}</math> matrix.
  
 
   >> B
 
   >> B
Line 1,871: Line 2,344:
  
 
[[File:Boundary-lda.png‎|frame|center| From this figure, we can see that the results of Logistic Regression and LDA are very similar.]]
 
[[File:Boundary-lda.png‎|frame|center| From this figure, we can see that the results of Logistic Regression and LDA are very similar.]]
 +
 +
=== Extra Matlab Examples ===
 +
 +
==== Example 1 ====
 +
 +
% This Matlab code provides a function that uses the Newton-Raphson algorithm
 +
% to calculate ML estimates of a simple logistic regression. Most of the
 +
% code comes from Anders Swensen, "Non-linear regression." There are two
 +
% elements in the beta vector, which we wish to estimate.
 +
 +
function [beta,J_bar] = NR_logistic(data,beta_start)
 +
x=data(:,1); % x is first column of data
 +
y=data(:,2); % y is second column of data
 +
n=length(x)
 +
diff = 1; beta = beta_start; % initial values
 +
while diff>0.0001 % convergence criterion
 +
  beta_old = beta;
 +
  p = exp(beta(1)+beta(2)*x)./(1+exp(beta(1)+beta(2)*x));
 +
  l = sum(y.*log(p)+(1-y).*log(1-p))
 +
  s = [sum(y-p); % scoring function
 +
  sum((y-p).*x)];
 +
  J_bar = [sum(p.*(1-p)) sum(p.*(1-p).*x); % information matrix
 +
  sum(p.*(1-p).*x) sum(p.*(1-p).*x.*x)]
 +
  beta = beta_old + J_bar\s % new value of beta
 +
  diff = sum(abs(beta-beta_old)); % sum of absolute differences
 +
end
 +
 +
==== Example 2 ====
 +
 +
% This Matlab program illustrates the use of the Newton-Raphson algorithm
 +
% to obtain maximum likelihood estimates of a logistic regression. The data
 +
% and much of the code are taken from Anders Swensen, "Non-linear regression,"
 +
% www.math.uio_no/avdc/kurs/ST110/materiale/opti_30.ps.
 +
% First, load and transform data:
 +
load 'beetle.dat'; % load data
 +
m=length(beetle(:,1)) % count the rows in the data matrix
 +
x=[]; % create empty vectors
 +
y=[];
 +
for j=1:m % expand group data into individual data
 +
  x=[x,beetle(j,1)*ones(1,beetle(j,2))];
 +
  y=[y,ones(1,beetle(j,3)),zeros(1,beetle(j,2)-beetle(j,3))];
 +
end
 +
beetle2=[x;y]';
 +
 +
% Next, specify starting points for iteration on parameter values:
 +
beta0 = [0; 0]
 +
 +
% Finally, call the function NR_logistic and use its output
 +
[betaml,Jbar] = NR_logistic(beetle2,beta0)
 +
covmat = inv(Jbar)
 +
stderr = sqrt(diag(covmat))
 +
 +
==== Example 3 ====
 +
 +
% function x = logistic(a, y, w)
 +
% Logistic regression.  Design matrix A, targets Y, optional
 +
% instance weights W.  Model is E(Y) = 1 ./ (1+exp(-A*X)).
 +
% Outputs are regression coefficients X.
 +
function x = logistic(a, y, w)
 +
epsilon = 1e-10;
 +
ridge = 1e-5;
 +
maxiter = 200;
 +
[n, m] = size(a);
 +
if nargin < 3
 +
  w = ones(n, 1);
 +
end
 +
x = zeros(m,1);
 +
oldexpy = -ones(size(y));
 +
for iter = 1:maxiter
 +
  adjy = a * x;
 +
  expy = 1 ./ (1 + exp(-adjy));
 +
  deriv = max(epsilon*0.001, expy .* (1-expy));
 +
  adjy = adjy + (y-expy) ./ deriv;
 +
  weights = spdiags(deriv .* w, 0, n, n);
 +
  x = inv(a' * weights * a + ridge*speye(m)) * a' * weights * adjy;
 +
  fprintf('%3d: [',iter);
 +
  fprintf(' %g', x);
 +
  fprintf(' ]\n');
 +
  if (sum(abs(expy-oldexpy)) < n*epsilon)
 +
  fprintf('Converged.\n');
 +
  break;
 +
  end
 +
  oldexpy = expy;
 +
end
  
 
===Lecture Summary===
 
===Lecture Summary===
  
Traditionally, logistic regression parameters are estimated using maximum likelihood. However, other optimization techniques may be used as well.
+
Traditionally, regression parameters are estimated using maximum likelihood. However, other optimization techniques may be used as well.
 
<br />
 
<br />
Since there is no closed-form solution for finding zero of the first derivative of the log likelihood function, the Newton Raphson algorithm is used instead. This problem is convex, so Newton's method is guaranteed to converge to a global optimum.
+
In the case of logistic regression, since there is no closed-form solution for finding zero of the first derivative of the log-likelihood function, the Newton-Raphson algorithm is typically used to estimate parameters. This problem is convex, so the Newton-Raphson algorithm is guaranteed to converge to a global optimum.
 
<br />
 
<br />
 
Logistic regression requires less parameters than LDA or QDA, which makes it favorable for high-dimensional data.
 
Logistic regression requires less parameters than LDA or QDA, which makes it favorable for high-dimensional data.
Line 1,884: Line 2,441:
 
A detailed proof that logistic regression is convex is available [http://people.csail.mit.edu/jrennie/writing/convexLR.pdf here]. See '1 Binary LR' for the case we discussed in lecture.
 
A detailed proof that logistic regression is convex is available [http://people.csail.mit.edu/jrennie/writing/convexLR.pdf here]. See '1 Binary LR' for the case we discussed in lecture.
  
== '''Multi-Class Logistic Regression & Perceptron - October 19, 2010''' ==
 
  
=== Lecture Summary ===
+
===[http://komarix.org/ac/lr Applications]===
 +
 
 +
1. Collaborative filtering.
 +
 
 +
2. Link Analysis.
 +
 
 +
3. Times Series with Logistic Regression.
 +
 
 +
4. Alias Detection.
 +
 
 +
===References===
 +
 
 +
1. Applied logistic regression
 +
[http://books.google.ca/books?hl=en&lr=&id=Po0RLQ7USIMC&oi=fnd&pg=PA1&dq=Logistic+Regression&ots=DmdTni_oGX&sig=PDYTPVdy3T115RtFbBN3_SzX5Vc#v=onepage&q&f=false]
 +
 
 +
2. External validity of predictive models: a comparison of logistic regression, classification trees, and neural networks
 +
[http://www.jclinepi.com/article/S0895-4356%2803%2900120-3/abstract]
 +
 
 +
3. Logistic Regression: A Self-Learning Text by David G. Kleinbaum, Mitchel Klein [http://books.google.ca/books?id=J7E0JQweHkoC&printsec=frontcover&dq=logistic+regression&hl=en&ei=7WECTcvqMp-KnAeaq6HlDQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CD8Q6AEwAg#v=onepage&q&f=false]
  
In this lecture, the topic of logistic regression was finalized by covering the multi-class logistic regression model and a new topic on the Perceptron was introduced. The Perceptron is a linear classifier for two-class problems. The main goal of Perceptron is to classify data into 2 classes by minimizing the distances between the misclassified points and the decision boundary. This will be continued in the following lectures.
+
4. Two useful ppt files introducing concepts of logistic regression
 +
[http://www.csun.edu/~ata20315/psy524/docs/Psy524%20lecture%2018%20logistic.pdf] [http://www.daniel-wiechmann.eu/downloads/logreg1.pdf]
 +
 
 +
== '''Multi-Class Logistic Regression & Perceptron - October 19, 2010''' ==
  
 
=== Multi-Class Logistic Regression ===
 
=== Multi-Class Logistic Regression ===
Recall that in two-class logistic regression, the posterior probability of one of the classes (say class 0) is modeled by a function in the form shown in figure 1.  
+
Recall that in two-class logistic regression, the class-conditional probability of one of the classes (say class 0) is modeled by a function in the form shown in figure 1.  
  
The posterior probability of the second class (say class 1) is the complement of the first class (class 0). <br /><br />
+
The class-conditional probability of the second class (say class 1) is the complement of the first class (class 0). <br /><br />
 
<math>\displaystyle P(Y=0 | X=x) = 1 - P(Y=1 | X=x)</math><br />
 
<math>\displaystyle P(Y=0 | X=x) = 1 - P(Y=1 | X=x)</math><br />
  
Line 1,901: Line 2,478:
 
<math>\displaystyle \sigma\,\!(a) = \frac {e^a}{1+e^a} = \frac {1}{1+e^{-a}}</math><br /><br />
 
<math>\displaystyle \sigma\,\!(a) = \frac {e^a}{1+e^a} = \frac {1}{1+e^{-a}}</math><br /><br />
  
In two-class logistic regression, we compare the posterior of one class to the other using this ratio:<br />
+
In two-class logistic regression, we compare the class-conditional probability of one class to the other using this ratio:<br />
  
 
:<math> \frac{P(Y=1|X=x)}{P(Y=0|X=x)}</math><br />
 
:<math> \frac{P(Y=1|X=x)}{P(Y=0|X=x)}</math><br />
Line 1,913: Line 2,490:
 
Using (*), we can extend the notion of logistic regression for the cases where we have more than two classes.<br />
 
Using (*), we can extend the notion of logistic regression for the cases where we have more than two classes.<br />
  
Assume we have <math>\,k</math> classes. Looking at the logarithm of the ratio of posteriors of each class and the <math>\,k</math><sup>th</sup> class, we have: <br />
+
Assume we have <math>\,k</math> classes, where <math>\,k</math> is greater than two. Putting an arbitrarily chosen class  (which for simplicity we shall assume is class <math>\,k</math>) aside, and then looking at the logarithm of the ratio of the class-conditional probability of each of the other classes and the class-conditional probability of class <math>\,k</math>, we have: <br />
  
 
+
:<math>\log\left(\frac{P(Y=1|X=x)}{P(Y=k|X=x)}\right)=\underline{\beta}_1^T\underline{x} </math> <br />
:<math>\log\left(\frac{P(Y=1|X=x)}{P(Y=k|X=x)}\right)=\underline{\beta_1}^T\underline{x} </math> <br />
+
:<math>\log\left(\frac{P(Y=2|X=x)}{P(Y=k|X=x)}\right)=\underline{\beta}_2^T\underline{x} </math> <br />
:<math>\log\left(\frac{P(Y=2|X=x)}{P(Y=k|X=x)}\right)=\underline{\beta_2}^T\underline{x} </math> <br />
 
 
::::<math> \vdots</math><br />
 
::::<math> \vdots</math><br />
:<math>\log\left(\frac{P(Y=k-1|X=x)}{P(Y=k|X=x)}\right)=\underline{\beta_{k-1}}^T\underline{x} </math> <br />
+
:<math>\log\left(\frac{P(Y=k-1|X=x)}{P(Y=k|X=x)}\right)=\underline{\beta}_{k-1}^T\underline{x} </math> <br />
  
  
Although the denominator, in the above posterior ratios, is chosen to be the posterior of the last class (class <math>\,k</math>), the choice of denominator is arbitrary in that the posterior estimates are equivariant under this choice - [http://www.springerlink.com/content/t45k620382733r71/ Linear Methods for Classification].<br /><br />
+
Although the denominator in the above class-conditional probability ratios is chosen to be the class-conditional probability of the last class (class <math>\,k</math>), the choice of the denominator is arbitrary in that the class-conditional probability estimates are equivariant under this choice - [http://www.springerlink.com/content/t45k620382733r71/ Linear Methods for Classification].<br /><br />
  
Each of these functions is linear in <math>\,x</math>. However, we have different <math>\underline{\,\beta_{i}}</math>'s. We have to make sure that the densities assigned to different classes sum to one.<br /><br />
+
Each of these functions is linear in <math>\,x</math>. However, we have different <math>\underline{\,\beta}_{i}</math>'s. We have to make sure that the densities assigned to all of the different classes sum to one.<br /><br />
  
 
In general, we can write:
 
In general, we can write:
<br /><math>P(Y=c | X=x) = \frac{e^{\underline{\beta_c}^T \underline{x}}}{1+\sum_{l=1}^{k-1}e^{\underline{\beta_l}^T \underline{x}}},\quad c \in \{1,\dots,k-1\} </math><br />
+
<br /><math>P(Y=c | X=x) = \frac{e^{\underline{\beta}_c^T \underline{x}}}{1+\sum_{l=1}^{k-1}e^{\underline{\beta}_l^T \underline{x}}},\quad c \in \{1,\dots,k-1\} </math><br />
<br /><math>P(Y=k | X=x) = \frac{1}{1+\sum_{l=1}^{k-1}e^{\underline{\beta_l}^T \underline{x}}}</math><br />
+
<br /><math>P(Y=k | X=x) = \frac{1}{1+\sum_{l=1}^{k-1}e^{\underline{\beta}_l^T \underline{x}}}</math><br />
These posteriors clearly sum to one. <br /><br />
+
These class-conditional probabilities clearly sum to one. <br /><br />
  
In the case of the two-class problem, it is pretty simple to find the <math>\,\underline{\beta}</math> parameter (the <math>\,\underline{\beta}</math> in two-class linear regression problems has dimension <math>\,(d+1)\times1</math>), as mentioned in previous lectures. In the multi-class case the iterative Newton method can be used, but here <math>\,\underline{\beta}</math> is of dimension <math>(d+1)\times(k-1)</math> and the weight matrix <math>W</math> is a dense and non-diagonal matrix. This results in a computationally inefficient yet feasible-to-be-solved algorithm. A trick would be to re-parametrize the logistic regression problem by expanding the input vector <math>\,\underline{x}</math> (as in Question 4 on Assignment #2).
+
In the case of the two-classes problem, it is pretty simple to find the <math>\,\underline{\beta}</math> parameter (the <math>\,\underline{\beta}</math> in two-class logistic regression problems has dimension <math>\,(d+1)\times1</math>), as mentioned in previous lectures. In the multi-class case the iterative Newton method can be used, but here <math>\,\underline{\beta}</math> is of dimension <math>\ (d+1)\times(k-1)</math> and the weight matrix <math>W</math> is a dense and non-diagonal matrix. This results in a computationally inefficient yet feasible-to-be-solved algorithm. A trick would be to re-parametrize the logistic regression problem. This is done by suitably expanding the following: the input vector <math>\,x</math>, the vector of parameters <math>\,\beta</math>, the vector of responses <math>\,y</math>, as well as the <math>\,\underline{P}</math> vector and the <math>\,W</math> matrix used in the Newton-Raphson updating rule. For interested readers, details regarding this re-parametrization can be found in [http://www.stat.psu.edu/~jiali/course/stat597e/notes2/logit.pdf Jia Li's "Logistic Regression" slides]. Another major difference between the two-classes logistic regression and the general multi-classes logistic regression is that, unlike the former which uses the logistic sigmoid function, the latter uses the softmax function instead. Details regarding the softmax function can be found in [http://www.cedar.buffalo.edu/~srihari/CSE574/Chap4/Chap4-Part3.pdf Sargur N. Srihari's "Logistic Regression" slides]. 
 +
The Newton-Raphson updating rule however, remains the same as it is in the two-classes case, i.e. it is still <math>\underline{\beta}^{new} \leftarrow \underline{\beta}^{old}+(XWX^T)^{-1}X(\underline{Y}-\underline{P})</math>. This key point is also addressed in [http://www.stat.psu.edu/~jiali/course/stat597e/notes2/logit.pdf Jia Li's slides] given above.  
 
<br /><br />
 
<br /><br />
  
Note that logistic regression does not assume a distribution for the prior. whereas LDA assumes the prior to be Bernoulli. <br /><br />
+
Note that logistic regression does not assume a distribution for the prior, whereas LDA assumes the prior to be Bernoulli. <br /><br />
  
===Neural Network Concept===
+
[http://en.wikipedia.org/wiki/Random_multinomial_logit Random multinomial logit] models combine a random ensemble of multinomial logit models for use as a classifier.
The concept of constructing an artificial neural network came from scientists who were interested in simulating the human neural network in their computers. They were trying to create computer programs that could learn like people. A neural network is a method in artificial intelligence, and it is a simplified model of neural processing in the brain, even though the relation between this model and brain biological architecture is not yet clear.
 
  
=== Perceptron ===
+
=== Multiple Logistic Regression in Matlab ===
  
[http://en.wikipedia.org/wiki/Perceptron Perceptron] was invented in 1957 by [http://en.wikipedia.org/wiki/Frank_Rosenblatt Frank Rosenblatt]. It is the basic building block of Feed-Forward neural networks<br /><br />
 
  
We know that the least-squares obtained by regression of -1/1 response variable <math>\displaystyle Y</math> on observation <math>\displaystyle x</math> leads to the same coefficients as LDA (recall that LDA minimizes the distance between discriminant function (decision boundary) and the data points). Least squares returns the sign of the linear combination of features as the class labels (Figure 2). This concept was called the Perceptron in Engineering literature during the 1950's. <br /><br />
+
% Examples: Multiple linear regression im Matlab
  
[[File:Perceptron.jpg|371px|thumb|right| Fig.2 Diagram of a linear perceptron ]]
+
% Load data on cars identify weight and horsepower as predictors, mileage as the response:
 +
load carsmall
 +
x1 = Weight;
 +
x2 = Horsepower; % Contains NaN data
 +
y = MPG;
  
There is a cost function <math>\,\displaystyle D</math> that the Perceptron tries to minimize:<br />
+
% Compute regression coefficients for a linear model with an interaction term:
  
<math>D(\underline{\beta},\beta_0)=-\sum_{i \in M}y_{i}(\underline{\beta}^T \underline{x_{i}}+\beta_0)</math><br />
+
X = [ones(size(x1)) x1 x2 x1.*x2];
 +
b = regress(y,X); % Removes NaN data
 +
 
 +
[[File:mra1.jpg]]
 +
 
 +
% Plot the data and the model:
 +
 
 +
scatter3(x1,x2,y,'filled','r')
 +
hold on
 +
x1fit = min(x1):100:max(x1);
 +
x2fit = min(x2):10:max(x2);
 +
[X1FIT,X2FIT] = meshgrid(x1fit,x2fit);
 +
YFIT = b(1) + b(2)*X1FIT + b(3)*X2FIT + b(4)*X1FIT.*X2FIT;
 +
mesh(X1FIT,X2FIT,YFIT);
 +
xlabel('Weight');
 +
ylabel('Horsepower');
 +
zlabel('MPG');
 +
view(50,10);
 +
 
 +
=== Matlab Code for Multiple Logistic Regression ===
 +
% Calculation of gradient and objective for Logistic
 +
% Multi-Class Classifcation.
 +
%
 +
% function [obj,grad] = mcclogistic(v,Y,V,lambda,l,varargin)
 +
% v - vector of parameters [n*p*l,1]
 +
% Y - rating matrix (labels) [n,m]
 +
% V - the feature matrix [m,p]
 +
% lambda - regularization parameter [scalar]
 +
% l - # of labels (1..l)
 +
% obj - value of objective at v [scalar]
 +
% grad - gradient at v [n*p*l,1]
 +
%
 +
% Written by Jason Rennie, April 2005
 +
% Last modified: Tue Jul 25 15:08:38 2006
 +
function [obj,grad] = mcclogistic(v,Y,V,lambda,l,varargin)
 +
  fn = mfilename;
 +
  if nargin < 5
 +
    error('insufficient parameters')
 +
  end
 +
  % Parameters that can be set via varargin
 +
  verbose = 1;
 +
  % Process varargin
 +
  paramgt;
 +
 
 +
  t0 = clock;
 +
  [n,m] = size(Y);
 +
  p = length(v)./n./l;
 +
  if p ~= floor(p) | p < 1
 +
    error('dimensions of v and Y don''t match l');
 +
  end
 +
  U = reshape(v,n,p,l);
 +
  Z = zeros(n,m,l);
 +
  for i=1:l
 +
    Z(:,:,i) = U(:,:,i)*V';
 +
  end
 +
  obj = lambda.*sum(sum(sum(U.^2)))./2;
 +
  dU = zeros(n,p,l);
 +
  YY = full(Y==0) + Y;
 +
  YI = sub2ind(size(Z),(1:n)'*ones(1,m),ones(n,1)*(1:m),YY);
 +
  ZY = Z(YI);
 +
  for i=1:l
 +
    obj = obj + sum(sum(h(ZY-Z(:,:,i)).*(Y~=i).*(Y>0)));
 +
  end
 +
  ZHP = zeros(n,m);
 +
  for i=1:l
 +
    ZHP = ZHP + hprime(ZY-Z(:,:,i)).*(Y~=i).*(Y>0);
 +
  end
 +
  for i=1:l
 +
    dU(:,:,i) = ((Y==i).*ZHP - (Y~=i).*(Y>0).*hprime(ZY-Z(:,:,i)))*V + lambda.*U(:,:,i);
 +
  end
 +
  grad = dU(:);
 +
  if verbose
 +
    fprintf(1,'lambda=%.2e obj=%.4e grad''*grad=%.4e time=%.1f\n',lambda,obj,grad'*grad,etime(clock,t0));
 +
  end
 +
 
 +
function [ret] = h(z)
 +
  ret = log(1+exp(-z));
 +
 
 +
function [ret] = hprime(z)
 +
  ret = -(exp(-z)./(1+exp(-z)));
 +
 
 +
% ChangeLog
 +
% 7/25/06 - Added varargin, verbose
 +
% 3/23/05 - made calcultions take better advantage of sparseness
 +
% 3/18/05 - fixed bug in objective (wasn't squaring fro norms)
 +
% 3/1/05 - added objective calculation
 +
% 2/23/05 - fixed bug in hprime()
 +
 
 +
The code is from [http://people.csail.mit.edu/jrennie/matlab/mcclogistic.m here].
 +
Click [http://people.csail.mit.edu/jrennie/matlab/ here] for more information.
 +
 
 +
===Neural Network Concept[http://en.wikipedia.org/wiki/Neural_network]===
 +
The concept of constructing an artificial neural network came from scientists who were interested in simulating the human neural network in their computers. They were trying to create computer programs that could learn like people. A neural network is a method in artificial intelligence, and it was thought to be a simplified model of neural processing in the brain. Later studies showed that the human neural network is much more complicated, and the structure described here is not a good model for the biological architecture of the brain. Although neural network was developed in an attempt to synthesize the human brain, in actuality it has nothing to do with the human neural system.
 +
 
 +
=== Perceptron ===
 +
 
 +
==== Content ====
 +
 
 +
[http://en.wikipedia.org/wiki/Perceptron Perceptron] was invented in 1957 by [http://en.wikipedia.org/wiki/Frank_Rosenblatt Frank Rosenblatt]. It is the basic building block of Feed-Forward neural networks. The perceptron quickly became very popular after it was introduced, because it was shown to be able to solve many classes of useful problems. However, in 1969, [http://en.wikipedia.org/wiki/Marvin_Minsky Marvin Minsky] and [http://en.wikipedia.org/wiki/Seymour_Papert Seymour Papert] published their book [http://en.wikipedia.org/wiki/Perceptrons_%28book%29 ''Perceptrons'' (1969)] in which the authors strongly criticized the perceptron regarding its inability of solving simple [http://en.wikipedia.org/wiki/XOR exclusive-or (XOR)] problems, which are not linearly separable. Indeed, the simple perceptron and the single hidden-layer perceptron neural network [http://homepages.gold.ac.uk/nikolaev/311perc.htm] are not able to solve any problem that is not linearly-separable. However, it was known to the authors of this book that the multi-layer perceptron neural network can in fact solve any type of problem, including ones that are not linearly separable such as exclusive-or problems, although no efficient learning algorithm was available at that time for this type of neural network. Because of the book ''Perceptrons'', interest regarding perceptrons and neural networks in general greatly declined to a much lower point as compared to before this book was published and things stayed that way until 1986 when the [http://en.wikipedia.org/wiki/Back-propagation back-propagation] learning algorithm (which is discussed in detail below) for neural networks was popularized. <br /><br />
 +
 
 +
We know that the least-squares obtained by regression of -1/1 response variable <math>\displaystyle Y</math> on observation <math>\displaystyle x</math> leads to the same coefficients as LDA (recall that LDA minimizes the distance between discriminant function (decision boundary) and the data points). Least squares returns the sign of the linear combination of features as the class labels (Figure 2). This concept was called the Perceptron in Engineering literature during the 1950's. <br /><br />
 +
 
 +
[[File:Perceptron.jpg|371px|thumb|right| Fig.2 Diagram of a linear perceptron ]]
 +
 
 +
There is a cost function <math>\,\displaystyle D</math> that the Perceptron tries to minimize:<br />
 +
 
 +
<math>D(\underline{\beta},\beta_0)=-\sum_{i \in M}y_{i}(\underline{\beta}^T \underline{x_{i}}+\beta_0)</math><br />
  
 
where <math>\,\displaystyle M</math> is the set of misclassified points. <br><br />
 
where <math>\,\displaystyle M</math> is the set of misclassified points. <br><br />
Line 1,982: Line 2,667:
  
 
However, this quantity is not always positive. Consider <math>\,y_{i}(\underline{\beta}^T \underline{x_{i}}+\beta_0)</math>. If <math>\underline{x_{i}}</math> is classified ''correctly'' then this product is positive, since both (<math>\underline{\beta}^T\underline{x_{i}}+\beta_0)</math> and <math>\displaystyle y_{i}</math> are positive or both are negative. However, if <math>\underline{x_{i}}</math> is classified ''incorrectly'', then one of <math>(\underline{\beta}^T\underline{x_{i}}+\beta_0)</math> and <math>\displaystyle y_{i}</math> is positive and the other one is negative; hence, the product <math>y_{i}(\underline{\beta}^T \underline{x_{i}}+\beta_0)</math> will be negative for a misclassified point. The "-" sign in <math>D(\underline{\beta},\beta_0)</math> makes this cost function always positive (since only misclassified points are passed to D). <br /><br />
 
However, this quantity is not always positive. Consider <math>\,y_{i}(\underline{\beta}^T \underline{x_{i}}+\beta_0)</math>. If <math>\underline{x_{i}}</math> is classified ''correctly'' then this product is positive, since both (<math>\underline{\beta}^T\underline{x_{i}}+\beta_0)</math> and <math>\displaystyle y_{i}</math> are positive or both are negative. However, if <math>\underline{x_{i}}</math> is classified ''incorrectly'', then one of <math>(\underline{\beta}^T\underline{x_{i}}+\beta_0)</math> and <math>\displaystyle y_{i}</math> is positive and the other one is negative; hence, the product <math>y_{i}(\underline{\beta}^T \underline{x_{i}}+\beta_0)</math> will be negative for a misclassified point. The "-" sign in <math>D(\underline{\beta},\beta_0)</math> makes this cost function always positive (since only misclassified points are passed to D). <br /><br />
 +
 +
=== Perceptron in Action ===
 +
Here is a Java applet [http://lcn.epfl.ch/tutorial/english/perceptron/html/index.html] which may help with the procedure of Perceptron perception. This applet has been developed in the Laboratory of Computational Neuroscience, University of EPFL, Lausanne, Switzerland.
 +
 +
This second applet [http://www.eee.metu.edu.tr/~alatan/Courses/Demo/AppletPerceptron.html] is developed in the Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara, Turkey.
 +
 +
This third Java applet [http://neuron.eng.wayne.edu/java/Perceptron/New38.html] has been provided by the Computation and Neural Networks Laboratory, College of Engineering, Wayne State University, Detroit, Michigan.
 +
 +
This fourth applet [http://husky.if.uidaho.edu/nn/jdemos/05/Fred%20Corbett/www.etimage.com/java/appletNN/NeuronTyper/MultiLayerPerceptron/perceptron.html] is provided on the official website of the University of Idaho at Idaho Falls.
 +
 +
=== Further Reading for Perceptron ===
 +
 +
1. Neural Network Classifiers Estimate Bayesian a posteriori Probabilities
 +
[http://www.mitpressjournals.org/doi/abs/10.1162/neco.1991.3.4.461]
 +
 +
2. A perceptron network for functional identification and control of nonlinear systems
 +
[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=286893]
 +
 +
3. Neural network classifiers estimate Bayesian a posteriori probabilities
 +
[http://www.mitpressjournals.org/doi/abs/10.1162/neco.1991.3.4.461]
  
 
==Perceptron Learning Algorithm and Feed Forward Neural Networks - October 21, 2010 ==
 
==Perceptron Learning Algorithm and Feed Forward Neural Networks - October 21, 2010 ==
 
===Lecture Summary===
 
===Lecture Summary===
In this lecture, we finalize our discussion of the Perceptron by reviewing its learning algorithm, which is based on gradient descent. We then begin the next topic, Neural Networks (NN), and we focus on a NN that is useful for classification: the Feed Forward Neural Network (FFNN). The mathematical model for the FFNN is shown, and we review one of its most popular learning algorithms: Back-Propagation.  
+
In this lecture, we finalize our discussion of the Perceptron by reviewing its learning algorithm, which is based on [http://en.wikipedia.org/wiki/Gradient_descent gradient descent]. We then begin the next topic, Neural Networks (NN), and we focus on a NN that is useful for classification: the Feed Forward Neural Network ([http://www.learnartificialneuralnetworks.com/robotcontrol.html#aproach1 FFNN]). The mathematical model for the FFNN is shown, and we review one of its most popular learning algorithms: Back-Propagation.  
  
To open the Neural Network discussion, we present a formulation of the universal function approximator. The mathematical model for Neural Networks is then built upon this formulation. We also discuss the trade-off between training error and testing error -- known as the generalization problem -- under the universal function approximator section.
+
To open the Neural Network discussion, we present a formulation of the [http://en.wikipedia.org/wiki/Universal_approximation_theorem universal function approximator]. The mathematical model for Neural Networks is then built upon this formulation. We also discuss the trade-off between training error and testing error -- known as the generalization problem -- under the universal function approximator section.
 +
 
 +
There is useful information in [http://page.mi.fu-berlin.de/rojas/neural/chapter/K4.pdf] by R. Rojas about Perceptron learning.
  
 
===Perceptron===
 
===Perceptron===
Line 2,005: Line 2,712:
 
                           -\displaystyle\sum_{i \in M}y_{i} \end{array} \right)</math>
 
                           -\displaystyle\sum_{i \in M}y_{i} \end{array} \right)</math>
 
 
However, the perceptron learning algorithm does not use the sum of the contributions from each observation to calculate the gradient for each step.  Instead, each step uses the gradient contribution from only a single observation, and each successive step uses a different observation. This slight modification is called stochastic gradient descent. That is, instead of subtracting some factor of <math>\nabla D(\underline{\beta},\beta_0)</math> at each step, we subtract a factor of
+
However, the perceptron learning algorithm does not use the sum of the contributions from all of the observations to calculate the gradient in each step.  Instead, each step uses the gradient contribution from only a single observation, and each successive step uses a different observation. This slight modification is called stochastic gradient descent. That is, instead of subtracting some factor of <math>\nabla D(\underline{\beta},\beta_0)</math> at each step, we subtract a factor of
  
<math>\left( \begin{array}{c} y_{i}\underline{x}_i \\     
+
<math>\left( \begin{array}{c} y_{i}\underline{x}_i^T \\     
 
                           y_{i} \end{array} \right)</math>
 
                           y_{i} \end{array} \right)</math>
  
 
As a result, the pseudo code for the Perceptron Learning Algorithm is as follows:
 
As a result, the pseudo code for the Perceptron Learning Algorithm is as follows:
  
:1) Choose a random initial value for <math>(\underline{\beta},\beta_0)</math>.
+
:1) Choose a random initial value <math>\begin{pmatrix}
 +
\underline{\beta}^0\\
 +
\beta_0^0
 +
\end{pmatrix}</math> for <math>(\underline{\beta},\beta_0)</math>.
  
 
:2) <math>\begin{pmatrix}
 
:2) <math>\begin{pmatrix}
Line 2,035: Line 2,745:
 
+\rho  
 
+\rho  
 
\begin{pmatrix}
 
\begin{pmatrix}
y_i \underline{x_i}\\
+
y_i \underline{x_i^T}\\
 
y_i
 
y_i
 
\end{pmatrix}</math> for some <math>\,i \in M</math>.
 
\end{pmatrix}</math> for some <math>\,i \in M</math>.
Line 2,042: Line 2,752:
  
 
The learning rate <math>\,\rho</math> controls the step size of convergence toward <math>\min_{\underline{\beta},\beta_0} D(\underline{\beta},\beta_0)</math>.  A larger value for <math>\,\rho</math> causes the steps to be larger.  If <math>\,\rho</math> is set to be too large, however, then the minimum could be missed (over-stepped).
 
The learning rate <math>\,\rho</math> controls the step size of convergence toward <math>\min_{\underline{\beta},\beta_0} D(\underline{\beta},\beta_0)</math>.  A larger value for <math>\,\rho</math> causes the steps to be larger.  If <math>\,\rho</math> is set to be too large, however, then the minimum could be missed (over-stepped).
In practice, <math>\,\rho</math> can be adaptive and not fixed, it means that, in the first steps <math>\,\rho</math> could be larger than the last steps. At the beginning, larger <math>\,\rho</math> helps to find the approximate answer sooner. And smaller <math>\,\rho</math> in last steps help to tune the final answer more accurately.
+
In practice, <math>\,\rho</math> can be adaptive and not fixed, it means that, in the first steps <math>\,\rho</math> could be larger than the last steps, with <math>\,\rho</math> gradually declining in size as the steps progress towards convergence. At the beginning, larger <math>\,\rho</math> helps to find the approximate answer sooner. And smaller <math>\,\rho</math> towards the last steps help to tune the final answer more accurately. Many works have been done relating to adaptive learning rates. For interested readers, an example of these works is [http://www.math.upatras.gr/~dgs/papers/reports/tr98-02.pdf this paper] by ''Plagianakos et al.'' and [http://cnl.salk.edu/~schraudo/pubs/Schraudolph99c.pdf this paper] by ''Schraudolph''.
  
  
 
As mentioned earlier, the learning algorithm uses just one of the data points at each iteration; this is the common practice when dealing with online applications. In an online application, datapoints are accessed one-at-a-time because training data is not available in batch form. The learning algorithm does not require the derivative of the cost function with respect to the previously seen points; instead, we just have to take into consideration the effect of each new point.
 
As mentioned earlier, the learning algorithm uses just one of the data points at each iteration; this is the common practice when dealing with online applications. In an online application, datapoints are accessed one-at-a-time because training data is not available in batch form. The learning algorithm does not require the derivative of the cost function with respect to the previously seen points; instead, we just have to take into consideration the effect of each new point.
  
One way that the algorithm could terminate is if there are no more mis-classified points (i.e. if set <math>\,M</math> is empty). As long as there are points in <math>\,M</math>, the algorithm continues until some other termination criterion is reached. Termination criterion for an optimization algorithm is usually convergence, but for numerical methods this is not well-defined. In theory, convergence is realized when the gradient of the cost function is zero; in numerical methods an answer close to zero within some margin of error is taken instead.
+
One way that the algorithm could terminate is if there are no more mis-classified points (i.e. if set <math>\,M</math> is empty). Another way that the algorithm could terminate is continuing until some other termination criterion is reached even if there are still points in <math>\,M</math>. The termination criterion for an optimization algorithm is usually convergence, but for numerical methods this is not well-defined. In theory, convergence is realized when the gradient of the cost function is zero; in numerical methods an answer close to zero within some margin of error is taken instead.
  
 
Since the data is linearly-separable, the solution is theoretically guaranteed to converge in a finite number of iterations.  This number of iterations depends on the  
 
Since the data is linearly-separable, the solution is theoretically guaranteed to converge in a finite number of iterations.  This number of iterations depends on the  
Line 2,053: Line 2,763:
 
* learning rate <math>\,\rho</math>
 
* learning rate <math>\,\rho</math>
  
* initial value <math>(\underline{\beta},\beta_0)</math>
+
* initial value <math>\begin{pmatrix}
 +
\underline{\beta}^0\\
 +
\beta_0^0
 +
\end{pmatrix}</math>
  
 
* difficulty of the problem. The problem is more difficult if the gap between the classes of data is very small.
 
* difficulty of the problem. The problem is more difficult if the gap between the classes of data is very small.
  
Note that we consider the offset term <math>\beta_0</math> separately from the <math>\underline{\beta}</math> to distinguish this formulation from those in which the direction of the hyperplane (<math>\underline{\beta}</math>) has been considered.
+
Note that we consider the offset term <math>\,\beta_0</math> separately from <math>\underline{\beta}</math> to distinguish this formulation from those in which the direction of the hyperplane (<math>\underline{\beta}</math>) has been considered.
  
A major concern about gradient descent is that it may get trapped in local optimal solutions.
+
A major concern about gradient descent is that it may get trapped in local optimal solutions. Many works such as [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=00298667 this paper] by ''Cetin et al.'' and [http://indian.cp.eng.chula.ac.th/cpdb/pdf/research/fullpaper/847.pdf this paper] by ''Atakulreka et al.'' have been done to tackle this issue.
  
 
====Some notes on the Perceptron Learning Algorithm====
 
====Some notes on the Perceptron Learning Algorithm====
  
* If there is access to the training data points in a batch form, we should better take advantage of a closed optimization technique like least-squares or maximum-likelihood estimation for linear classifiers. (These closed solutions has been around many years before invention of the Perceptron).
+
* If there is access to the training data points in a batch form, it is better to take advantage of a closed optimization technique like least-squares or maximum-likelihood estimation for linear classifiers. (These closed form solutions have been around many years before the invention of Perceptron).
  
* Just like the linear classifier, a Perceptron can discriminate between only two classes at a time, and one can generalize its performance for multi-class problems by using one of the <math>k-1</math>, <math>k</math>, or <math>k(k-1)/2</math>-hyperplane methods.
+
* Just like a linear classifier, a Perceptron can discriminate between only two classes at a time, and one can generalize its performance for multi-class problems by using one of the <math>k-1</math>, <math>k</math>, or <math>k(k-1)/2</math>-hyperplane methods.
  
 
* If the two classes are linearly separable, the algorithm will converge in a finite number of iterations to a hyperplane, which makes the error of training data zero. The convergence is guaranteed if the learning rate is set adequately.
 
* If the two classes are linearly separable, the algorithm will converge in a finite number of iterations to a hyperplane, which makes the error of training data zero. The convergence is guaranteed if the learning rate is set adequately.
  
* If the two classes are not linearly separable, the algorithm will never converge. So, one may think of a termination criterion in these cases. (e.g. a maximum number of iterations in which convergence is expected, or the rate of changes in both a cost function and its derivative).
+
* If the two classes are not linearly separable, the algorithm will never converge. So, one may think of a termination criterion in these cases (e.g. a maximum number of iterations in which convergence is expected, or the rate of changes in both a cost function and its derivative).
  
* In the case of linearly separable classes, the final solution and number of iterations will be dependent on the initial conditions, learning rate, and the gap between the two classes. In general, a smaller gap between classes requires a greater number of iterations for the algorithm to converge.
+
* In the case of linearly separable classes, the final solution and the number of iterations will be dependent on the initial values (which are arbitrarily chosen), the learning rate (for example, fixed or adaptive), and the gap between the two classes. In general, a smaller gap between classes requires a greater number of iterations for the algorithm to converge.
  
* Learning rate --or updating step-- has a direct impact on both number of iterations and the accuracy of the solution for the optimization problem. Smaller quantities for this factor make convergence slower, even though we will end up with a more accurate solution. In the opposite way, larger values for learning rate make the process faster, even though we may lose some precision. So, one may make a balance for this trade-off in order to get fast enough to an accurate enough solution. (exploration vs. exploitation)
+
* Learning rate --or updating step-- has a direct impact on both the number of iterations and the accuracy of the solution for the optimization problem. Smaller quantities of this factor make convergence slower, even though we will end up with a more accurate solution. In the opposite way, larger values of the learning rate make the process faster, even though we may lose some precision. So, one may make a balance for this trade-off in order to get to an accurate enough solution fast enough (exploration vs. exploitation). In addition, an adaptive learning rate that starts off with a large value and then gradually decreases to a small value over the steps toward convergence can be used in place of a fixed learning rate.
  
In the upcoming lectures, we introduce the Support Vector Machines (SVM), which use a method similar in iterational optimization scheme to what the Perceptron suggests, but have a different definition for the cost function.
+
In the upcoming lectures, we introduce the Support Vector Machines (SVM), which use a method similar to the iteration optimization scheme to what the Perceptron suggests, but have a different definition for the cost function.
 +
 
 +
===An example of the determination on learning rate===
 +
( Based on J. Amini  Optimum Learning Rate in Back-Propagation Neural Network for Classification
 +
of Satellite Images (IRS-1D) Scientia Iranica, Vol. 15, No. 6, pp. 558-567 )
 +
 
 +
Learning rate plays an important role in the application of Neural Network (NN). Choosing an optimum learning rate helps us to obtain the best regression model with the fastest possible speed. In the application of NN by different algorithms, the optimum learning rate tends to be determined differently. In the paper, Optimum Learning Rate in Back-Propagation Neural Network for Classification of Satellite Images (IRS-1D), the author applied one hidden layer and two hidden layers as networks to satellite images by Variable Learning Rate (VLR) algorithms and compared their optimum learning rates based on the various networks. In practice, the number of neurons should not be very small or very large. Since the network with too few neurons does not have enough degrees of freedom to train the data, but the network with too many neurons is more likely to lead to over fitting, the range of the number of neurons in the experiment is from 3 to 40. Finally, the optimum learning rate under various cases keeps 0.001-0.006. In practice, we could use a similar way to estimate the optimum learning rate to improve our models. For more details, please see the article mentioned above.
  
 
===Universal Function Approximator===
 
===Universal Function Approximator===
 +
In mathematics, the [http://en.academic.ru/dic.nsf/enwiki/10694320 Universal Approximation Theorem] states that the standard multilayer feed-forward neural network with a single hidden layer that contains a finite and sufficient number of hidden neurons and having an arbitrary activation function for each neuron is a universal approximator on a compact subset of <math>\mathbb{R}^n</math> under the assumption that the output units are always linear. George Cybenko first proved this theorem in 1989 for a sigmoid activation function, and thus the Universal Approximation Theorem is also called Cybenko's Theorem. For interested readers, a detailed proof of Cybenko's Theorem is given in [http://cs.haifa.ac.il/~hhazan01/Advance%20Seminar%20on%20Neuro-Computation/2010/nn1.pdf this presentation] by Yousef Shajrawi and Fadi Abboud. In 1991, Kurt Hornik showed that the potential of a particular neural network of being a universal approximator does not depend on the specific choice of the activation function used by the neurons, rather it depends on the multilayer feedforward architecture itself that is used by that neural network.
 +
 +
 
The universal function approximator is a mathematical formulation for a group of estimation techniques. The usual formulation for it is
 
The universal function approximator is a mathematical formulation for a group of estimation techniques. The usual formulation for it is
  
 
<math>\hat{Y}(x)=\sum\limits_{i=1}^{n}\alpha_i\sigma(\omega_i^Tx+b_i),</math>
 
<math>\hat{Y}(x)=\sum\limits_{i=1}^{n}\alpha_i\sigma(\omega_i^Tx+b_i),</math>
  
where <math>\hat{Y}(x)</math> is an estimation for a function like <math>\,Y(x)</math>. According to the universal approximation theorem we have
+
where <math>\hat{Y}(x)</math> is an estimation for a function <math>\,Y(x)</math>. According to the universal approximation theorem we have
  
 
<math>|\hat{Y}(x) - Y(x)|<\epsilon,</math>
 
<math>|\hat{Y}(x) - Y(x)|<\epsilon,</math>
  
which means that <math>\hat{Y}(x)</math> can get as close to <math>\,Y(x)</math>, as necessary.
+
which means that <math>\hat{Y}(x)</math> can get as close to <math>\,Y(x)</math> as necessary.
  
This formulation assumes that the output, <math>\,Y(x)</math>, is a linear combination of a set of functions like <math>\,\sigma(.)</math> where <math>\,\sigma(.)</math> is a nonlinear function of the inputs or <math>\,x_i</math>s.
+
This formulation assumes that the output, <math>\,Y(x)</math>, is a linear combination of a set of functions like <math>\,\sigma(.)</math> where <math>\,\sigma(.)</math> is a nonlinear function of the inputs or <math>\,x_i</math>'s.
  
 
====Generalization Factors====
 
====Generalization Factors====
 
Even though this formulation represents a universal function approximator, which means that it can be fitted to a set of data as closely as demanded, the closeness of fit must be carefully decided upon. In many cases, the purpose of the model is to target unseen data. However, the fit to this unseen data is impossible to determine before it arrives.
 
Even though this formulation represents a universal function approximator, which means that it can be fitted to a set of data as closely as demanded, the closeness of fit must be carefully decided upon. In many cases, the purpose of the model is to target unseen data. However, the fit to this unseen data is impossible to determine before it arrives.
  
To overcome this dilemma, a common practice is to divide the test data points into two sets: training data and validation data.  We use the training data to estimate the fixed parameters for the model, and then use the validation data to find values for the construction-dependent parameters. How these construction-dependent parameters vary depends on the model. In the case of a polynomial, the construction-dependent parameter would be its highest degree, and for a neural network, the construction-dependent parameter could be the number of hidden layers and the number of neurons in each layer.
+
To overcome this dilemma, a common practice is to divide the set of available data points into two sets: training data and validation (test) data.  We use the training data to estimate the fixed parameters for the model, and then use the validation data to find values for the construction-dependent parameters. How these construction-dependent parameters vary depends on the model. In the case of a polynomial, the construction-dependent parameter would be its highest degree, and for a neural network, the construction-dependent parameter could be the number of hidden layers and the number of neurons in each layer.
  
 
These matters on model generalization vs. complexity matters will be discussed with more detail in the lectures to follow.
 
These matters on model generalization vs. complexity matters will be discussed with more detail in the lectures to follow.
  
 
===Feed-Forward Neural Network===
 
===Feed-Forward Neural Network===
The Neural Network (NN) is one application of the universal function approximator. It can be thought of as a system of Perceptrons linked together as units of a network.  One particular NN useful for classification is the Feed-Forward Neural Network (FFNN), which consists of multiple "hidden layers" of Perceptron units. Our discussion here is based around the FFNN, which has a toplogy shown in Figure 1. The first hidden layer of units receive input from the original features. Between the hidden layers, connections from each unit are always directed to units in the next adjacent layer.  The output layer, which receives input only from the last hidden layer, each unit produces a target measurement for a distinct class (i.e. <math>\,K</math> classes require <math>\,K</math> units). In Figure 1, the units in a single layer are distributed vertically, and the inputs and outputs of the network are shown as the far left and right layers respectively.
+
Neural Network (NN) is one instance of the universal function approximator. It can be thought of as a system of Perceptrons linked together as units of a network.  One particular NN useful for classification is the Feed-Forward Neural Network ([http://www.learnartificialneuralnetworks.com/robotcontrol.html#aproach1 FFNN]), which consists of multiple "hidden layers" of Perceptron units (also known as neurons). Our discussion here is based around the FFNN, which has a topology shown in Figure 1. The neurons in the first hidden layer take their inputs, the original features (the <math>\,x_i</math>'s), and pass their inputs unchanged as their outputs to the first hidden layer. From the first layer (the input layer) to the last hidden layer, connections from each neuron are always directed to the neurons in the next adjacent layer.  In the output layer, which receives input only from the last hidden layer, each neuron produces a target measurement for a distinct class. <math>\,K</math> classes typically require <math>\,K</math> output neurons in the output layer. In the case where the target variable has two values, it suffices to have one output node in the output layer, although it is generally necessary for the single output node to have a sigmoid activation function so as to restrict the output of the neural network to be a value between 0 and 1. As shown in Figure 1, the neurons in a single layer are typically distributed vertically, and the inputs and outputs of the network are shown as the far left layer and the far right layer, respectively. Furthermore, as shown in Figure 1, it is often useful to add an extra hidden node to each hidden layer that represents the bias term (or the intercept term) of that hidden layer's hyperplane. Each bias node usually outputs a constant value of -1. The purpose of adding a bias node to each hidden layer is to ensure that the hyperplane of that hidden layer does not necessarily have to pass through the origin. In Figure 1, the bias node in the single hidden layer is the topmost hidden node in that layer. 
  
 
[[File:FFNN.png|300px|thumb|right|Fig.1 A common architecture for the FFNN]]
 
[[File:FFNN.png|300px|thumb|right|Fig.1 A common architecture for the FFNN]]
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====Mathematical Model of the FFNN with One Hidden Layer====
 
====Mathematical Model of the FFNN with One Hidden Layer====
  
{{Cleanup|date=October 29 2010|reason= It is a common practice to consider bias terms for the inputs of the all hidden layers, as considered <math>x_0</math>, for the inputs of the first layer. In the second layer (the first hidden layer) the first neuron from the top is a bias term.}}
+
The FFNN with one hidden layer for a <math>\,K</math>-class problem is defined as follows:<br /> Let <math>\,d</math> be the number of input features, <math>\,p</math> be the number of neurons in the hidden layer, and <math>\,K</math> be the number of classes which is also typically the number of neurons in the output layer in the case where <math>\,K</math> is greater than 2.
{{Cleanup|date=November 2 2010|reason= I can't understand your question, could you please explain it in more details? Are you referring to Figure 1?}}
 
  
The FFNN with one hidden layer for a <math>\,K</math>-class problem is defined as follows:<br /> Let <math>\,d</math> be the number of input features, <math>\,p</math> be the number of units in the hidden layer, and <math>\,K</math> be the number of classes (i.e. the number of units in the output layer).
+
Each neuron calculates its derived feature (i.e. output) using a linear combination of its inputs. Suppose <math>\,\underline{x}</math> is the <math>\,d</math>-dimensional vector of input features. Then, each hidden neuron uses a <math>\,d</math>-dimensional vector of weights to combine these input features. For the <math>\,i</math>th hidden neuron, let <math>\underline{u}_i</math> be this neuron's vector of weights. The linear combination calculated by the <math>\,i</math>th hidden neuron is then given by
  
Each neural unit calculates its derived feature (i.e. output) using a linear combination of its inputs. Suppose <math>\,\underline{x}</math> is the <math>\,d</math>-dimensional vector of input features. Then, each neural unit uses a <math>\,d</math>-dimensional vector of weights to combine these input features: for the <math>\,i</math>th neural unit, let <math>\underline{u}_i</math> be this vector of weights. The linear combination calculated by the <math>\,i</math>th unit is then given by
+
<math>a_i = \sum_{j=1}^{d}\underline{u}_{ij}^T\underline{x}_j, i={1,...,p}</math>
  
<math>a_i = \underline{u}_i^T\underline{x}</math>
 
  
{{Cleanup|date=November 2 2010|reason= It is more clear to use the following notation, instead of the one used: <math>a_i = \sum_{j=1}^{d}\underline{u}_{ij}^T\underline{x}_j, j={1,...,p}</math>.}}
+
However, we want the derived feature of each hidden neuron and each output neuron to lie between 0 and 1, so we apply an ''activation function'' <math>\,\sigma(a)</math> to each hidden or output neuron. The derived feature of each hidden or output neuron <math>\,i</math> is then given by
However, we want the derived feature to lie between 0 and 1, so we apply an ''activating function'' <math>\,\sigma(a)</math>. The derived feature for the <math>\,i</math>th unit is then given by
 
  
<math>\,z_i = \sigma(a_i)</math> where <math>\,\sigma</math> is typically the logistic function
+
<math>\,z_i = \sigma(a_i)</math> where <math>\,\sigma</math> is typically the logistic sigmoid function <math>\sigma(a) = \cfrac{1}{1+e^{-a}}</math>.
  
<math>\sigma(a) = \cfrac{1}{1+e^{-a}}</math>
 
  
 
Now, we place each of the derived features <math>\,z_i</math> from the hidden layer into a <math>\,p</math>-dimensional vector:
 
Now, we place each of the derived features <math>\,z_i</math> from the hidden layer into a <math>\,p</math>-dimensional vector:
Line 2,124: Line 2,842:
 
<math>\underline{z} = \left[ \begin{array}{c} z_1 \\ z_2 \\ \vdots \\ z_p \end{array}\right]</math>
 
<math>\underline{z} = \left[ \begin{array}{c} z_1 \\ z_2 \\ \vdots \\ z_p \end{array}\right]</math>
  
Like in the hidden layer, each unit in the output layer calculates its derived feature using a linear combination of its inputs. Each neural unit uses a <math>\,p</math>-dimensional vector of weights to combine the input features derived from the hidden layer. Let <math>\,\underline{w}_k</math> be this vector of weights used in the <math>\,k</math>th unit. The linear combination calculated by the <math>\,k</math>th unit is then given by
+
As in the hidden layer, each neuron in the output layer calculates its derived feature using a linear combination of its inputs which are the elements of <math>\underline{z}</math>. Each output neuron uses a <math>\,p</math>-dimensional vector of weights to combine its inputs derived from the hidden layer. Let <math>\,\underline{w}_k</math> be the vector of weights used by the <math>\,k</math>th output neuron. The linear combination calculated by the <math>\,k</math>th output neuron is then given by
 +
<math>\hat{y}_k = \sum_{j=1}^{p}\underline{w}_{kj}^T\underline{z}_j, k={1,...,K}</math>.
  
<math>\hat{y}_k = \underline{w}_k^T\underline{z}</math>
+
<math>\,\hat y_k</math> is thus the target measurement for the <math>\,k</math>th class. It is not necessary to use an activation function <math>\,\sigma</math> for each of the hidden and output neurons in the case of regression since the outputs are continuous, though it is necessary to use an activation function <math>\,\sigma</math> for each of the hidden and output neurons in the case of classification so as to ensure that the outputs are in the <math> [0, 1]</math> interval.
{{Cleanup|date=November 2 2010|reason= Similarly, for the output layer, it is better to use the following: <math>\hat{y}_k = \sum_{j=1}^{p}\underline{w}_{kj}^T\underline{z}_j, j={1,...,K}</math>.}}
+
{{Cleanup|date=December 2010|reason=The sentence above is misleading, I think. The outputs will not be discrete, we need the activation function in order to keep them in the {0,1} interval. Please correct me if I'm wrong.}}
<math>\,y_k</math> is thus the target measurement for the <math>\,k</math>th class. Note that an activation function <math>\,\sigma</math> is not used here.
 
  
Notice that in each of the units, two operations take place:
+
Notice that in each neuron, two operations take place one after the other:
  
 
* a linear combination of the neuron's inputs is calculated using corresponding weights
 
* a linear combination of the neuron's inputs is calculated using corresponding weights
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These two calculations are shown in Figure 2.  
 
These two calculations are shown in Figure 2.  
  
The nonlinear function <math>\,\sigma(.)</math> is called the activation function. Activation functions, like the logistic function shown earlier, are usually continuous and have a limited range. Another common activation function used in neural networks is <math>\,tanh(x)</math> (Figure 3).  
+
The nonlinear function <math>\,\sigma(.)</math> is called the activation function. Activation functions, like the logistic function shown earlier, are usually continuous and usually have a finite range with regard to their outputs. Another common activation function used in neural networks is the hyperbolic tangent function <math>\,\sigma(a) = tanh(a)</math> (Figure 3). The logistic sigmoid activation function <math>\sigma(a) = \cfrac{1}{1+e^{-a}}</math> and the hyperbolic tangent activation function are very similar to each other. One major difference between them is that, as shown in their illustrations, the output range of the the logistic sigmoid activation function is <math>\,[0,1]</math> while that of the hyperbolic tangent activation function is <math>\,[-1,1]</math>. Typically, in a neural network used for classification tasks, the logistic sigmoid activation function is used rather than any other type of activation function. The reason is that, as explained in detail in [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=809075&tag=1 this paper] by ''Helmbold et al.'', the logistic sigmoid activation function results in the least [http://www.soe.ucsc.edu/classes/cmps290c/Spring09/lect/7/pap_slides.pdf matching loss] as compared to other types of activation functions.
 
 
{{Cleanup|date=October 28 2010|reason= Why there is just  <math>\,tanh(x)</math> activation function mentioned here? Does this function have any benefit with comparison to the other activation functions for this case?}}
 
{{Cleanup|date=October 29 2010|reason= It has been mentioned, along with the logistic function which might be a more common one (to be referred to the text). It may have not any advantages over the logistic regression for the classification tasks.}}
 
 
 
{{Cleanup|date=November 3 2010|reason= I can say that with respect to its matching loss function, it gives less noise please check  [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=809075 this paper] }}
 
  
 
[[File:neuron2.png|300px|thumb|right|Fig.2 A general construction for a single neuron]]
 
[[File:neuron2.png|300px|thumb|right|Fig.2 A general construction for a single neuron]]
 
[[File:actfcn.png|300px|thumb|right|Fig.3 <math>tanh</math> as activation function]]
 
[[File:actfcn.png|300px|thumb|right|Fig.3 <math>tanh</math> as activation function]]
  
The NN can be applied as a regression method or as a classifier, and the output layer differs depending on the application. The major difference between regression and classification is in the output space of the model, which is continuous in the case of regression, and discrete in the case of classification. For a regression task, no consideration is needed beyond what has already been mentioned earlier, since the outputs of the network would already be continuous. However, to use the neural network as a classifier, a threshold stage is necessary.
+
The NN can be applied as a regression method or as a classifier, and the output layer differs depending on the application. The major difference between regression and classification is in the output space of the model, which is continuous in the case of regression and discrete in the case of classification. For a regression task, no consideration is needed beyond what has already been mentioned earlier, since the outputs of the network would already be continuous. However, to use the neural network as a classifier, as mentioned above, it is necessary to have a threshold stage for each of the hidden and output neurons using an activation function.
  
 
====Mathematical Model of the FFNN with Multiple Hidden Layers====
 
====Mathematical Model of the FFNN with Multiple Hidden Layers====
In the FFNN model with a single hidden layer, the derived features were represented as elements of the vector <math>\underline{z}</math>, and the original features were represented as elements of the vector <math>\underline{x}</math>.  In the FFNN model with more than one hidden layer, <math>\underline{z}</math> is processed by the second hidden layer in the same way that <math>\underline{x}</math> was processed by the first hidden layer. Perceptrons in the second layer each use their own combination of weights to calculate a new set of derived features.  These new derived features are processed by the third hidden layer in a similar way, and the cycle repeats for each additional hidden layer. This progression of processing is depicted in Figure 4.
+
In the FFNN model with a single hidden layer, the derived features were represented as elements of the vector <math>\underline{z}</math>, and the original features were represented as elements of the vector <math>\underline{x}</math>.  In the FFNN model with more than one hidden layer, <math>\underline{z}</math> is processed by the second hidden layer in the same way that <math>\underline{x}</math> was processed by the first hidden layer. Perceptrons in the second hidden layer each use their own combination of weights to calculate a new set of derived features.  These new derived features are processed by the third hidden layer in a similar way, and the cycle repeats for each additional hidden layer. This progression of processing is depicted in Figure 4.
  
 
====Back-Propagation Learning Algorithm====
 
====Back-Propagation Learning Algorithm====
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[[File:bpl.png|300px|thumb|right|Fig.4 Labels for weights and derived features in the FFNN.]]
 
[[File:bpl.png|300px|thumb|right|Fig.4 Labels for weights and derived features in the FFNN.]]
  
Every linear-combination calculation in the FFNN involves weights that need to be set, and these weights are set using training data and an algorithm called Back-Propagation. This algorithm is similar to the gradient-descent algorithm introduced in the discussion of the Perceptron. The primary difference is that the gradient used in Back-Propagation is calculated in a more complicated way.
+
Every linear-combination calculation in the FFNN involves weights that need to be updated after they are initialized to be small random values, and these weights are updated using an algorithm called Back-Propagation when each data point in the training data-set is fed into the neural network. This algorithm is similar to the gradient-descent algorithm introduced in the discussion of the Perceptron. The primary difference is that the gradient used in Back-Propagation is calculated in a more complicated way.
  
First of all, we want to minimize the error between the estimated and true target measurements for the training data. That is, if <math>\,U</math> is the set of all weights in the FFNN, then we want to determine
+
First of all, we want to minimize the error between the estimated target measurement and the true target measurement of each input from the training data-set. That is, if <math>\,U</math> is the set of all weights in the FFNN, then we want to determine
  
<math>\arg\min_U \left|y - \hat{y}\right|^2</math>
+
<math>\arg\min_U \left|y - \hat{y}\right|^2</math> for each data point in the training data-set.
  
 
Now, suppose the hidden layers of the FFNN are labelled as in Figure 4. Then, we want to determine the derivative of <math>\left|y - \hat{y}\right|^2</math> with respect to each weight in the hidden layers of the FFNN. For weights <math>\,u_{jl}</math> this means we will need to find
 
Now, suppose the hidden layers of the FFNN are labelled as in Figure 4. Then, we want to determine the derivative of <math>\left|y - \hat{y}\right|^2</math> with respect to each weight in the hidden layers of the FFNN. For weights <math>\,u_{jl}</math> this means we will need to find
Line 2,193: Line 2,906:
  
 
<math>\delta_k = \cfrac{\partial \left|y - \hat{y}\right|^2}{\partial a_k}</math>
 
<math>\delta_k = \cfrac{\partial \left|y - \hat{y}\right|^2}{\partial a_k}</math>
 +
 +
{{Cleanup|date=November 2 2010|reason= It is true that an activation function is not applied to each output neuron if the neural network is used for regression. But, if the neural network is used for classification, I think it is necessary to apply an activation function to each output neuron. I believe that this is correct. In Chapter 5.2 of Pattern Recognition and Machine Learning by Christopher Bishop  , it is written that for 2 class classification sigmoid output functions are used and for multi-class the [http://en.wikipedia.org/wiki/Softmax_activation_function Softmax]function is used.}}
 +
 +
{{Cleanup|date=November 2 2010|reason= To avoid an extra stage of thresholding, it is suggested for classification task to use the outputs of the hidden units themselves, instead of a linear combination of them. This does not make any sense to me. It is likely that there are more hidden units than output units , so how would you use these to do the classification? }}
  
 
However, <math>\,a_k = \hat{y}</math> because an activation function is not applied in the output layer. So, our calculation becomes
 
However, <math>\,a_k = \hat{y}</math> because an activation function is not applied in the output layer. So, our calculation becomes
Line 2,199: Line 2,916:
 
= -2(y - \hat{y})</math>
 
= -2(y - \hat{y})</math>
  
Now that we have <math>\,\delta_k</math> and a recursive definition for <math>\,\delta_j</math>, it is clear that our weights can be deduced by starting from the output layer and working through the hidden layers through toward the input layer.
+
Now that we have <math>\,\delta_k</math> and a recursive definition for <math>\,\delta_j</math>, it is clear that our weights can be deduced by starting from the output layer and working leftwards through the hidden layers one layer at a time towards the input layer.
  
Based on the above derivation, our algorithm for determining weights in the FFNN is as follows
+
Based on the above derivation, our algorithm for determining weights in the FFNN is as follows:
  
:1) Choose a random initial weights.
+
First, choose small random values to initialize the network weights. Then, during each epoch (a single pass through all of the training data points), all of the training data points are sequentially fed into the FFNN one at a time. The network weights are updated using the back-propagation algorithm when each training data point <math>\underline{x}</math>is fed into the FFNN. This update procedure is done using the following steps: 
  
:2) Apply each datapoint <math>\underline{x}</math> in the dataset to the FFNN as the input layer, and calculate the values for all units.
 
  
:3) Compute <math>\,\delta_k = -2(y_k - \hat{y}_k)</math>.
+
* Apply <math>\underline{x}</math> to the FFNN's input layer, and calculate the outputs of all input neurons.
  
:4) Back-propagate layer-by-layer by computing <math>\delta_j = \sigma'(a_j)\sum_{i=1}^p \delta_i \cdot u_{ij}</math> for all units.
 
  
:5) Compute <math>\cfrac{\partial \left|y - \hat{y}\right|^2}{\partial u_{jl}} = \delta_{j}z_l</math> for all weights <math>\,u_{jl}</math>.
+
* Propagate the outputs of each hidden layer forward, one hidden layer at a time, and calculate the outputs of all hidden neurons.
  
:6) Update <math>u_{jl}^{\mathrm{new}} \leftarrow u_{jl}^{\mathrm{old}}
 
- \rho \cdot \cfrac{\partial \left|y - \hat{y}\right|^2}{\partial u_{jl}} </math> for all weights <math>\,u_{jl}</math> where <math>\,\rho</math> is the learning rate.
 
  
:7) If the termination criterion has not been met, go back to step 2 and apply another datapoint (ie. begin a new "epoch").
+
* Once <math>\underline{x}</math> reaches the output layer, calculate the output(s) of all output neuron(s) given the outputs of the previous hidden layer.
 +
 
 +
 
 +
* At the output layer, compute <math>\,\delta_k = -2(y_k - \hat{y}_k)</math> for each output neuron(s), then compute <math>\cfrac{\partial \left|y - \hat{y}\right|^2}{\partial u_{jl}} = \delta_{j}z_l</math> for all weights <math>\,u_{jl}</math>, and then update <math>u_{jl}^{\mathrm{new}} \leftarrow u_{jl}^{\mathrm{old}} - \rho \cdot \cfrac{\partial \left|y - \hat{y}\right|^2}{\partial u_{jl}} </math> for all weights <math>\,u_{jl}</math>. Here, <math>\,\rho</math> is the learning rate.
 +
 
 +
 
 +
* Starting from the last hidden layer, back-propagate layer-by-layer to the first hidden layer. At each hidden layer, compute <math>\delta_j = \sigma'(a_j)\sum_{i=1}^p \delta_i \cdot u_{ij}</math> for all hidden neurons in that layer, then compute <math>\cfrac{\partial \left|y - \hat{y}\right|^2}{\partial u_{jl}} = \delta_{j}z_l</math> for all weights <math>\,u_{jl}</math>, and then update <math>u_{jl}^{\mathrm{new}} \leftarrow u_{jl}^{\mathrm{old}} - \rho \cdot \cfrac{\partial \left|y - \hat{y}\right|^2}{\partial u_{jl}} </math> for all weights <math>\,u_{jl}</math>. Here, <math>\,\rho</math> is the learning rate.
 +
 
 +
 
 +
Usually, a fairly large number of epochs is necessary for training the FFNN so that the network weights would be close to being their optimal values. The learning rate <math> \,\rho </math> should be chosen carefully. Usually, <math> \,\rho </math> should satisfy <math> \,\rho \rightarrow 0 </math> as the iteration times <math> i \rightarrow \infty </math>. [http://www.youtube.com/watch?v=fJ7eH0Y7xEM This] is an interesting video depicting the training procedure of the weights of an FFNN using the back-propagation algorithm.
 +
 
 +
A Matlab implementation of the pseudocode above is given as an example in the Weight Decay subsection under the [[Regularization for Neural Network - November 4, 2010|Regularization]] title.
  
 
====Alternative Description of the Back-Propagation Algorithm====
 
====Alternative Description of the Back-Propagation Algorithm====
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Where <math>\,W_i</math> is a matrix of the connection's weights, between two layers of <math>\,i</math> and <math>\,i+1</math>, and has <math>\,n_i</math> columns and <math>\,n_i+1</math> rows, where <math>\,n_i</math> is the number of neurons of the <math>\,i^{th}</math> layer.
 
Where <math>\,W_i</math> is a matrix of the connection's weights, between two layers of <math>\,i</math> and <math>\,i+1</math>, and has <math>\,n_i</math> columns and <math>\,n_i+1</math> rows, where <math>\,n_i</math> is the number of neurons of the <math>\,i^{th}</math> layer.
  
Considering this matrix equations, one can imagine a closed form for the derivative of the error in respect to the weights of the network. For a neural network with two hidden layers, the equations are as follows.
+
Considering this matrix equations, one can imagine a closed form for the derivative of the error with respect to the weights of the network. For a neural network with two hidden layers, the equations are as follows:
  
 
<math>\begin{align}
 
<math>\begin{align}
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where <math>\,\sigma'(.)</math> is the derivative of the activation function <math>\,\sigma(.)</math>.
 
where <math>\,\sigma'(.)</math> is the derivative of the activation function <math>\,\sigma(.)</math>.
  
Using this closed form derivative, it is possible to code the procedure for any number of layers and neurons. Here is a Matlab code for backpropagation algorithm. (<math>\,tanh</math> is utilized as the activation function.)
+
Using this closed form derivative, it is possible to code the procedure for any number of layers and neurons. Given below is the Matlab code for the back-propagation algorithm (<math>\,tanh</math> is utilized as the activation function).  
  
 
{{Cleanup|date=November 2 2010|reason= This MATLAB code is not clear (no description for the variable and steps is provided). I am not sure, if the code in its current version, which is provided here is of any use.}}
 
{{Cleanup|date=November 2 2010|reason= This MATLAB code is not clear (no description for the variable and steps is provided). I am not sure, if the code in its current version, which is provided here is of any use.}}
 +
 +
{{Cleanup|date=November 2 2010|reason= This code might be more useful, if one consider it along with the above approach for taking derivatives of the error in respect to the weights.}}
 +
 +
{{Cleanup|date=November 2 2010|reason= I also think that some descriptions or comments should be added to the code to make it more clear.}}
 +
 +
 +
% This code might be used to train a neural network, using backpropagation algorithm
 +
% ep: maximum number of epochs
 +
% io: matrix of all the inputs and outputs of the network's layers, given the weights matrix, w.
 +
% w: w is the weights matrix
 +
% gp: is the derivatives matrix
 +
% shuffle: a function for changing the permutation of the data
 +
%
 
  while i < ep
 
  while i < ep
 
     i = i + 1;
 
     i = i + 1;
Line 2,268: Line 3,005:
 
     end
 
     end
 
  end
 
  end
 +
 +
=== The Neural Network Toolbox in Matlab ===
 +
 +
% Here is a problem consisting of inputs P and targets T that we would like to solve with a network.
 +
P = [0 1 2 3 4 5 6 7 8 9 10];
 +
T = [0 1 2 3 4 3 2 1 2 3 4];
 +
 +
% Here a network is created with one hidden layer of 5 neurons.
 +
net = newff(P,T,5);
 +
 +
% Here the network is simulated and its output plotted against the targets.
 +
Y = sim(net,P);
 +
plot(P,T,P,Y,’o’)
 +
 +
[[File:nn1.jpg]]
 +
 +
% Here the network is trained for 50 epochs. Again the network’s output is plotted.
 +
net.trainParam.epochs = 50;
 +
net = train(net,P,T);
 +
Y = sim(net,P);
 +
plot(P,T,P,Y,’o’)
 +
 +
[[File:nn2.jpg]]
  
 
====Some notes on the neural network and its learning algorithm====
 
====Some notes on the neural network and its learning algorithm====
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* Given a set of data for training a neural network, one should keep aside a ratio of it as the validation dataset, to obtain a sufficient number of layers and number of neurons in each of the layers. The best construction may be the one which leads to the least error for the validation dataset. Validation data may not be used as the training data of the network (refer to cross-validation and k-fold validation explained in the next lecture).
 
* Given a set of data for training a neural network, one should keep aside a ratio of it as the validation dataset, to obtain a sufficient number of layers and number of neurons in each of the layers. The best construction may be the one which leads to the least error for the validation dataset. Validation data may not be used as the training data of the network (refer to cross-validation and k-fold validation explained in the next lecture).
 
  
 
* We can also use the validation-training scheme to estimate how many epochs is enough for training the network.
 
* We can also use the validation-training scheme to estimate how many epochs is enough for training the network.
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Neural Network with Back-propagation faces some subtleties.
 
Neural Network with Back-propagation faces some subtleties.
  
Deep Neural Networks became popular two or three years ago, when introduced by Dr. Geoffrey E. Hinton. Deep Neural Network training algorithm deals with the training of a Neural Network with a large number of layers.
+
Deep Neural Networks became popular two or three years ago, when introduced by Dr. Geoffrey E. Hinton, a Professor in computer science at the University of Toronto. Deep Neural Network training algorithm [http://www.cs.toronto.edu/~hinton/absps/ncfast.pdf] deals with the training of a Neural Network with a large number of layers.
  
 
The approach of training the deep network is to assume the network has only two layers first and train these two layers. After that we train the next two layers, so on and so forth.
 
The approach of training the deep network is to assume the network has only two layers first and train these two layers. After that we train the next two layers, so on and so forth.
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Take a specific marketing case for example. A feedforward neural network was trained using back-propagation to assist the marketing control of airline seat allocations. The neural approach was adaptive to the rule. The system is used to monitor and recommend booking advice for each departure.
 
Take a specific marketing case for example. A feedforward neural network was trained using back-propagation to assist the marketing control of airline seat allocations. The neural approach was adaptive to the rule. The system is used to monitor and recommend booking advice for each departure.
 +
 +
Neural networks have been applied to almost every field that one can think of. For the interested reader, a detailed description with links that discusses some of the many application of neural networks is available [http://www.faqs.org/faqs/ai-faq/neural-nets/part7/section-2.html here].
  
 
=== Issues with Neural Network ===
 
=== Issues with Neural Network ===
 
When Neural Networks was first introduced they were thought to be modeling human brains, hence they were given the fancy name "Neural Network". But now we know that they are just logistic regression layers on top of each other but have nothing to do with the real function principle in the brain.
 
When Neural Networks was first introduced they were thought to be modeling human brains, hence they were given the fancy name "Neural Network". But now we know that they are just logistic regression layers on top of each other but have nothing to do with the real function principle in the brain.
  
We do not know why deep networks turn out to work quite well in practice. Some people claim that they mimic the human brains, but this is unfounded. As a result of these kinds of claims it is important to keep the right perspective on what this field of study is trying to accomplish. For example, the goal of machine learning may be to mimic the 'learning' function of the brain, but necessarily the processes the brain uses to learn.
+
We do not know why deep networks turn out to work quite well in practice. Some people claim that they mimic the human brains, but this is unfounded. As a result of these kinds of claims it is important to keep the right perspective on what this field of study is trying to accomplish. For example, the goal of machine learning may be to mimic the 'learning' function of the brain, but not necessarily the processes that the brain uses to learn.
  
As for the algorithm, since it does not have a convex form, we still face the problem of local minimum, although people have devised other techniques to avoid this dilemma.
+
As for the algorithm, as discussed above, since it does not have a convex form, it still faces the problem of getting trapped in local minima, although people have devised techniques to help it avoid this problem.
  
In sum, Neural Network lacks a strong learning theory to back up its "success", thus it's hard for people to wisely apply and adjust it. Having said that, it is not an active research area in machine learning. NN still has wide applications in the engineering field such as in control.
+
In sum, Neural Network lacks a strong learning theory to back up its "success", thus it's hard for people to wisely apply and adjust it. Having said that, it is still an active research area in machine learning. NN still has wide applications in the engineering field such as in control.
  
 
===Business Applications of Neural Networks===
 
===Business Applications of Neural Networks===
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* Money Laundering Detection with a Neural-Network  
 
* Money Laundering Detection with a Neural-Network  
 
* Utilizing Fuzzy Logic and Neurofuzzy for Business Advantage
 
* Utilizing Fuzzy Logic and Neurofuzzy for Business Advantage
 +
 +
=== Further readings ===
 +
Bishop,C. "Neural Networks for Pattern Recognition"
 +
 +
Haykin, Simon. "Neural Networks. A Comprehensive Foundation" Available [http://www.esnips.com/doc/83becbe7-0fa6-4f90-a7c4-34697b63a8cb/Neural-Networks---A-Comprehensive-Foundation---Simon-Haykin here]
 +
 +
Nilsson,N. "Introduction to Machine Learning", Chapter 4: Neural Networks. Available [http://robotics.stanford.edu/people/nilsson/mlbook.html here]
 +
 +
Brian D. Ripley "Pattern Recognition and Neural Networks" Available [http://books.google.com/books?id=m12UR8QmLqoC&printsec=frontcover&dq=Neural+Networks+for+Pattern+Recognition&hl=en&ei=r3YCTbOlDMiYnAfh_JXmDQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDYQ6AEwAg#v=onepage&q&f=false here]
 +
 +
G. Dreyfus "Neural networks: methodology and applications" Available [http://books.google.com/books?id=g2J4J2bLgRQC&printsec=frontcover&dq=Neural+Networks&hl=en&ei=WncCTaimM86lngeg-OzlDQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CD4Q6AEwAg#v=onepage&q&f=false here]
  
 
===References===
 
===References===
<references/>
+
 
 +
1. On fuzzy modeling using fuzzy neural networks with the back-propagation algorithm
 +
[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=159069]
 +
 
 +
2. Thirty years of adaptive neural networks: perceptron, madaline and backpropagation
 +
[http://onlinelibrary.wiley.com/doi/10.1002/9780470231616.app7/pdf]
  
 
==Complexity Control  - October 26, 2010==
 
==Complexity Control  - October 26, 2010==
  
 
=== Lecture Summary ===  
 
=== Lecture Summary ===  
Selecting the model structure with an appropriate complexity is a standard problem in pattern recognition and machine learning. Systems with the optimal complexity have a good generalization to unseen data.
+
Selecting the model structure with an appropriate complexity is a standard problem in pattern recognition and machine learning. Systems with the optimal complexity have a good [http://www.csc.kth.se/~orre/snns-manual/UserManual/node16.html generalization] to yet unobserved data.
 +
 
 +
A wide range of techniques may be used which alter the system complexity. In this lecture, we present the concepts of over-fitting & under-fitting, and an example to illustrate how we choose a good classifier and how to avoid over-fitting.
  
A wide range of techniques may be used which alter the system complexity. In this lecture, we present over & under-fitting  concepts with an example to illustrate, how do we choose a good classifier? and how to avoid Over-fitting?
+
Moreover, [http://en.wikipedia.org/wiki/Cross-validation_%28statistics%29 cross-validation] has been introduced during the lecture which is a method for estimating generalization error based on "re-sampling" (Weiss and Kulikowski 1991; Plutowski, Sakata, and White 1994; Shao and Tu 1995)[1],[2],[3]. The resulting estimates of generalization error are often used for choosing among various models. A model which is associated with the smallest estimated generalization error would be selected. Finally, the common types of cross-validation have been addressed.
  
Moreover, Cross-validation has been introduced during the leacture which is a method for estimating generalization error based on "resampling" (Weiss and Kulikowski 1991; Plutowski, Sakata, and White 1994; Shao and Tu 1995)[1],[2],[3]. The resulting estimates of generalization error are often used for choosing among various models. Also, it can be used for model selection by choosing one of several models that has the smallest estimated generalization error. Finally, the common types of cross-validation has been addressed.
+
Before starting the next section, a short description of model complexity is necessary. As the name suggests, model complexity somehow describes how complicated our model is. Suppose we have a feed forward neural network -- if we increase the number of hidden layers or the number of nodes in a specific layer, it makes sense that our model is becoming more complex. Or, suppose we want to fit a polynomial function on some data points -- if we add to the degree of this polynomial it seems that we are choosing a more complex model. Intuitively, it seems that fitting a more complex model would be better, since we have more degrees of freedom and can get a more exact answer. The next section will explain why this is not the case, and why there is a trade-off between model complexity and optimal result. This makes it necessary to find methods for controlling complexity in model selection. We will see this procedure in an example.
  
 
=== Over-fitting and Under-fitting ===
 
=== Over-fitting and Under-fitting ===
[[File:overfitting-model.png|500px|thumb|right|Figure 1. The overfitting model passes through all the points of the training set, but has poor predictive power for new points.
+
[[File:overfitting-model.png|500px|thumb|right|Figure 1. The overfitting model that uses kernel regression and smoothing splines passes through all of the points of the training set, but has poor predictive power for new data points that are not in the training set.
In exchange the line model has some error on the training points but has extracted the main characteristic of the training points, and has good predictive power.]]
+
 
 +
On the other hand, the line model makes more errors on the training points but it is better at extracting the main characteristic of the training points, i.e. the underlying function. Consequently, it has better predictive power for new data points that are not in the training set.]]
 
There are [http://academicearth.org/lectures/underfitting-and-overfitting two issues] that we have to avoid in Machine Learning:
 
There are [http://academicearth.org/lectures/underfitting-and-overfitting two issues] that we have to avoid in Machine Learning:
 
#[http://en.wikipedia.org/wiki/Overfitting Overfitting]
 
#[http://en.wikipedia.org/wiki/Overfitting Overfitting]
 
#Underfitting
 
#Underfitting
  
Overfitting occurs when our model is heavily complex with so many degrees of freedom, that we can learn every detail of the training set. Such a model will have very high precision on the training set but will show very poor ability to predict outcomes of new instances, especially outside the domain of the training set.Dangerous for the overfitting:it will easily lead the predictions to the range that is far beyond the training data, and produce wild predictions in multilayer perceptrons even with noise-free data.The best way to avoid overfitting is to use lots of training data.  
+
Suppose there is no noise in the training data, then we would face no problem with over-fitting, because in this case every training data point lies on the underlying function, and the only goal is to build a model that is as complex as needed to pass through every training data point.  
  
In a Neural Network if the depth is too much, the network will have many degrees of freedom and will learn every characteristic of the training data set. That means it will show a very precise outcome of the training set but will not be able to generalize the commonality of the training set to predict the outcome of new cases.
+
However, in the real-world, the training data are [http://en.wikipedia.org/wiki/Statistical_noise noisy], i.e. they tend to not lie exactly on the underlying function, instead they may be shifted to unpredictable locations by random noise. If the model is more complex than what it needs to be in order to accurately fit the underlying function, then it would end up fitting most or all of the training data. Consequently, it would be a poor approximation of the underlying function and have poor prediction ability on new, unseen data.  
  
Underfitting occurs when the model we picked to describe the data is not complex enough, and has high error rate on the training set.
+
The danger of overfitting is that the model becomes susceptible to predicting values outside of the range of training data. It can cause wild predictions in multilayer perceptrons, even with noise-free data. The best way to avoid overfitting is to use lots of training data. Unfortunately, that is not always useful. Increasing the training data alone does not guarantee that over-fitting will be avoided. The best strategy is to use a large-enough size training set, and control the complexity of the model. The training set should have a sufficient number of data points which are sampled appropriately, so that it is representative of the whole data space.
There is always a trade-off. If our model is too simple, underfitting could occur and if it is too complex, overfitting can occur.
+
 
 +
In a Neural Network, if the number of hidden layers or nodes is too high, the network will have many degrees of freedom and will learn every characteristic of the training data set. That means it will fit the training set very precisely, but will not be able to generalize the commonality of the training set to predict the outcome of new cases.
 +
 
 +
Underfitting occurs when the model we picked to describe the data is not complex enough, and has a high error rate on the training set.
 +
There is always a trade-off. If our model is too simple, underfitting could occur and if it is too complex, overfitting can occur.
  
 
'''Example'''
 
'''Example'''
#Consider the example showed in the figure. We have a training set and we want to find a model which fits it the best. We can find a polynomial of high degree which almost passes through all the points in the training set. But, in fact the training set is coming from a line model. Now the problem is although the complex model has less error on the training set it diverges from the line in other ranges which we have no training point. Because of that the high degree polynomail has very poor predictive result on test cases. This is an example of overfitting model.
+
#Consider the example shown in the figure. We have a training set and want to find a model which fits it best. We can find a polynomial of high degree which passes through almost all points in the training set. But in reality, the training set comes from a linear model. Although the complex model has little error on the training set, it diverges from the line in other ranges in which we have no training points. As a result, the high degree polynomial has very poor prediction power on the test cases. This is an example of overfitted model.
 
#Now consider a training set which comes from a polynomial of degree two model. If we model this training set with a polynomial of degree one, our model will have high error rate on the training set, and is not complex enough to describe the problem.
 
#Now consider a training set which comes from a polynomial of degree two model. If we model this training set with a polynomial of degree one, our model will have high error rate on the training set, and is not complex enough to describe the problem.
#Consider a simple classification example.  If our classification rule takes as input only the colour of a fruit and concludes that it is a banana, then it is not a good classifier.   The reason is that just because a fruit is a yellow, does not mean that it is a banana.   We can add complexity to our model to make it a better classifier by considering more features typical of bananas, such as size and shape.   If we continue to make our model more and more complex in order to improve our classifier, we will eventually reach a point where the quality of our classifier no longer improves, ie., we have overfit the data.  This occurs when we have considered so many features that we have perfectly described the existing bananas, but if presented with a new banana of slightly different shape than the existing bananas, for example, it cannot be detected.  This is the tradeoff; what is the right level of complexity?
+
#Consider a simple classification example: if our classification rule takes as input only the colour of a fruit and concludes that it is a banana, then it is not a good classifier. The reason is that just because a fruit is yellow, does not mean that it is a banana. We can add complexity to our model to make it a better classifier by considering more features, such as size and shape. If we continue to make our model more and more complex in order to improve our classifier, we will eventually reach a point where the quality of our classifier no longer improves, ie. we have overfit the data.  This occurs when we have considered so many features that we have perfectly described our existing banana that we training on, but if presented with a new banana of a slightly different shape for example, it may not be detected.  This is the tradeoff: what is the right level of complexity?
  
Overfitting occurs when the model becomes too complex and underfitting occurs when it is not complex enough, both of which are not desirable.  To control complexity, it is necessary to make assumptions for the model before fitting the data. Assumptions that we can make for a model are with polynomials or a neural network. There are other ways as well.
+
Overfitting occurs when the model is too complex and underfitting occurs when it is not complex enough, both of which are not desirable.  To control complexity, it is necessary to make assumptions for the model before fitting the data. Some of the assumptions that we can make for a model are with polynomials or a neural network. There are other ways as well.
  
 
[[File:Family_of_polynomials.jpg|200px|thumb|right|Figure 2: An example of a model with a family of polynomials]]
 
[[File:Family_of_polynomials.jpg|200px|thumb|right|Figure 2: An example of a model with a family of polynomials]]
 
We do not want a model to get too complex, so we control it by making an assumption on the model. With complexity control, we want a model or a classifier with a low error rate. The lecture will explain the [http://academicearth.org/lectures/bias-variance-tradeoff tradeoff between Bias and variance] for model complexity control.
 
We do not want a model to get too complex, so we control it by making an assumption on the model. With complexity control, we want a model or a classifier with a low error rate. The lecture will explain the [http://academicearth.org/lectures/bias-variance-tradeoff tradeoff between Bias and variance] for model complexity control.
 +
 +
'''Overfitted model and Underfitted model:'''
 +
 +
[[File:extrem_model.jpg|400px|thumb|right|Figure 3]]
 +
After the structure of the model is determined, the next step is do the model selection. The problem encountered is how to estimate the parameters effectively, especially when we use iteration methods to do the estimation. In the iteration method, the key point is to determine the best time to stop updating parameters.
 +
Let us see a very simple example; assume the dotted line on the graph can be expressed as a function <math>\,h(x)</math>, and the data points (the circles) are generated by the function with added noise.
 +
 +
 +
'''Model 1'''(as shown on the left of Figure 3)
 +
A line <math>\,g(x)</math> can be used to describe the data points, where two parameters are needed to construct the estimated function.  However, it is clear that it performs badly. This model is a typical example of an underfitted model. In this case, the model will perform well in prediction, but a large bias could be generated.
 +
 +
'''Model 2''' (as shown on the right of Figure 3)
 +
In this model, lots of parameters are used to fit the data. Although it looks like a fairly good fit, the prediction performance could be very bad. This means that this model will generate a large variance when we use it on points not part of the training data.
 +
The models above are the extreme cases in the model selection, we do not want to choose any of them in our classification task. The key is to stop our training process at the optimal time, such that a balance of bias and variance is obtained, that is, the time t in the following graph.
 +
 +
[[File:optimal_time.jpg|300px|thumb|right|Figure 4]]
 +
 +
To achieve this goal, one approach we can use is to divide our data points into two groups: one (training set) is used in the training process to obtain parameters, the other one (validation set) is used for determining the optimal model. After every update of parameters, the test in the validation set is implemented and the error curve is plotted to find the optimal point <math>\,t</math>. Here, the validation test is a good measure of generalization. Remember to not update the parameters in the validation test. If another, independent test is needed to follow validation, three independent groups should be determined at the beginning. In addition, this approach is suitable for the case of more data points, especially a finite data set, since the effect of noise could be decreased to the lowest level.
 +
 +
So far, we have learned two of the most popular ways to estimate the expected level of fit of a model to a test data set that is independent of the data used to train the model:
 +
:1. Cross validation
 +
:2. Regularization: refers to a series of techniques we can use to suppress overfitting, that is, making our function not so curved that it performs badly in prediction. The specific way is to add a new penalty term into the error function, this prevents increasing the weights too much when they are updated at each iteration.
 +
 +
Indeed, there are many techniques could be used, such as:
 +
:1.[http://en.wikipedia.org/wiki/Akaike_information_criterion Akaike information criterion]
 +
:2.[http://en.wikipedia.org/wiki/Bayesian_information_criterion Bayesian information criterion]
 +
:3.[http://en.wikipedia.org/wiki/Mallows'_Cp Mallows' Cp]]
 +
 +
===='''Note'''====
 +
When the model is linear, the true error form AIC approach is identical to that from Cp approach; when the model is nonlinear, they are different.
  
 
=== '''How do we choose a good classifier?''' ===
 
=== '''How do we choose a good classifier?''' ===
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Recall the empirical error rate
 
Recall the empirical error rate
  
<math>\ \hat L_{h}= \frac{1}{n} \sum_{i=1}^{n} I(h(x_{i}) \neq y_{i})</math>
+
<math>\ \hat L(h)= \frac{1}{n} \sum_{i=1}^{n} I(h(x_{i}) \neq y_{i})</math>
 
 
<math>\,h</math> is a classifier and we want to minimize the error rate. So we apply <math>\displaystyle x_1</math> to <math>\displaystyle x_n</math> to <math>\displaystyle h</math>, and take the average to get the empirical true error rate estimation of probability that
 
<math>h(x_{i}) \neq y_{i}</math>.
 
  
 
<span id="prediction-error">[[File:Prediction_Error.jpg|200px|thumb|right|Figure 3]]</span>
 
<span id="prediction-error">[[File:Prediction_Error.jpg|200px|thumb|right|Figure 3]]</span>
There is a downward bias to this estimate meaning that it is always less than the true error rate.  
+
There is a downward bias to the training error estimate, it is always less than the true error rate.  
  
If there is a change in our complexity from low to high, our error rate is always decreasing. When we apply our model to the test data, our error rate will start to decrease to a point, but then it will increase since the model hasn't seen it before.  This can be explained since training error will decrease when we fit the model better by increasing its complexity, but as we have seen, this complex model will not generalize well, resulting in a larger test error.  
+
If there is a change in our complexity from low to high, our training (empirical) error rate is always decreased. When we apply our model to the test data, our error rate will decrease to a point, but then it will increase because the model has not seen the test data points before.  This results in a convex test error curve as a function of learning model complexity. The training error will decrease when we keep fitting increasingly complex models, but as we have seen, a model too complex will not generalize well, resulting in a large test error.  
  
We use our test data (from the test sample line shown on Figure 2) to get our empirical error rate.
+
We use our test data (from the test sample line shown on Figure 2) to get our true error rate.
Right complexity is defined as where error rate of the test data is minimum; and this is one idea behind complexity control.
+
Right complexity is defined as the point where the true error rate ( the error rate associated with the test data) is minimum; this is one idea behind complexity control.
  
 
[[File:Bias.jpg|200px|thumb|left|Figure 4]]
 
[[File:Bias.jpg|200px|thumb|left|Figure 4]]
  
We assume that we have samples <math>\,X_1, . . . ,X_n</math> that follow some (possibly unknown) distribution. We want to estimate a parameter <math>\,f</math> of the unknown distribution. This parameter may be the mean <math>\,E(X_i)</math>, the variance <math>\,var(X_i)</math> or some other quantity.
+
We assume that we have samples <math>\,x_1, . . . ,x_n</math> that follow some (possibly unknown) distribution. We want to estimate a parameter <math>\,f</math> of the unknown distribution. This parameter may be the mean <math>\,E(x_i)</math>, the variance <math>\,var(x_i)</math> or some other quantity.
  
 
The unknown parameter <math>\,f</math> is a fixed real number <math>f\in R</math>. To estimate it, we use an estimator which is a
 
The unknown parameter <math>\,f</math> is a fixed real number <math>f\in R</math>. To estimate it, we use an estimator which is a
function of our observations, <math>\hat{f}(X_1,...,X_n)</math>.  
+
function of our observations, <math>\hat{f}(x_1,...,x_n)</math>.  
  
 
<math>Bias (\hat{f}) = E(\hat{f}) - f</math>
 
<math>Bias (\hat{f}) = E(\hat{f}) - f</math>
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<math>Variance (\hat{f}) = E[(\hat{f} - E(\hat{f}))^2]</math>
 
<math>Variance (\hat{f}) = E[(\hat{f} - E(\hat{f}))^2]</math>
  
One property we desire of the estimator is that it is correct on average, that is, it is unbiased. <math>Bias (\hat{f}) = E(\hat{f}) - f=0</math>.
+
One desired property of the estimator is that it is correct on average, that is, it is unbiased. <math>Bias (\hat{f}) = E(\hat{f}) - f=0</math>.
However, there is a more important property for an estimator than just being unbiased: the mean squared error. In statistics, there are problems for which it may be good to use an estimator with a small bias. In some cases, an estimator with a small bias may have lesser mean squared error or be median-unbiased (rather than mean-unbiased, the standard unbiasedness property). The property of median-unbiasedness is invariant under transformations while the property of mean-unbiasedness may be lost under nonlinear transformations. For example, while using an unbiased estimator with large mean square error to estimate the parameter, we highly risk a big error. In contrast, a biased estimator with small mean square error will well improve the precision of our prediction.
+
However, there is a more important property for an estimator than just being unbiased: low mean squared error. In statistics, there are problems for which it may be good to use an estimator with a small bias. In some cases, an estimator with a small bias may have lesser mean squared error or be median-unbiased (rather than mean-unbiased, the standard unbiasedness property). The property of median-unbiasedness is invariant under transformations while the property of mean-unbiasedness may be lost under nonlinear transformations. For example, while using an unbiased estimator with large mean square error to estimate the parameter, we risk a big error. In contrast, a biased estimator with small mean square error will improve the precision of our predictions.
  
 
Hence, our goal is to minimize <math>MSE (\hat{f})</math>.
 
Hence, our goal is to minimize <math>MSE (\hat{f})</math>.
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<math>MSE (\hat{f})=Variance (\hat{f})+Bias ^2(\hat{f}) </math>. Thus given the Mean Squared Error (MSE), if we have a low bias, then we will have a high variance and vice versa.
 
<math>MSE (\hat{f})=Variance (\hat{f})+Bias ^2(\hat{f}) </math>. Thus given the Mean Squared Error (MSE), if we have a low bias, then we will have a high variance and vice versa.
  
A Test error is a good estimation on MSE. We want to have a somewhat balanced bias and variance (not high on bias or variance), although it will have some bias.
+
'''Algebraic Proof''':
  
Referring to Figure 3, overfitting happens after the point where training data (training sample line) starts to decrease and test data (test sample line) starts to increase.
+
<math>MSE (\hat{f}) = E[(\hat{f} - f)^2] = E[(\hat{f} - E(\hat{f}) + E(\hat{f}) - f)^2]</math>
 +
 
 +
<math>E[(\hat{f} - E(\hat{f}))^2+(E(\hat{f}) - f)^2 + 2(\hat{f} - E(\hat{f}))(E(\hat{f}) - f)]</math>
 +
 
 +
<math>E(\hat{f} - E(\hat{f}))^2 + E(E(\hat{f}) - f)^2 + E(2(\hat{f} - E(\hat{f}))(E(\hat{f}) - f))</math>
 +
 
 +
By definition,  
 +
 
 +
<math>E(\hat{f} - E(\hat{f}))^2 = Var(\hat{f})</math> 
 +
 
 +
<math>(E(\hat{f}) - f)^2 = Bias^2(\hat{f})</math>
 +
 
 +
So we must show that:
 +
 
 +
<math>E(2(\hat{f} - E(\hat{f}))(E(\hat{f}) - f)) = 0</math>
 +
 
 +
<math>E(2(\hat{f} - E(\hat{f}))(E(\hat{f}) - f)) = 2E(\hat{f}E(\hat{f})) - \hat{f}f - E(\hat{f})E(\hat{f}) + E(\hat{f})f)</math>
 +
 
 +
<math>2(E(\hat{f})E(\hat{f}) - E(\hat{f})f - E(\hat{f})E(\hat{f}) + E(\hat{f})f) = 0</math>
 +
 
 +
 
 +
A test error is a good estimation of MSE. We want to have a somewhat balanced bias and variance (not high on bias or variance), although it will have some bias.
 +
 
 +
=== References ===
 +
 
 +
1. A Comparison of Prediction Accuracy, Complexity, and Training Time of Thirty-Three Old and New Classification Algorithms
 +
[http://www.springerlink.com/content/u751321011502645.pdf]
 +
 
 +
2. Model complexity control and statistical learning theory
 +
[http://www.springerlink.com/content/wh40jlnrbr6cnh9x/]
 +
 
 +
3. On Dimensionality, Sample Size, Classification Error, and Complexity of Classification Algorithm in Pattern Recognition
 +
[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4767011]
 +
 
 +
4. Overfitting, Underfitting and Model Complexity
 +
[http://www.chemometrie.com/phd/2_8_1.html]
  
 
=== Avoid Overfitting ===
 
=== Avoid Overfitting ===
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<math>\hookrightarrow</math> Decrease the chance of overfitting by controlling the weights
 
<math>\hookrightarrow</math> Decrease the chance of overfitting by controlling the weights
 +
 +
<math>\hookrightarrow</math> Weight Decay: bound the complexity and non-linearity of the output by a new regularized cost function.
  
 
=== Cross-Validation ===
 
=== Cross-Validation ===
 +
 +
'''Cross-validation''', sometimes called '''rotation estimation''', is a technique for assessing how the results of a statistical analysis will generalize to an independent data set.  It is mainly used in settings where the goal is prediction, and one wants to estimate how accurately a predictive model will perform in practice.  One round of cross-validation involves partitioning a sample of data into complementary subsets, performing the analysis on one subset (called the ''training set''), and validating the analysis on the other subset (called the ''validation set'' or ''testing set'').  To reduce variability, multiple rounds of cross-validation are performed using different partitions, and the validation results are averaged over the rounds.
  
 
[[File:Cv.jpg|200px|thumb|right|Figure 1: Illustration of Cross-Validation]]
 
[[File:Cv.jpg|200px|thumb|right|Figure 1: Illustration of Cross-Validation]]
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:2) Train the classifier using only data in T
 
:2) Train the classifier using only data in T
  
:3) Estimate the error rate, <math>\begin{align}\hat L(h)\end{align}</math>, using only data in V
+
:3) Estimate the true error rate, <math>\begin{align}\hat L(h)\end{align}</math>, using only data in V
  
 
:<math>\hat L(h) = \frac{1}{|\mathrm{V}|}\sum_{x_i \in \mathrm{V}}I(h(x_i) \neq y_i)</math>, where <math>\begin{align}\,|\mathrm{V}|\end{align}</math> is the cardinality of the validation set and
 
:<math>\hat L(h) = \frac{1}{|\mathrm{V}|}\sum_{x_i \in \mathrm{V}}I(h(x_i) \neq y_i)</math>, where <math>\begin{align}\,|\mathrm{V}|\end{align}</math> is the cardinality of the validation set and
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Note that the validation set will be totally unknown to the trained model but the proper label of all elements in this set are known. Therefore, it is easy to count the number of misclassified points in V.
 
Note that the validation set will be totally unknown to the trained model but the proper label of all elements in this set are known. Therefore, it is easy to count the number of misclassified points in V.
  
The best classifier is the model with minimum error, <math>\begin{align}\hat L(h)\end{align}</math>.
+
The best classifier is the model with minimum true error, <math>\begin{align}\hat L(h)\end{align}</math>.
  
 
=== K-Fold Cross-Validation ===
 
=== K-Fold Cross-Validation ===
 
[[File:k-fold.png|350px|thumb|right|Figure 2: K-fold cross-validation]]
 
[[File:k-fold.png|350px|thumb|right|Figure 2: K-fold cross-validation]]
 
The results from the method above may differ significantly based on the initial choice of T and V. Therefore, we improve simple cross-validation by introducing K-fold cross-validation.  
 
The results from the method above may differ significantly based on the initial choice of T and V. Therefore, we improve simple cross-validation by introducing K-fold cross-validation.  
The advantage of K-fold cross validation is that all the values in the dataset are eventually used for both training and testing.  
+
The advantage of K-fold cross validation is that all the values in the dataset are eventually used for both training and testing. When using K-fold cross validation the number of folds must be considered. If the user has a large data set then more folds can be used because a smaller portion of the total data is needed to train the classifier. This leaves more test data and therefore a better estimate on the test error. Unfortunately, the more folds one uses the longer the cross-validation will run. If the user has a small data set then fewer, larger folds must be taken to properly train the classifier.
  
 
In this case, the algorithm is:
 
In this case, the algorithm is:
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: 4) Compute the error rate, <math>\begin{align}\hat L_k(h)\end{align}</math>, using only data in part k
 
: 4) Compute the error rate, <math>\begin{align}\hat L_k(h)\end{align}</math>, using only data in part k
  
: <math>\hat L_k(h) = \frac{1}{m} \sum_{i=1}^{n} I(h(x_{i}) \neq y_{i})</math>, where <math>m</math> is the number of data points in part k
+
: <math>\hat L_k(h) = \frac{1}{m} \sum_{i=1}^{m} I(h(x_{i}) \neq y_{i})</math>, where <math>m</math> is the number of data points in part k
  
 
: 5) End loop
 
: 5) End loop
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=== Leave-One-Out Cross-Validation - October 28, 2010 ===
 
=== Leave-One-Out Cross-Validation - October 28, 2010 ===
  
Leave-one-out cross-validation is similar to k-fold validation by selecting sets of equal size for error estimation. Leave-one-out cross-validation instead removes a single data point, with n-partitions. For example, we estimate the <math>\,n-1</math> data points with <math>\,m</math> linear models over the <math>\,n</math> sets, and compare the average error rates.<br />
+
Leave-one-out cross validation is used to determine how accurately a learning algorithm will be able to predict data that it was not trained on. When using the leave-one-out method, the learning algorithm is trained multiple times, using all but one of the training set data points. The form of the algorithm is as follows:
  
In the above example, we can see that k-fold cross-validation can be computationally expensive: for every possible value of the parameter, we must train the model <math>\,K</math> times. This deficiency is even more obvious in leave-one-out cross-validation, where we must train the model <math>\,n</math> times, where <math>\,n</math> is the number of data points in the data set.<br />
+
For k = 1 to n (where n is the number of points in our dataset)
  
However, in the linear model, we can save complexity analytically. A model is ''correct'' if the mean response is the linear combination of subsets of a vector and the columns of <math>X_n</math>. Let <math>A_n</math> be a finite set of proposed models. Let <math>a_n^L</math> be the model minimizing average squared error, then the selection procedure is ''consistent'' if the probability of the model selected being <math>a_n^L</math> approaches 1. Leave-one-out is correct, can be inconsistent, and given
+
•Temporarily remove the kth data point.
  
* <math>\max_{i <= n} x_i^t (X_n^tX_n)^{-1} x_i \to 0</math>
+
•Train the learning algorithm on the remaining n - 1 points.
  
is asymptotically equivalent to AIC, which performs slightly worse than k-fold <ref>Shao, J. ''An asymptotic theory for linear model selection,'' Statistica Sineca, 7, 221-264 (1997).</ref>
+
•Test the removed data point and note your error.
<br />
+
 
 +
Calculate the mean error over all n data points.
 +
 
 +
Leave-one-out cross validation is useful because it does not waste data. When training, all but one of the points are used, so the resulting regression or classification rules are essentially the same as if they had been trained on all the data points. The main drawback to the leave-one-out method is that it is expensive - the computation must be repeated as many times as there are training set data points.
 +
 
 +
 
 +
Leave-one-out cross-validation is similar to k-fold validation by selecting sets of equal size for error estimation. Leave-one-out cross-validation instead removes a single data point, with n-partitions. Each partition is used systematically for testing exactly once whereas the remaining partitions are used for training. For example, we estimate the <math>\,n-1</math> data points with <math>\,m</math> linear models over the <math>\,n</math> sets, and compare the average error rates of the m linear model.The leave-one-out error is the average error over all partitions.<br />
 +
 
 +
 
 +
In the above example, we can see that k-fold cross-validation can be computationally expensive: for every possible value of the parameter, we must train the model <math>\,K</math> times. This deficiency is even more obvious in leave-one-out cross-validation, where we must train the model <math>\,n</math> times, where <math>\,n</math> is the number of data points in the data set.<br />
 +
 
 +
But an expensive computational load does not tell the whole story. Why do we need this validation? The key factor is not having enough data points! In some real world problems gathering data points can be very expensive or time consuming. Suppose we want to study the effect of a new drug on the human body. To do this, we must test the drug on some patients. However, it is very hard to convince a person to take part in this procedure since there may be risks and side effects with testing the new drug on him/her. As well, a long-term study needs to be done to observe any long-term effects. In a similar manner we lack data points or observations in some problems. But if we use K-fold cross-validation and divide the data points into a training and test data set then we may not have enough data to train the neural network or fit any other model, and under fitting may occur. To avoid this the best thing that can be done is to do leave-one-out cross-validation. In this way we will take advantage of the data points we have and yet still be able to test the model. 
 +
 
 +
Leave-one-out cross-validation often works well for estimating generalization error for continuous error functions such as the mean squared error, but it may perform poorly for discontinuous error functions such as
 +
the number of misclassified cases. In the latter case, k-fold cross-validation is preferred. But if k gets too small, the error estimate is pessimistically biased because of the difference in training-set size between the full-sample analysis and the cross-validation analyses.
 +
 
 +
However, in the linear model, we can save complexity analytically. A model is ''correct'' if the mean response is the linear combination of subsets of a vector and the columns of <math>X_n</math>. Let <math>A_n</math> be a finite set of proposed models. Let <math>a_n^L</math> be the model minimizing average squared error, then the selection procedure is ''consistent'' if the probability of the model selected being <math>a_n^L</math> approaches 1. Leave-one-out is correct, can be inconsistent, and given
 +
 
 +
* <math>\max_{i <= n} x_i^t (X_n^tX_n)^{-1} x_i \to 0</math>
 +
 
 +
is asymptotically equivalent to AIC, which performs slightly worse than k-fold <ref>Shao, J. ''An asymptotic theory for linear model selection,'' Statistica Sineca, 7, 221-264 (1997).</ref>.AIC has an asymptotic probability of one of choosing a "good" subset, but less than one of choosing the "best" subset. Many simulation studies have also found that AIC overfits badly in small samples. Hence, these results suggest that leave-one-out
 +
cross-validation should overfit in small samples.
 +
<br />
  
 
Leave-one-out cross-validation can perform poorly in comparison to k-fold validation. A paper by Breiman compares k-fold (leave-many-out) cross-validation to leave-one-out cross-validation, noting that average prediction loss and downward bias increase from k-fold to leave-one-out <ref>Breiman, L. ''Heuristics of instability and stabilization in model selection,'' Annals of Statistics, 24, 2350-2383 (1996).</ref>. This can be explained by the lower bias of leave-one-out validation, causing an increase in variance. The bias is relative to the size of the sample set compared to the training set [http://en.wikipedia.org/wiki/Cross-validation_%28statistics%29#Leave-one-out_cross-validation]. As such, as k becomes larger, it becomes more biased and has less variance. Similarly, larger data sets will direct the bias toward zero.<br /><br />
 
Leave-one-out cross-validation can perform poorly in comparison to k-fold validation. A paper by Breiman compares k-fold (leave-many-out) cross-validation to leave-one-out cross-validation, noting that average prediction loss and downward bias increase from k-fold to leave-one-out <ref>Breiman, L. ''Heuristics of instability and stabilization in model selection,'' Annals of Statistics, 24, 2350-2383 (1996).</ref>. This can be explained by the lower bias of leave-one-out validation, causing an increase in variance. The bias is relative to the size of the sample set compared to the training set [http://en.wikipedia.org/wiki/Cross-validation_%28statistics%29#Leave-one-out_cross-validation]. As such, as k becomes larger, it becomes more biased and has less variance. Similarly, larger data sets will direct the bias toward zero.<br /><br />
  
==== Further Reading ====
+
====k × 2 cross-validation====
 +
This is a variation on k-fold cross-validation. For each fold, we randomly assign data points to two sets d0 and d1, so that both sets are equal size (this is usually implemented as shuffling the data array and then splitting in two). We then train on d0 and test on d1, followed by training on d1 and testing on d0.
 +
This has the advantage that our training and test sets are both large, and each data point is used for both training and validation on each fold. In general, k = 5 (resulting in 10 training/validation operations) has been shown to be the optimal value of k for this type of cross-validation.
  
One-item-out: [http://biomet.oxfordjournals.org/content/64/1/29.abstract Asymptotics for and against cross-validation]
+
* One-item-out: [http://biomet.oxfordjournals.org/content/64/1/29.abstract Asymptotics for and against cross-validation]
 +
* [http://www.springerlink.com/content/tfvyva1cqvtqacvy/fulltext.pdf Leave-one-out style crossvalidation bound for Kernel methods applied to some classification and regression problems]
 +
 
 +
=== Matlab Code for Cross Validation ===
 +
1. Generate cross validation index using matlab build-in function 'crossvalind.m'. Click [http://www.mathworks.com/help/toolbox/bioinfo/ref/crossvalind.html here] for details.
 +
 
 +
2. Use 'cvpartition.m' to partition data. Click [http://www.mathworks.com/help/toolbox/stats/cvpartition.html here].
 +
 
 +
=== Further Reading ===
 +
1. Two useful pdf's introducing concepts of cross validation. [http://www.autonlab.org/tutorials/overfit10.pdf] [http://www.autonlab.org/tutorials/overfit10.pdf]
  
 
=== References ===
 
=== References ===
Line 2,545: Line 3,428:
  
 
3. Shao, J. and Tu D. (1995). The Jackknife and Bootstrap. Springer, New York.
 
3. Shao, J. and Tu D. (1995). The Jackknife and Bootstrap. Springer, New York.
 +
 +
4. http://en.wikipedia.org/wiki/Cross-validation_(statistics)
  
 
== Radial Basis Function (RBF) Network  - October 28, 2010==
 
== Radial Basis Function (RBF) Network  - October 28, 2010==
{{Cleanup|date=October 2010|reason= It might be interesting to explain the reason for over fitting in MLP. Based on what we learnt on the class we might think that over-fitting occurs due to complexity in structure but the most common way that occurs in MLP has to do nothing with the structure. In MLP over-fitting occurs to do iterations more than what is needed. Since if you this then weights of neural network get very adapted to training data and the training error rate will decrease, this will cause over-fitting. To avoid this problem training of neural network is stopped heuristically. Making this automatic for MLP's can be a good topic for research! }}
 
{{Cleanup|date=November 2010|reason= Could you please clarify what you mean? }}
 
=== Lecture Summary ===
 
  
* Radial Basis Function (RBF) networks is an artificial neural network consisting of an output layer, a hidden layer, and weights from the hidden layer to the output; it has a closed form solution and can be solved without back-propagation.
 
* During the model selection process, the complexity (the number of neurons in the hidden layer), the basis function, and the basis function parameters are estimated.
 
** A common basis function is the RBF (or Gaussian) function;
 
** Function parameters can be estimated by clustering the data into as many clusters as there are nodes and using the sample mean and variance
 
** The complexity of the model is determined during the training process (methods like K-Fold Cross-Validation or Leave-One-Out Cross Validation can be used)
 
 
[[File:Rbf_net.png|350px|thumb|right|Figure 1: Radial Basis Function Network]]
 
[[File:Rbf_net.png|350px|thumb|right|Figure 1: Radial Basis Function Network]]
  
Line 2,567: Line 3,444:
 
* and no weights from the first layer to the hidden layer.
 
* and no weights from the first layer to the hidden layer.
  
An RBF network can be trained without back propagation since it has a closed-form solution. The neurons in the hidden layer contain basis functions.
+
An RBF network can be trained without back propagation since it has a closed-form solution. The neurons in the hidden layer contain basis functions. A common basis function for  RBF network is a kind of Gaussian function without the scaling factor.
  
* Note: [http://ibiblio.org/e-notes/Splines/Intro.htm Spline], RBF, Fourier, and similar methods differ only in the basis function.<br />  
+
* Note: [http://ibiblio.org/e-notes/Splines/Intro.htm Spline], RBF, [http://www.aaai.org/Papers/Workshops/1999/WS-99-04/WS99-04-008.pdf Fourier], and similar methods differ only in the basis function.<br />  
  
RBFN were first used in solving multivariate interpolation problems and numerical analysis. Their prospect is similar in neural network applications, where the training and query targets are rather continuous. RBFN are artificial neural networks and it can be applied to Regression, Classification and Time series prediction.
+
RBF networks were first used in solving multivariate interpolation problems and in numerical analysis. Their prospect is similar in neural network applications, where the training and query targets are continuous. RBF networks are artificial neural networks and they can be applied to Regression, Classification and Time series prediction.
  
For example, if we consider <math>\,n</math> data points along a one dimensional line and <math>\,m</math> clusters. An RBF network with radial basis (Gaussian) functions will cluster points around the <math>\,m</math> means, <math>\displaystyle\mu_{j}</math>. The other data points will be distributed normally around these centers.
+
For example, if we consider <math>\,n</math> data points along a one dimensional line and <math>\,m</math> clusters. An RBF network with radial basis (Gaussian) functions will cluster points around the <math>\,m</math> means, <math>\displaystyle\mu_{j}</math> for <math>j= 1, ..., m</math>. The other data points will be distributed normally around these centers.
  
* Note: The hidden layer has a variable number of neurons (the optimal number is determined by the training process). As usual the more neurons in the hidden layer, the higher the model complexity.<br />
+
* Note: The hidden layer can have a variable number of basis functions (the optimal number of basis function can be determined using the complexity control techniques discussed in the previous section). As usual, the more basis functions are in the hidden layer, the higher the model complexity will be.<br />
  
=== Model Detail ===
+
RBF networks, K-Means clustering, Probabilistic Neural Networks(PNN) and General Regression Neural Networks(GRNN) are almost the same. The main difference is that PNN/GRNN networks have one neuron for each point in the training file, whereas the number of RBF networks  neurons (basis functions) is not set, and it is usually much less than the number of training points. When the size of the training set is not very large, PNN and GRNN perform well. But for large size data sets RBF will be more useful, since PNN/GRNN are impractical.
  
==== Hidden Layer ====
+
====A brief introduction to the K-means algorithm====
 +
K-means is a commonly applied technique in clustering, which aims to divide <math>\,n</math> observations into <math>\,k</math> groups by computing the distance from each of individual observations to the <math>\,k</math> cluster centers. A typical K-means algorithm can be described as follows:
  
 +
Step1: Select <math>\,k</math> as the number of clusters
  
The hidden layer has <math>\, m</math> neurons, where the optimal number (complexity) is determined by the training process. Each neuron consists of a basis function of an input layer point <math>\underline x_{i}</math> refered to as <math>\,\Phi_{j}(\underline x_{i}) </math> where <math>\, j \in \{1 ... m\}</math> and <math>\, i \in \{1 ... n\}</math>. <br>
+
Step2: Randomly select <math>\,k</math> observations from the <math>\,n</math> observations, to be used as <math>\,k</math> initial centers.  
  
* Note: In the following section, <math>k</math> is the number of outputs, <math>n</math> is the number of data points, and <math>m</math> is the number of hidden units.  If <math>\,k = 1</math>, <math>\,\hat Y</math> and <math>\,W</math> are column vectors. <br>
+
Step3: For each data point from the rest of observations, compute the distance to each of the <math>\,k</math> initial centers and classify it into the cluster with the minimum distance.
  
A common basis function is the RBF (or Gaussian) function: <br>
+
Step4: Obtain updated <math>\,k</math> cluster centers by computing the mean of all the observations in the corresponding clusters.
<math>\Phi_{i}(\hbar x_i) = e^{\frac{-\Vert\underline x_i - \mu_{j}\Vert ^2}{2\gamma_{j}}}</math><br />
 
  
The radii of the RBF functions may be different. The centers and radii can be determined by clustering the data into <math>\, m</math> groups and taking each group mean <math>\,\mu_{j}</math>. <br>
+
Step5: Repeat Step 3 and Step 4 until all of the differences between the old cluster centers and new cluster centers are acceptable.
 +
====Typical Radial Function====
  
:<math>\Phi_{n,m} = \left[ \begin{matrix}
+
Gaussian :
\Phi_{1}(\underline x_{1}) & \Phi_{2}(\underline x_{1}) & \cdots & \Phi_{m}(\underline x_{1}) \\
 
\Phi_{1}(\underline x_{2}) & \Phi_{2}(\underline x_{2}) & \cdots & \Phi_{m}(\underline x_{2}) \\
 
\vdots & \vdots & \ddots & \vdots \\
 
\Phi_{1}(\underline x_{n}) & \Phi_{2}(\underline x_{n}) & \cdots & \Phi_{m}(\underline x_{n})
 
\end{matrix}\right] </math> is the matrix of Radial Basis Functions.
 
  
==== Weights ====
+
<math>\ \phi(r) = e^{- \frac{r^{2}}{2 \sigma^2}} </math>
  
The weights <math>\, w_k</math> used in calculating the output layer can be optimally calculated. Let
+
Hardy Multi-quadratic :
  
:<math>W_{m,k} = \left[ \begin{matrix}
+
<math>\ \phi(r) = \frac{\sqrt{r^2+c^2}}{c} , c>0 </math>
w_{1,1} & w_{1,2} & \cdots & w_{1,k} \\
 
w_{2,1} & w_{2,2} & \cdots & w_{2,k} \\
 
\vdots & \vdots & \ddots & \vdots \\
 
w_{m,1} & w_{m,2} & \cdots & w_{m,k}
 
\end{matrix}\right] </math> be the matrix of weights. <br>
 
  
 +
Hardy Multi-quadratic :
  
==== Output Layer ====
+
<math>\ \phi(r) = \frac{c}{\sqrt{r^2+c^2}} ,  c>0 </math>
  
The output layer can be multi-dimensional.
+
==== Reference for the above paragraph ====
 +
1. Improving the performance of k-means clustering algorithm to position the centers of RBF network
 +
[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.97.548&rep=rep1&type=pdf]
 +
 
 +
2. GA-RBF: A self-optimising RBF network
 +
[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.60.7406&rep=rep1&type=pdf]
 +
 
 +
3. A closer look at the radial basis function (RBF) networks
 +
[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=342544]
 +
 
 +
4. Probabilistic neural networks
 +
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6T08-485RHV9-67&_user=10&_coverDate=12%2F31%2F1990&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1539932149&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=c94cf11ba958be01e865443bf9f3fcb3&searchtype=a]
 +
 
 +
5. A general regression neural network
 +
[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=97934]
 +
 
 +
6. Experience with adaptive probabilistic neural networks and adaptive general regression neural networks
 +
[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=374355]
 +
 
 +
7. Using General Regression and Probabilistic Neural Networks To Predict Human Intestinal Absorption with Topological Descriptors Derived from Two-Dimensional Chemical Structures
 +
[http://pubs.acs.org/doi/abs/10.1021/ci020013r]
 +
 
 +
=== Model Detail ===
 +
==== RBF Network's Hidden Layer ====
 +
 
 +
The hidden layer has <math>\, m</math> neurons, where the optimal number for <math>\, m</math> can be determined using cross validation techniques discussed in the previous section.
 +
For example, if the data is generated from mixture of Gaussian distribution, you can cluster the data and estimate each Gaussian distribution mean and variance by [http://en.wikipedia.org/wiki/Expectation-maximization_algorithm EM algorithm]. Their mean and variance can be used for constructing the basis functions. Each neuron consists of a basis function of an input layer point <math>\underline x_{i}</math> referred to as <math>\,\Phi_{j}(\underline x_{i}) </math> where <math>\, j \in \{1 ... m\}</math> and <math>\, i \in \{1 ... n\}</math>. <br>
 +
 
 +
* Note: In the following section, <math>k</math> is the number of outputs, <math>n</math> is the number of data points, and <math>m</math> is the number of hidden units.  If <math>\,k = 1</math>, <math>\,\hat Y</math> and <math>\,W</math> are column vectors. <br>
 +
 
 +
A common basis function is the radial basis Gaussian function: <br>
 +
<math>\Phi_{j}(\underline x_i) = e^{\frac{-\Vert\underline x_i - \mu_{j}\Vert ^2}{2\gamma_{j}}}</math><br />
 +
 
 +
* Note: An RBF function <math>\Phi</math> is a real-valued function whose value depends only on the distance from a centre <math>\underline c</math>, such that <math>\Phi(\underline x,\underline c) = \Phi(\|\underline x - \underline c \|)</math>. Other commonly used radial basis functions are Multiquadric, Polyharmonic spline, and Thin plate spline.
 +
 
 +
 
 +
:<math>\Phi_{n,m} = \left[ \begin{matrix}
 +
\Phi_{1}(\underline x_{1}) & \Phi_{2}(\underline x_{1}) & \cdots & \Phi_{m}(\underline x_{1}) \\
 +
\Phi_{1}(\underline x_{2}) & \Phi_{2}(\underline x_{2}) & \cdots & \Phi_{m}(\underline x_{2}) \\
 +
\vdots & \vdots & \ddots & \vdots \\
 +
\Phi_{1}(\underline x_{n}) & \Phi_{2}(\underline x_{n}) & \cdots & \Phi_{m}(\underline x_{n})
 +
\end{matrix}\right] </math> is the matrix of Radial Basis Functions.
 +
 
 +
==== Weights ====
 +
 
 +
The weights <math>\, w_k</math> used in calculating the output layer can be optimally calculated. Let
 +
 
 +
:<math>W_{m,k} = \left[ \begin{matrix}
 +
w_{1,1} & w_{1,2} & \cdots & w_{1,k} \\
 +
w_{2,1} & w_{2,2} & \cdots & w_{2,k} \\
 +
\vdots & \vdots & \ddots & \vdots \\
 +
w_{m,1} & w_{m,2} & \cdots & w_{m,k}
 +
\end{matrix}\right] </math> be the matrix of weights. <br>
 +
 
 +
 
 +
==== Output Layer ====
 +
 
 +
The output layer can be multi-dimensional.
  
 
:<math>Y_{n,k} = \left[ \begin{matrix}
 
:<math>Y_{n,k} = \left[ \begin{matrix}
Line 2,642: Line 3,569:
 
Assume that given data <math>\, D=\{x_i, y_i\} </math> for <math>\, i \in \{1 ... n\}</math>,<br>
 
Assume that given data <math>\, D=\{x_i, y_i\} </math> for <math>\, i \in \{1 ... n\}</math>,<br>
 
<math>\, y_i = f(x_i) + \epsilon_i </math><br>
 
<math>\, y_i = f(x_i) + \epsilon_i </math><br>
<math> \, \epsilon</math> is what essentially contributes to the complexity of the model. If there were no noise then model selection would be trivial since there would exist many functions of various degrees of complexity that would perfectly fit the data. We assume that <math>\, \epsilon_i \sim N(0, \sigma^2)</math>.<br>
+
<math> \, \epsilon</math> is what essentially contributes to the complexity of the model. If there were no noise then model selection would be trivial since there would exist many functions of various degrees of complexity that would perfectly fit the data. We assume that <math> \, \epsilon</math> is an additive Gaussian noise: <math>\, \epsilon_i \sim N(0, \sigma^2)</math>.<br>
 
<math> \, err = E[(y - \hat y)^2]</math><br>
 
<math> \, err = E[(y - \hat y)^2]</math><br>
 
<math> \,= E[(f(x) + \epsilon - \hat f(x))^2]</math><br>
 
<math> \,= E[(f(x) + \epsilon - \hat f(x))^2]</math><br>
 
<math> \,= E[(f(x) - \hat f(x))^2 + \epsilon^2 - 2\epsilon(f(x) - \hat f(x))]</math><br>
 
<math> \,= E[(f(x) - \hat f(x))^2 + \epsilon^2 - 2\epsilon(f(x) - \hat f(x))]</math><br>
The part of the error term we want to approximate is <math>\, E[(f(x) - \hat f(x))^2] </math>. We will try to estimate this by finding the other terms of the above expression.
+
The part of the error term we want to approximate is <math>\, E[(f(x) - \hat f(x))^2] </math>. We will try to estimate this by finding the other terms of the above expression. See lecture titled "Model Selection for an RBF network", November 2, 2010, below.
  
 
==== Conceptualizing RBF Networks ====
 
==== Conceptualizing RBF Networks ====
Line 2,664: Line 3,591:
 
An Example of RBF Networks [http://reference.wolfram.com/applications/neuralnetworks/ApplicationExamples/12.1.2.html]
 
An Example of RBF Networks [http://reference.wolfram.com/applications/neuralnetworks/ApplicationExamples/12.1.2.html]
  
== '''Model Selection (Stein's Unbiased Risk Estimator) - November 2nd, 2010''' ==
+
This paper suggests an objective approach in determining proper samples to find good RBF networks with respect to accuracy[http://www.wseas.us/e-library/conferences/2009/hangzhou/MUSP/MUSP41.pdf].
 +
 
 +
=====Improvement for RBF Neural Networks Based on Cloud Theory=====
 +
Base on cloud theory, an improved algorithm for RBF neural networks was introduced to transfer the problem of determining the center and its corresponding bandwidth of cluster of RBF to determine the parameters of normal cloud model in order to make the output of each of hidden layers having vague and random properties and the randomness of each of data are kept and passed to the output layer. The conclusion shows that the improved algorithm is superior to the classical RBF in prediction and the actual result is well. Simultaneously, the improved algorithm can be transplanted to the improvement of RBF neural networks algorithms. For more information, see Lingfang Sun, Shouguo Wang, Ce Xu, Dong Ren, Jian Zhang, Research on the improvement for RBF neural networks based on cloud theory, Proceedings of the World Congress on Intelligent Control and Automation (WCICA), pp. 3110-3113, 2008.
 +
 
 +
=== Radial Basis Approximation Implementation ===
 +
 
 +
% This code uses the built-in NEWRB MATLAB function to create a radial basis network that
 +
% approximates a function defined by a set of data points.
 +
 
 +
% Define 21 inputs P and associated targets T.
 +
 
 +
P = -1:.1:1;
 +
T = [-.9602 -.5770 -.0729  .3771  .6405  .6600  .4609 ...
 +
      .1336 -.2013 -.4344 -.5000 -.3930 -.1647  .0988 ...
 +
      .3072  .3960  .3449  .1816 -.0312 -.2189 -.3201];
 +
plot(P,T,'+');
 +
title('Training Vectors');
 +
xlabel('Input Vector P');
 +
ylabel('Target Vector T');
 +
 
 +
% We would like to find a function which fits the 21 data points.  One way to do this is with a radial % basis network. A radial basis network is a network with two layers.  A hidden layer of radial basis  % neurons and an output layer of linear neurons.  Here is the radial basis transfer function used by  % the hidden layer.
 +
 
 +
p = -3:.1:3;
 +
a = radbas(p);
 +
plot(p,a)
 +
title('Radial Basis Transfer Function');
 +
xlabel('Input p');
 +
ylabel('Output a');
 +
 
 +
% The weights and biases of each neuron in the hidden layer define the position
 +
% and width of a radial basis function.  Each linear output neuron forms a
 +
% weighted sum of these radial basis functions.  With the correct weight and
 +
% bias values for each layer, and enough hidden neurons, a radial basis network
 +
% can fit any function with any desired accuracy.  This is an example of three
 +
% radial basis functions are scaled and summed to produce a function
 +
 
 +
a2 = radbas(p-1.5);
 +
a3 = radbas(p+2);
 +
a4 = a + a2*1 + a3*0.5;
 +
plot(p,a,'b-',p,a2,'b--',p,a3,'b--',p,a4,'m-')
 +
title('Weighted Sum of Radial Basis Transfer Functions');
 +
xlabel('Input p');
 +
ylabel('Output a');
 +
 
 +
[[File:rbf1.jpg]]
 +
 
 +
% The function NEWRB quickly creates a radial basis network which approximates
 +
% the function defined by P and T.  In addition to the training set and targets,
 +
% NEWRB takes two arguments, the sum-squared error goal and the spread constant.
 +
 
 +
eg = 0.02; % sum-squared error goal
 +
sc = 1;    % spread constant
 +
net = newrb(P,T,eg,sc);
 +
 
 +
% To see how the network performs, replot the training set.  Then simulate the
 +
% network response for inputs over the same range.  Finally, plot the results on
 +
% the same graph.
 +
 
 +
plot(P,T,'+');
 +
xlabel('Input');
 +
X = -1:.01:1;
 +
Y = sim(net,X);
 +
hold on;
 +
plot(X,Y);
 +
hold off;
 +
legend({'Target','Output'})
 +
 
 +
=== Linear Basis Network ===
 +
A piece-wise linear trajectory maybe modeled more easily and meaningfully using linear bases, rather than Gaussians. To replace Gaussian with linear, one may need to replace the clustering algorithm and modify the basis function. C-varieties algorithm developed by H.H.Bock and J.C.Bezdek is a good way for finding linear clusters (for details on this algorithm you may want to refer to the Fuzzy Cluster Analysis, by Frank Hoeppner et al). And then the resulting linear functions should be accompanied with some Gaussian functions, or any other localizing functions to localize the working area of each one of the linear clusters. The combination is a general function approximator, just like the RBF, NN, etc.
 +
 
 +
<math>\begin{align}\hat{y}=\sum_{i=1}^{P}f_i(x)\Phi_i(x) \end{align}</math>
 +
 
 +
Where in this equation, <math>\ f_i(x)</math> is a linear function, coming the clustering stage and corresponding to the ith cluster and <math>\ \Phi_i</math> is the validity or localizing function, corresponding to the ith cluster.
 +
 
 +
== '''Model Selection for RBF Network (Stein's Unbiased Risk Estimator) - November 2nd, 2010''' ==
  
 
===Model Selection===
 
===Model Selection===
Line 2,671: Line 3,673:
  
  
However, training error and testing error do not demonstrate a linear relationship. In particular, a smaller training error does not necessarily result in a smaller testing error. In practice, one often observes that up to a certain point the model error on testing data tends to decrease as the training error decreases. However, if one attempts to decrease the training error too much by increasing the model complexity, the testing error often can take a dramatic turn and begin to increase. This was explained and a graphic provided in the lecture on October 26th concerning complexity control.  
+
However, training error and testing error do not demonstrate a linear relationship. In particular, a smaller training error does not necessarily result in a smaller testing error. In practice, one often observes that up to a certain point the model error on testing data tends to decrease as the training error decreases. However, if one attempts to decrease the training error too much by increasing the model complexity, the testing error often can take a dramatic turn and begin to increase. This behavior was explained and related figures illustrating this concept were provided in the lecture on complexity control on October 26th.  
  
[[File:data_noise.jpg|500px|thumb|right|Figure 1. Data sampled from a smooth function (in black) cannot be over-fit. Data sampled from a smooth function with noise (in red) can be over-fit when the noise is modelled along with the smooth function.]]
+
[[File:data_noise.jpg|500px|thumb|right|Figure 1. Data sampled from a smooth function (in black) cannot be over-fit. Data sampled from a smooth function with noise (in red) can be over-fit when the noise is modeled along with the smooth function.]]
  
The basic reason behind this phenomenon of the training and testing errors is that in the process of minimizing training error, after a certain point, the model begins to over-fit the training set. Over-fitting in this context means fitting the model to the training data at the expense of losing generality. As seen in Figure 1, the red data points have been over-fit as the general form of the underlying smooth function has been lost in the red-curve model. In the extreme case, a set of <math>\displaystyle N</math> training data points can be modeled exactly with <math>\displaystyle N</math> radial basis functions. Such a model will fit the training data set perfectly. However, the perfectly-fit model fails to be as accurate or perform as well on the training data set because it has modelled not only the true function <math>\displaystyle f(X)</math> but the random noise as well, and thus has over-fit the data (as the red curve in Figure 1 has done). It is interesting to note that in the case of no noise, over-fitting will not occur and hence the complexity of the model can be increased without bound. However, this is not realistic in practice as random noise is almost always present in the data.  
+
The basic reason behind this phenomenon of the training and testing errors is that in the process of minimizing training error, after a certain point, the model begins to over-fit the training set. Over-fitting in this context means fitting the model to the training data at the expense of losing generality. As seen in Figure 1, the red data points have been over-fit as the general form of the underlying smooth function has been lost in the red-curve model. In the extreme case, a set of <math>\displaystyle N</math> training data points can be modeled exactly with <math>\displaystyle N</math> radial basis functions. Such a model will fit the training data set perfectly. However, the perfectly-fit model fails to be as accurate or perform as well on the training data set because it has modeled not only the true function <math>\displaystyle f(X)</math> but the random noise as well, and thus has over-fit the data (as the red curve in Figure 1 has done). It is interesting to note that in the case of no noise, over-fitting will not occur and hence the complexity of the model can be increased without bound. However, this is not realistic in practice as random noise is almost always present in the data.  
  
In general, the training error rate will be less than the testing error on the new data. A model typically adapts to the training data, and hence the training error will be an overly optimistic estimate of the testing error. An obvious way to estimate testing error is to add a penalty term to the training error to compensate for the difference. SURE is developed based on this idea.
+
All in all, we can expect the training error will be an overly optimistic estimate of the testing error. An obvious way to estimate testing error is to add a penalty term to the training error to compensate for the difference. SURE, a technique developed by Charles Stein, a professor of statistics at Stanford University, is based on this idea.
  
 
===Stein's unbiased risk estimate (SURE)===
 
===Stein's unbiased risk estimate (SURE)===
  
 +
Stein's unbiased risk estimate (SURE) is an unbiased estimator of the mean-squared error of a given estimator in a deterministic estimation scenario. In other words, it provides an indication of the accuracy of a given estimator. This is important since, in deterministic estimation, the true mean-squared error of an estimator generally depends on the value of the unknown parameter, and thus cannot be determined completely. A standard application of SURE is to choose a parametric form for an estimator, and then optimize the values of the parameters to minimize the risk estimate.
  
Note that the material presented here is applicable to model selection in general, and is not specific to RBF networks.  
+
Stein’s unbiased risk estimation (SURE) theory gives a rigorous definition of the degrees of freedom for any fitting procedure. [http://www.ams.org/mathscinet-getitem?mr=0630098]. For more information about the relation between Stein's unbiased risk estimator and Stein's lemma refer to[http://www.cc.gatech.edu/~lebanon/notes/sure.pdf]. The following is the description of Stein's lemma and its use to derive Stein's unbiased risk estimator (SURE).
 +
 
 +
Note that the material presented here is applicable to model selection in general, and is not specific to RBF networks.
 +
 
 +
===Applications of Stein's unbiased risk estimate===
 +
A standard application of SURE is to choose a parametric form for an estimator, and then optimize the values of the parameters to minimize the risk estimate. This technique has been applied in several settings. For example, a variant of the James–Stein estimator[http://en.wikipedia.org/wiki/James%E2%80%93Stein_estimator] can be derived by finding the optimal shrinkage estimator. The technique has also been used by Donoho and Johnstone to determine the optimal shrinkage factor in a wavelet denoising setting [http://www.jstor.org/sici?sici=0162-1459%28199512%2990%3A432%3C1200%3AATUSVW%3E2.0.CO%3B2-K].
 +
SURE has also been used for optical flow estimation by Mingren Shi [http://www.sci.usq.edu.au/research/seminars/files//seminar1/OpSureTalk.pdf].
  
 
====Important Notation [http://en.wikipedia.org/wiki/Stein's_unbiased_risk_estimate]====
 
====Important Notation [http://en.wikipedia.org/wiki/Stein's_unbiased_risk_estimate]====
Line 2,775: Line 3,784:
  
 
This means that equation <math>\displaystyle (1)</math> now becomes, for one data point:
 
This means that equation <math>\displaystyle (1)</math> now becomes, for one data point:
<math>\displaystyle E[(\hat y_i-y_i)^2 ]=E[(\hat f_i-f_i)^2]+\sigma^2-2\sigma^2E\left[\frac {\partial \hat f}{\partial y_i}\right]</math>.
+
<math>\displaystyle E[(\hat y-y)^2 ]=E[(\hat f-f)^2]+\sigma^2-2\sigma^2E\left[\frac {\partial \hat f}{\partial y}\right]</math>.
  
  
Line 2,793: Line 3,802:
  
 
Consider only one node of the network. In this case we can write:
 
Consider only one node of the network. In this case we can write:
<math>\hat f=\,H_{i1}y_1+\,H_{i2}y_2+\cdots+\,H_{ii}y_i+\cdots+\,H_{in}y_n</math>.
+
<math>\hat f_i=\,H_{i1}y_1+\,H_{i2}y_2+\cdots+\,H_{ii}y_i+\cdots+\,H_{in}y_n</math>.
 
Denote this as equation <math>\displaystyle (4)</math>.
 
Denote this as equation <math>\displaystyle (4)</math>.
  
Line 2,804: Line 3,813:
  
 
Here we recall that <math>\sum_{i=1}^n \,H_{ii}= \,Trace(H)</math>, the sum of the diagonal elements of <math>\,H</math>. Using the permutation property of the trace function we can further simplify the expression as follows:
 
Here we recall that <math>\sum_{i=1}^n \,H_{ii}= \,Trace(H)</math>, the sum of the diagonal elements of <math>\,H</math>. Using the permutation property of the trace function we can further simplify the expression as follows:
<math>\,Trace(H)= Trace(\Phi(\Phi^{T}\Phi)^{-1}\Phi^{T})= Trace(\Phi^{T}\Phi(\Phi^{T}\Phi)^{-1})=m</math>, where <math>\displaystyle m</math> is the number of basis functions in the RBF network (and hence <math>\displaystyle \Phi</math> has dimension <math>\displaystyle n \times m</math>).  
+
<math>\,Trace(H)= Trace(\Phi(\Phi^{T}\Phi)^{-1}\Phi^{T})= Trace(\Phi^{T}\Phi(\Phi^{T}\Phi)^{-1})=m</math>, by the trace cyclical permutation property, where <math>\displaystyle m</math> is the number of basis functions in the RBF network (and hence <math>\displaystyle \Phi</math> has dimension <math>\displaystyle n \times m</math>).<br>
  
 +
====Sketch of Trace Cyclical Property Proof:====
 +
For <math>\, A_{mn}, B_{nm}, Tr(AB) = \sum_{i=1}^{n}\sum_{j=1}^{m}A_{ij}B_{ji} = \sum_{j=1}^{m}\sum_{i=1}^{n}B_{ji}A_{ij} = Tr(BA)</math>.<br>
 +
With that in mind, for <math>\, A_{nn}, B_{nn} = CD, Tr(AB) = Tr(ACD) = Tr(BA)</math> (from above) <math>\, = Tr(CDA)</math>.<br><br>
  
 
Note that since <math>\displaystyle \Phi</math> is a projection of the input matrix <math>\,X</math> onto a basis set spanned by <math>\,m</math>, the number of basis functions, that sometimes an extra <math>\displaystyle \Phi_0</math> term is included without any input to represent the intercept of a fitted model. In this case, if considering an intercept, then <math>\,Trace(H)= m+1</math>.
 
Note that since <math>\displaystyle \Phi</math> is a projection of the input matrix <math>\,X</math> onto a basis set spanned by <math>\,m</math>, the number of basis functions, that sometimes an extra <math>\displaystyle \Phi_0</math> term is included without any input to represent the intercept of a fitted model. In this case, if considering an intercept, then <math>\,Trace(H)= m+1</math>.
  
  
Substituing <math>\sum_{i=1}^n \,H_{ii} = m+1</math> into equation <math>\displaystyle (3)</math> gives the following:
+
Substituting <math>\sum_{i=1}^n \,H_{ii} = m+1</math> into equation <math>\displaystyle (3)</math> gives the following:
 
<math>\displaystyle Err=err-n\sigma^2+2\sigma^2(m+1)</math>.
 
<math>\displaystyle Err=err-n\sigma^2+2\sigma^2(m+1)</math>.
  
  
 
Computationally, to obtain an estimate for the true error <math>\displaystyle Err</math> the above expression is repeatedly evaluated beginning at <math>\displaystyle m = 1 </math>, then at <math>\displaystyle m = 2 </math>, then <math>\displaystyle m = 3 </math>, and so on until the minimum value for <math>\displaystyle Err</math> is determined. The value of ''m'' that gives the minimum true error estimate is the optimal number of basis functions to be implemented in the RBF network, and hence is also the optimal degree of complexity of the model.
 
Computationally, to obtain an estimate for the true error <math>\displaystyle Err</math> the above expression is repeatedly evaluated beginning at <math>\displaystyle m = 1 </math>, then at <math>\displaystyle m = 2 </math>, then <math>\displaystyle m = 3 </math>, and so on until the minimum value for <math>\displaystyle Err</math> is determined. The value of ''m'' that gives the minimum true error estimate is the optimal number of basis functions to be implemented in the RBF network, and hence is also the optimal degree of complexity of the model.
 +
 +
=== Lecture Summary ===
 +
 +
Stein's unbiased risk estimate (SURE) is an unbiased estimator of the mean-squared error of a given estimator, in a deterministic estimation scenario. It provides an indication of the accuracy of a given estimator.
 +
 +
In RBF network, the problem of selecting the appropriate number of basis functions is a critical issue. An RBF network with an overly restricted basis gives poor predictions on new data. But if an RBF network with too many basis functions, it also gives poor generalization performance.
 +
 +
This lecture introduce a criterion for selecting the number of radial basis functions in an RBF network, using the generalization of Stein’s unbiased risk estimator (SURE).
 +
 +
====Reference:====
 +
 +
* Automatic basis selection for RBF networks using Stein’s unbiased risk estimator [http://www.google.ca/url?sa=t&source=web&cd=2&sqi=2&ved=0CB4QFjAB&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.5.5344%26rep%3Drep1%26type%3Dpdf&rct=j&q=Stein%27s%20Unbiased%20Risk%20Estimator%29%20RBF&ei=YsHSTKzgDYausAO-4IWrCw&usg=AFQjCNHO9oFBQ6tljsEqdLOjFgtiQz9gxQ&sig2=Cx9Sh0Uk-h8pDgihKkU_HA&cad=rja.pdf]
 +
* J. Moody and C. J. Darken, "Fast learning in networks of locally tuned processing units," Neural Computation, 1, 281-294 (1989). Also see [http://www.ki.inf.tu-dresden.de/~fritzke/FuzzyPaper/node5.html Radial basis function networks according to Moody and Darken]
 +
* T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE 78(9), 1484-1487 (1990).
 +
* Roger D. Jones, Y. C. Lee, C. W. Barnes, G. W. Flake, K. Lee, P. S. Lewis, and S. Qian, [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=137644 Function approximation and time series prediction with neural networks], Proceedings of the International Joint Conference on Neural Networks, June 17-21, p. I-649 (1990).
 +
* John R. Davies, Stephen V. Coggeshall, Roger D. Jones, and Daniel Schutzer, "Intelligent Security Systems,"
 +
* S. Chen, C. F. N. Cowan, and P. M. Grant, "Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks", IEEE Transactions on Neural Networks, Vol 2, No 2 (Mar) 1991.
  
 
====Further Reading:====
 
====Further Reading:====
Line 2,820: Line 3,849:
 
From Stein's unbiased risk estimates to the method of generalized cross validation [http://www.jstor.org/pss/2241359]
 
From Stein's unbiased risk estimates to the method of generalized cross validation [http://www.jstor.org/pss/2241359]
  
(This paper concerns the method of generalized cross validation (GCV), based on Stein estimates and the associated unbiased risk estimates.)
+
''(This paper concerns the method of generalized cross validation (GCV), based on Stein estimates and the associated unbiased risk estimates.)''
  
 
Adaptive denoising based on SURE risk [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=720560]
 
Adaptive denoising based on SURE risk [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=720560]
  
(In this paper, a new adaptive denoising method is presented based on Stein's (1981) unbiased risk estimate (SURE) and on a new class of thresholding functions.)
+
''(In this paper, a new adaptive denoising method is presented based on Stein's (1981) unbiased risk estimate (SURE) and on a new class of thresholding functions.)''
  
 
Wavelet shrinkage denoising using the non-negative garrote [http://www.jstor.org/pss/1390677]
 
Wavelet shrinkage denoising using the non-negative garrote [http://www.jstor.org/pss/1390677]
  
 
Estimation of the Mean of a Multivariate Normal Distribution [http://www.jstor.org/pss/2240405]
 
Estimation of the Mean of a Multivariate Normal Distribution [http://www.jstor.org/pss/2240405]
 +
 +
=====Generalized SURE for Exponential Families=====
 +
As we know, Stein’s unbiased risk estimate (SURE) is limited to be applied for the independent, identically distributed (i.i.d.) Gaussian model. However, in some recent work, some researchers tried to work on obtaining a SURE counterpart for general, instead of deriving estimate by dominating least-squares estimation, and this technique made SURE extend its application to a wider area. In 2009, Yonina C. Eldar from Department of Electrical Engineering Technion, Israel Institute of Technology published her paper, in which a new method for choosing regularization parameters in penalized LS estimators was introduced to design estimates without predefining their structure and its application can be proved to have superior performance over the conventional generalized cross validation and discrepancy approaches in the context of image deblurring and deconvolution. For more information, see Yonina C. Eldar, Generalized SURE for Exponential Families: Applications to Regularization, IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 2, FEBRUARY 2009.
 +
 +
== '''Regularization for Neural Network - November 4, 2010'''==
 +
 +
Large weights in a neural network can hurt generalization in two different ways. First, excessively large weights leading to hidden units can cause the output function to be too rough, possibly with near discontinuities. Second, excessively large weights leading to output units can cause wild outputs far beyond the range of the data if the output activation function is not bounded to the same range as the data. To put it another way, large weights can cause excessive variance of the output. The effort to reduce the size of these weights is called regularization.
 +
 +
Training of the weights in a neural network is usually accomplished by iteratively developing them from some regularized set of small initial values.  The weights tend to increase in absolute values as training proceeds.  When neural networks were first developed, weights were prevented from getting too large by simply stopping the training session early; to determine when to stop training the neural network, a set of test data was used to detect overfitting. Using this method, the stopping point would be determined by finding the length of training time that results in minimal classification error for the test set.  However, in this section, a somewhat different for regularization method is presented that does not require the training session to be terminated early; rather, this method directly penalizes overfitting in the optimization calculation.
 +
 +
=== ''' Weight decay'''===
 +
Weight decay is a subset of regularization methods, which aim to prevent overfitting in model selection. The penalty term in
 +
weight decay, by definition, penalizes large weights. Other regularization
 +
methods may involve not only the weights but various derivatives of the
 +
output function [http://research.microsoft.com/en-us/um/people/cmbishop/nnpr.htm].
 +
The weight decay penalty term causes the weights to converge to smaller
 +
absolute values than they otherwise would.
 +
 +
[[File:figure 2.png|350px|thumb|right|Figure 3: activation function]]
 +
Weight decay training is suggested as a method useful in achieving a robust [http://en.wikipedia.org/wiki/Neural_network neural network] which is insensitive to noise. Since the number of hidden layers in a NN is usually decided by certain domain knowledge, it may easily get into the problem of overfitting.
 +
 +
The weight–decay method is an effective way to improve the generalization ability of neural networks. In general, the trained weights are constrained to be small when the weight-decay method is applied. Large weights in the output layer can cause outputs that are far beyond the range of the data (when test data is used); in other words, large weights can result in high output variance.
 +
 +
It can be seen from Figure 3 that when the weight is in the vicinity of zero, the operative part of the activation function shows linear behavior. That is, the operative part of a sigmoid function is almost linear for small weights. The NN then collapses to an approximately linear model. Note that a linear model is the simplest model, and we can avoid overfitting by constraining the weights to be small. This gives us a hint on why we initialize the random weights to be close to zero. If the weights are large, the model is more complex and the activation function tends to be nonlinear.
 +
 +
Note that it is not necessarily bad to go to the nonlinear section of the activation function. In fact, we use nonlinear activation functions to increase the ability of neural networks and make it possible to estimate nonlinear functions. What we must avoid is using the nonlinear section more than required, which would result in overfitting of the training data. To achieve this we add a penalty term to the error function.
 +
 +
The usual penalty is the sum of squared weights times a decay constant. In a linear model, this form of weight decay is equivalent to ridge regression [http://komarix.org/ac/papers/thesis/thesis_html/node15.html]. Now the regularized error function becomes:
 +
 +
 +
<math>\,REG = err + \lambda( \sum_{ij}|u_{ij}|^2)</math>, where <math>\,err</math> is the original error in back-propagation;and it decreases all the time; <math>\,u_{ij}</math> is the weights of the hidden layers.
 +
 +
Usually, we use <math>\,\lambda( \sum_{ij}|u_{ij}|^2)</math> to control the value of the weights. We can use cross validation to estimate <math>\,\lambda</math>.Another approach to choosing the <math>\,\lambda</math> is to train several networks with different amounts of decay and estimates the generalization error for each; then choose the <math>\,\lambda</math> that minimizes the estimated generalization error.
 +
 +
 +
A similar penalty, weight elimination, is given by,
 +
 +
<math>\,REG = err + \lambda(\sum_{jk}\frac{|u_{jk}|^2}{1+|u_{jk}|^2})</math>.
 +
 +
As in back-propagation, we take partial derivative with respect to the weights:
 +
 +
<math>\frac{\partial REG}{\partial u_{ij}} = \frac{\partial err}{\partial u_{ij}} + 2\lambda u_{ij}</math>
 +
 +
<math>u^{new} \leftarrow u^{old} - \rho\left(\frac{\partial err}{\partial u} + 2\lambda u^{old}\right)</math>
 +
 +
 +
To conclude, the weight decay penalty term lead the weights to converge to smaller
 +
absolute values than they otherwise would. Large weights can effect generalization negatively in two different ways. Excessively large weights leading to
 +
hidden units can cause the output function to be too rough, possibly with
 +
near discontinuities. Excessively large weights leading to output units can
 +
cause wild outputs far beyond the range of the data if the output activation
 +
function is not bounded to the same range as the data. In another words, large weights can cause large variance of the output [http://portal.acm.org/citation.cfm?id=148062]. According to [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.2302], the size (L1-
 +
norm) of the weights is more important than the number of weights in
 +
determining generalization.
 +
 +
 +
Note:<br />
 +
here <math>\,\lambda</math> serves as a trade-off parameter, tuning between the error rate and the linearity. Actually, we may also set <math>\,\lambda</math> by cross-validation.  The tuning parameter is important since weights of zero will lead to zero derivatives and the algorithm will not change.  On the other hand, starting with weights that are too large means starting with a nonlinear model which can often lead to poor solutions. <ref>Trevor Hastie, Robert Tibshirani, Jerome Friedman, Elements of Statistical Learning (Springer 2009) pp.398</ref><br />
 +
We can standardize or normalize the inputs and targets, or adjust the penalty term for the standard deviations of all the inputs and targets in order to omit the biases and get good result from weight decay.<br />
 +
<math>\,\lambda</math>is different for different types of weights in the NN. We can have different <math>\,\lambda</math> for input-to-hidden, hidden-to-hidden, and hidden-to-output weights.
 +
 +
The following Matlab code implements a neural network with weight decay using the backpropagation method:
 +
 +
    %initialize
 +
    m;         %features
 +
    n;         %datapoints
 +
    nodes; %number of hidden nodes
 +
    u1=rand(m,nodes)-0.5; %input-to-hidden weights
 +
    u2=rand(nodes,1)-0.5; %hidden-to-output weights
 +
    ro; %learning rate
 +
    weightPenalty;
 +
   
 +
    Z_output=zeros(n,1);
 +
    Z=zeros(nodes,n);
 +
   
 +
    %% TRAIN DATA
 +
    for epoch=1:100
 +
   
 +
        for i=1:n
 +
            %% Forward Pass
 +
            %determine inputs to hidden layer
 +
            A=u1'*training_X(:,i);
 +
           
 +
        %apply activation function to hidden layer weighted inputs
 +
            for j=1:nodes
 +
                Z(j,i)=1/(1+exp(-A(j)));
 +
            end
 +
               
 +
        %apply weights to get fitted outputs;
 +
            Z_output(i,1) = u2'*Z(:,i);
 +
           
 +
            %% Backward Pass
 +
            %output delta
 +
            delta_O = -2*(training_Y(i)-Z_output(i,1));
 +
               
 +
        %tweak the hidden-output weights
 +
            for j=1:nodes
 +
                u2(j)=u2(j)-ro*(delta_O*Z(j)+2*weightPenalty*u2(j));
 +
            end
 +
            for j=1:nodes
 +
                sigmaPrime=exp(-A(j))/(1+exp(-A(j)))^2;
 +
                delta_H = sigmaPrime*delta_O*u2(j);
 +
                u1(:,j)=u1(:,j)-ro*(delta_H*training_X(:,i)+2*weightPenalty*u1(:,j));
 +
            end
 +
        end
 +
        yhat(:,1) = Z_output(:,1) > 0.5;
 +
    end
 +
 +
 +
For more reading about the effect of weight decay training for backpropagation on noisy data sets please refer to [http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6T08-3TYVWK9-F-P&_cdi=4856&_user=1067412&_pii=S089360809800046X&_origin=search&_coverDate=08%2F31%2F1998&_sk=999889993&view=c&wchp=dGLbVzW-zSkzS&md5=52846ec8e0ba54b28000ef1de34c7bc5&ie=/sdarticle.pdf] and how weight decay can improve generalization in feed forward network refer to [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.4221]
 +
 +
 +
A fundamental problem with weight decay is that different types of weights
 +
in the network will usually require different decay constants for good
 +
generalization. At the very least, you need three different decay constants
 +
for input-to-hidden, hidden-to-hidden, and hidden-to-output weights.
 +
Adjusting all these decay constants to produce the best estimated
 +
generalization error often requires vast amounts of computation.
 +
 +
Fortunately, there is a superior alternative to weight decay: hierarchical
 +
Bayesian learning. Bayesian learning makes it possible to estimate
 +
efficiently numerous decay constants.
 +
 +
Weight decay is proposed to reduce overfitting as it often
 +
appears in the learning tasks of artificial neural networks. For example, in [http://www.springerlink.com/content/f21781218007l750/fulltext.pdf] weight decay is applied to a well
 +
defined model system based on a single
 +
layer perceptron, which exhibits strong overfitting. Since the optimal
 +
non-overfitting solution is known for this system, they have compared the effect
 +
of the weight decay with this solution. A strategy to find the optimal
 +
weight decay strength is proposed, which leads to the optimal solution
 +
for any number of examples.
 +
 +
 +
====Methods to estimate the weight decay parameter====
 +
One of the biggest problems in weight decay regularization of neural networks is how to estimate its parameter. There are many ways proposed in the literature to estimate the weight decay parameter.
 +
 +
Typically,the weight decay parameter is set between 0.001 and 0.1  that is based on that is based on network training .An inappropriate estimate of the decay parameter may cause over-fitting or over smoothing . Determining the correct value of the parameter is a very tedious process which needs a lot of trial and error . Typically, the optimal value of the weight decay is determined by training the network many times .That is, performing network training based on the same set of initial weights ,same network configuration with fixed  number of neutral layers , and fit the network with various weight decay parameters . Then determine the optimal value of weight decay values by the smallest generalization error.
 +
 +
The following papers are good start for some one who is looking for further reading.
 +
 +
1- On the selection of weight decay parameter for faulty networks [http://portal.acm.org/citation.cfm?id=1862025 here]
 +
 +
2- A Simple Trick for Estimating the Weight Decay Parameter [http://www.springerlink.com/content/0889d07ufuwgql03/ here]
 +
 +
===Regularization invariant under transformation===
 +
 +
A major drawback of the simple weight decay is that it is inconsistent with regard to transformations done to the input variables and / or the target variables of the training data. This fact is remarkably easy to show. For the interested reader, a simple derivation of it is available on page 8 of Sargur Srihari's [http://www.cedar.buffalo.edu/~srihari/CSE574/Chap5/Chap5.5-Regularization.pdf ''Regularization in Neural Networks''] slides.
 +
 +
It is desirable for regularization to have the property of being invariant under transformation of the inputs and / or the outputs of the training data. This is so that, if one is given a set of training data and uses regularization during training, and trains one network after transforming the training data using one transformation and trains another network after transforming the training data using another transformation, then the two solutions represented by the two trained networks should only differ from each other with regard to the weights as given so that neither solution would be arbitrarily favored over the other. 
 +
 +
Many approaches have been devised so that, when regularization is used during the training process of a network, the resulting predictions would be invariant under any transformation(s) made to the input variable(s). One such approach is to add a regularization term to the error function that serves to penalize any possible changes to the outputs resulting from any transformation(s) applied to the inputs. A common example of this approach is [http://arts.uwaterloo.ca/~cnrglab/?q=system/files/tangent_prop.pdf tangent propagation], which is described in Sargur Srihari's [http://www.cedar.buffalo.edu/~srihari/CSE574/Chap5/Chap5.5-Regularization.pdf slides] and which is discussed in detail in Simard ''et al.'''s 1998 [http://yann.lecun.com/exdb/publis/pdf/simard-98.pdf paper] regarding transformation invariance. Several other approaches are also described in Sargur Srihari's slides. 
 +
 +
=== Other Alternatives for NN Regularization ===
 +
As enumerated before, there are some drawbacks for the weight decay, which makes it necessary to think of some other regularization methods. Consistent Gaussian priors, early Stopping, invariances, tangent propagation, training with transformed data, convolutional networks, and soft weight sharing are some other alternatives for neural network regularization. A zealous reader may find a great deal of information about these topics in the Pattern Recognition and Machine Learning by Christopher M. Bishop (chapter 5, section 5).
 +
 +
==='''Further reading'''===
 +
The generalization ability of the network can depend crucially on the decay constant, especially with small training sets. One approach to choosing the decay constant is to train several networks with different amounts of decay and estimate the generalization error for each; then choose the decay constant that minimizes the estimated generalization error.
 +
 +
There are other important considerations for getting good results from weight decay. You must either standardize the inputs and targets, or adjust the penalty term for the standard deviations of all the inputs and targets. It is usually a good idea to omit the biases from the penalty term.
 +
 +
A fundamental problem with weight decay is that different types of weights in the network will usually require different decay constants for good generalization. At the very least, you need three different decay constants for input-to-hidden, hidden-to-hidden, and hidden-to-output weights. Adjusting all these decay constants to produce the best estimated generalization error often requires vast amounts of computation.
 +
 +
Fortunately, there is a superior alternative to weight decay: hierarchical Bayesian learning. Bayesian learning makes it possible to estimate efficiently numerous decay constants.For information about bayesian learning, please refer to [http://en.wikipedia.org/wiki/Bayesian_inference Bayesian inference]
 +
 +
[http://books.google.ca/books?id=jFAbzhrDqRcC&pg=PA1125&lpg=PA1125&dq=regularization+in+neural+networks+weight+decay&source=bl&ots=6YX8KIhxyO&sig=Dcwk5Y1_LPvtLhukEx3gDcVNEik&hl=en&ei=b0HzTLbfBYmgnwfv-5mXCg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CCIQ6AEwATgK#v=onepage&q&f=false]
 +
 +
===='''References'''====
 +
 +
1. A Simple Weight Decay Can Improve Generalization
 +
[http://www.google.ca/url?sa=t&source=web&cd=6&sqi=2&ved=0CEQQFjAF&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.47.4221%26rep%3Drep1%26type%3Dpdf&rct=j&q=weight%20decay&ei=gY7gTOuQDOGJ4Ab9uID7Bw&usg=AFQjCNGUYTQrmgjbjIrxMhi8iAAeFLfwpQ&sig2=bp7lSPCEA4ksu4EOHwwEIg&cad=rja.pdf]
 +
 +
2. Weight decay backpropagation for noisy data
 +
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6T08-3TYVWK9-F&_user=10&_coverDate=08%2F31%2F1998&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1540464228&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=f921fbc5786f7774f5fcc34a3df1c6a6&searchtype=a]
 +
 +
3. Learning with ensembles: How overfitting can be useful
 +
[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.51.9792&rep=rep1&type=pdf.pdf]
 +
 +
4. Sargur Srihari. ''Regularization in Neural Networks'' slides. [http://www.cedar.buffalo.edu/~srihari/CSE574/Chap5/Chap5.5-Regularization.pdf]
 +
 +
5. Neural Network Modeling using SAS Enterprise Miner [http://www.sasenterpriseminer.com/neural_networks.htm]
 +
 +
6. Valentina Corradi, Halbert White ''Regularized neural networks: some convergence rate results'' [http://portal.acm.org/citation.cfm?id=211706]
 +
 +
7. Bayesian Regularization in a Neural Network Model to Estimate Lines of Code Using Function Points [http://www.scipub.org/fulltext/jcs/jcs14505-509.pdf]
 +
 +
7. An useful pdf introducing regularization of neural network: [http://www.cedar.buffalo.edu/~srihari/CSE574/Chap5/Chap5.5-Regularization.pdf]
 +
 +
=='''Support Vector Machine - November 09, 2010'''==
 +
 +
===Introduction===
 +
 +
 +
Through the course we have seen different methods for solving linearly separable problems, e.g.: Linear regression, LDA, Neural Networks. In most cases, we can find many linear boundaries for a problem which separate classes (see figure 1) and all have the same training error. A question arises: which of these boundaries is optimal and has minimum true error? The answer to this question leads to a new type of classifiers called [http://en.wikipedia.org/wiki/Support_vector_machine Support Vector Machines (SVM)].
 +
SVMs are a set of supervised learning methods.
 +
 +
 +
The original algorithm was proposed by Vladimir  [http://en.wikipedia.org/wiki/Vapnik Vapnik] and later formulated to what is in current literature by Corinna Cortes and Vapnik. The modern history of SVM can be traced to 1974 when the field of [http://www.econ.upf.edu/~lugosi/mlss_slt.pdf statistical learning theory] was pioneered by [http://en.wikipedia.org/wiki/Vladimir_Vapnik Vladimir Vapnik] and [http://en.wikipedia.org/wiki/Alexey_Chervonenkis Alexey Chervonenkis]. In 1979, SVM was established when Vapnik further developed statistical learning theory and wrote a book in 1979 documenting his works. Since Vapnik's 1979 book was written in Russian, SVM did not become popular until Vapnik immigrated to the US and, in 1982, translated his 1979 book into English. More of SVM's history can be found in this [http://www.svms.org/history.html link].
 +
 +
 +
The current standard incarnation of SVM is known as "soft margin" and was proposed by Corinna Cortes and Vladimir Vapnik [http://en.wikipedia.org/wiki/Vladimir_Vapnik]. In practice the data is not usually linearly separable. Although theoretically we can make the data linearly separable by mapping it into higher dimensions, the issues of how to obtain the mapping and how to avoid overfitting are still of concern. A more practical approach to classifying non-linearly separable data is to add some error tolerance to the separating hyperplane between the two classes, meaning that a data point in class A can cross the separating hyperplane into class B by a certain specified distance. This more generalized version of SVM is the so-called "soft margin" support vector machine and is generally accepted as the standard form of SVM over the hard margin case in practice today. [http://en.wikipedia.org/wiki/Support_vector_machine#Soft_margin]
 +
 +
 +
SVM was introduced after neural networks and gathered attention by outperforming neural networks in many applications e.g. bioinformatics, text, image recognition, etc. It retained popularity until recently, when the notion of deep network (introduced by Hinton) outperformed SVM in some applications. A support vector machine constructs a hyperplane which can be used as a classification boundary. These linear decision boundaries explicitly try to separate the data into different classes while maximizing the margin of separation. Intuitively, if we are dealing with separable data clusters, a good separation is achieved by the hyperplane that has the largest distance to the nearest training data point(s) from each of the classes since in general the larger the margin the lower is the generalization error of the classifier, i.e. the lower is the probability that a new data point would be misclassified into the wrong class.
 +
 +
 +
The techniques that make the extensions to the non-linearly separable case, where the classes overlap no matter what linear boundary is created, are generalized to what is known as the "kernel support vector machine". Kernel SVM produces a nonlinear boundary by constructing a linear boundary in a higher-dimensional space and transformed feature space. This non-linear boundary is a linear boundary in the transformed feature space obtained by application of kernel, making kernel SVM a linear classifier just as the original form of SVM.
 +
 +
 +
No matter whether the training data are linearly-separable or not, the linear boundary produced by any of the versions of SVM is calculated using only a small fraction of the training data rather than using all of the training data points. This is much like the difference between the median and the mean. SVM can also be considered a special case of [http://en.wikipedia.org/wiki/Tikhonov_regularization Tikhonov regularization]. A special property is that they simultaneously minimize the empirical classification error and maximize the geometric margin; hence they are also known as maximum margin classifiers. The key features of SVM are the use of kernels, the absence of local minima, the sparseness of the solution (i.e. few training data points are needed to construct the linear decision boundary) and the capacity control obtained by optimizing the margin.(Shawe-Taylor and Cristianini (2004)). Another key feature of SVM, as discussed below, is the use of [http://en.wikipedia.org/wiki/Slack_variable slack variables] to control the amount of tolerable misclassification on the training data, which form the soft margin SVM. This key feature can serve to improve the generalization of SVM to new data. SVM has been used successfully in many real-world problems:
 +
 +
- Pattern Recognition (Face Detection [17], Face Verification [18], Object Recognition [19], Handwritten Character/Digit Recognition  [20], Speaker/Speech Recognition [21], Image Retrieval [22], Prediction [23])
 +
 +
- Text (and hypertext) categorization
 +
 +
- Image classification
 +
 +
- Bioinformatics (Protein classification, Cancer classification)
 +
 +
 +
For a complete list of SVM application please refer to [http://www.clopinet.com/isabelle/Projects/SVM/applist.html].
 +
 +
===Optimal Separating Hyperplane===
 +
 +
As can be seen in Figure 1, there exists an infinite number of linear hyperplanes between the two classes. A Support Vector Machine (SVM) performs classification by constructing an N-dimensional hyperplane that optimally separates the data into two categories.
 +
The data points which are indicated in the green circles in Figure 2 are the data points that are on the boundary of the margin and are called Support Vectors.
 +
[[File:Yyy.png|250px|thumb|right|Fig. 1 Linear Classifiers]]
 +
[[File:Xxx.png|250px|thumb|right|Fig. 2 Maximum Margin]]
 +
[[File:444.png|300px|thumb|right|Fig. 3 The linear algebra of a hyperplane]]
 +
 +
=== Some facts about the geometry of hyperplane===
 +
 +
Figure 3 shows the linear algebra of the hyperplane, where <math>\,d_i</math> is the distance between the origin and a point <math>\,x_i</math> on the hyperplane.
 +
 +
Suppose a hyperplane is defined as <math>\displaystyle \beta^{T}x+\beta_0=0</math>, as shown in figure 3, and suppose that the data is linearly separable and <math>\displaystyle y_i \in \{-1,1 \} </math>. Where <math>\displaystyle \beta_0</math> is the distance of the hyperplane to the origin.
 +
 +
 +
Property 1:  <math>\displaystyle \beta </math> is orthogonal to the hyperplane
 +
 +
Suppose that <math>\displaystyle x_1,x_2</math> are <br />lying on the hyperplane. Then we have
 +
 +
: <math>\displaystyle \beta^{T}x_1+\beta_0=0</math> , and
 +
 +
: <math>\displaystyle \beta^{T}x_2+\beta_0=0</math> .
 +
 +
Therefore,
 +
 +
: <math>\displaystyle \beta^{T}x_1+\beta_0 - (\beta^{T}x_2+\beta_0)=0</math> , and
 +
 +
: <math>\displaystyle \beta^{T}(x_1-x_2)=0</math> .
 +
 +
Hence,
 +
 +
: <math>\displaystyle \beta \bot \displaystyle (x_1 - x_2)</math> .
 +
 +
But <math>\displaystyle x_1-x_2</math> is a vector lying in the hyperplane, since the two points were arbitrary. So, <math>\displaystyle \beta </math> is orthogonal to every vector lying in the hyperplane and by definition orthogonal to hyperplane.
 +
 +
 +
Property 2:
 +
 +
For any point <math>\displaystyle  x_0 </math> on the hyperplane, we can say that
 +
 +
: <math>\displaystyle \beta^{T}x_0+\beta_0=0</math>&nbsp; and
 +
 +
: <math>\displaystyle \beta^{T}x_0=-\beta_0</math> .
 +
 +
For any point on the hyperplane, multiplying by <math>\displaystyle \beta^{T}</math> gives negative value of the intercept of the hyperplane.
 +
<br />
 +
 +
 +
Property 3:
 +
 +
For any point <math>\displaystyle x_i</math>, let the distance of the point to the hyperplane be denoted by <math>\displaystyle d_i</math>, which is the projection of (<math>\displaystyle x_i - x_0</math>) onto <math>\displaystyle\beta</math>. The signed distance for any point <math>\displaystyle x_i </math> to the hyperplane is <math> \displaystyle d_i = \beta^{T}(x_i - x_0)</math>. Since the length of <math>\displaystyle \beta </math> changes the value of the distance, we can normalize it by dividing <math>\displaystyle \beta </math> into its length. Thus, we get
 +
 +
: <math>\displaystyle d_i=\frac{\beta^{T}(x_i-x_0)}{\|\beta\|}  </math> <math>\displaystyle i=1,2,....,N </math> ,
 +
 +
: <math>\displaystyle d_i=\frac{\beta^{T}x_i-\beta^{T}x_0}{\|\beta\|} </math>&nbsp; by property 2, and
 +
 +
: <math>\displaystyle d_i=\frac{\beta^{T}x_i+\beta_0}{\|\beta\|} </math> .
 +
 +
Therefore, for any point if we want to find it's distance to the hyperplane we simply put it in the above equation.
 +
 +
 +
Property 4:
 +
 +
We use labels to make the distance positive. Therefore, let <math>\displaystyle Margin=(y_id_i)</math>. Since we would like to maximize the Margin, we have
 +
 +
: <math>\displaystyle  Margin=max(y_id_i)</math>      <math>\displaystyle i=1,2,....,N </math> .
 +
 +
Since we now know how to compute <math>\displaystyle  d_i </math> , by property 3,
 +
 +
: <math>\displaystyle  Margin=max\{y_i\frac{\beta^{T}x_i+\beta_0}{\|\beta\|}\}    \quad    (1)</math> , and
 +
 +
: <math>\displaystyle y_i(\beta^{T}x_i+\beta_0)\ge 0</math> .
 +
 +
Since the margin is a distance it is always non-negative. If the point is on the hyperplane, it is zero. Otherwise, it is greater than zero.
 +
For all training data points <math>\,i</math> that are not on the hyperplane,
 +
 +
: <math>\displaystyle y_i(\beta^{T}x_i+\beta_0)>0 </math> .
 +
 +
Let <math> \displaystyle c>0 </math> be the minimum distance between the hyperplane and the training data points not on the hyperplane. We have
 +
 +
: <math>\, y_i(\beta^{T}x_i+\beta_0)\ge c </math>
 +
 +
for all training data points <math> \displaystyle i </math> that are not on the hyperplane. Thus,
 +
 +
: <math>\displaystyle y_i(\frac{\beta^{T}x_i}{c}+\frac{\beta_0}{c})\ge 1</math> .
 +
 +
This is known as the canonical representation of the decision hyperplane. For <math>\displaystyle \beta^{T} </math> only the direction is important, so <math>\displaystyle \frac{\beta^{T}}{c} </math> does not change its direction and the hyperplane will be the same. Thus,
 +
 +
: <math>\displaystyle y_i(\beta^{T}x_i+\beta_0)\ge 1  \quad  (2)</math> ,<br />
 +
 +
equivalently, as we care only about the direction of the <math>\displaystyle\beta</math>, we can write:<br />
 +
 +
: <math>\displaystyle  y_i\frac{\beta^{T}x_i+\beta_0}{\|\beta\|}\geq1  </math> <br /><br />
 +
 +
Considering (2) and (1), for the the closest datapoints to the margin (those datapoints, which are placed at the distance 1 to the margin as shown above), (1) becomes:<br />
 +
 +
: <math>\displaystyle  Margin=max\{\frac{1}{\|\beta\|}\}    </math>
 +
 +
 +
Therefore, in order to maximize the margin we have to minimize the norm of <math>\,\beta</math>. So, we get
 +
 +
: minimize <math>\displaystyle\|\beta\|^2</math>&nbsp; and
 +
 +
: minimize <math>\displaystyle\frac{1}{2}\|\beta\|^2</math>    s.t <math> \displaystyle y_i(\beta^T x_i + \beta_0) \geq 1  \forall</math> i
 +
 +
for the <math>\displaystyle\beta</math> s which have distance greater than or equal to one.
 +
<br />
 +
we choose to minimize norm 2 of <math>\displaystyle\beta</math> mainly for the sake of simplified optimization.
 +
We have used <math>\displaystyle\frac{1}{2}</math> factor only for convenience in derivation of the derivative.
 +
 +
===Writing Lagrangian Form of Support Vector Machine===
 +
 +
The Lagrangian form using [http://en.wikipedia.org/wiki/Lagrange_multipliers Lagrange multipliers] and constraints that are discussed below is introduced to ensure that the optimization conditions are satisfied, as well as finding an optimal solution (the optimal saddle point of the Lagrangian for the [http://en.wikipedia.org/wiki/Quadratic_programming classic quadratic optimization]). The problem will be solved in dual space by introducing  <math>\,\alpha_i</math> as dual constraints, this is in contrast to solving the problem in primal space as function of the betas.  A [http://www.cs.wisc.edu/dmi/lsvm/ simple algorithm] for iteratively solving the Lagrangian has been found to run well on very large data sets, making SVM more usable.  Note that this algorithm is intended to solve Support Vector Machines with some tolerance for errors - not all points are necessarily classified correctly.  Several papers by Mangasarian explore different algorithms for solving SVM.
 +
 +
Dual form of the optimization problem:
 +
 +
: <math>\,L(\beta,\beta_0,\alpha) = \frac{1}{2}\|\beta\|^2 - \sum_{i=1}^n{\alpha_i\left(y_i(\beta^Tx_i+\beta_0)-1\right)}</math> . 
 +
 +
To find the optimal value, we set the derivative equal to zero:
 +
 +
: <math>\,\frac{\partial L}{\partial \beta} = 0</math> and <math>\,\frac{\partial L}{\partial \beta_0} = 0</math> .
 +
 +
Note that <math>\,\frac{\partial L}{\partial \alpha_i}</math> is equivalent to the constraints <math>\left(y_i(\beta^Tx_i+\beta_0)-1\right) \geq 0, \,\forall\, i</math>.
 +
 +
First, setting <math>\,\frac{\partial L}{\partial \beta} = 0</math>:
 +
 +
: <math>\,\frac{\partial L}{\partial \beta} = \frac{\partial}{\partial \beta}\frac{1}{2}\|\beta\|^2 - \sum_{i=1}^n{\left\{\frac{\partial}{\partial \beta}(\alpha_iy_i\beta^Tx_i)+\frac{\partial}{\partial \beta}\alpha_iy_i\beta_0-\frac{\partial}{\partial \beta}\alpha_iy_i\right\}}</math> ,
 +
 +
: <math>\frac{\partial}{\partial \beta}\frac{1}{2}\|\beta\|^2 = \beta</math> ,
 +
 +
: <math>\,\frac{\partial}{\partial \beta}(\alpha_iy_i\beta^Tx_i) = \alpha_iy_ix_i</math> ,
 +
 +
: <math>\,\frac{\partial}{\partial \beta}\alpha_iy_i\beta_0 = 0</math> , and
 +
 +
: <math>\,\frac{\partial}{\partial \beta}\alpha_iy_i = 0</math> .
 +
 +
So this simplifies to <math>\,\frac{\partial L}{\partial \beta} = \beta - \sum_{i=1}^n{\alpha_iy_ix_i} = 0</math> . In other words,
 +
 +
: <math>\,\beta = \sum_{i=1}^n{\alpha_iy_ix_i}</math> and <math>\,\beta^T = \sum_{i=1}^n{\alpha_iy_ix_i^T}</math> .
 +
 +
Similarly, <math>\,\frac{\partial L}{\partial \beta_0} = \sum_{i=1}^n{\alpha_iy_i} = 0</math> .
 +
 +
Thus, our objective function becomes <math>\,\frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\alpha_i\alpha_jy_iy_jx_i^Tx_j}} - \sum_{i=1}^n{\sum_{j=1}^n{\alpha_i\alpha_jy_iy_jx_i^Tx_j}} + \sum_{i=1}^n{\alpha_i}</math>, which is a dual representation of the maximum margin. Since <math>\,\alpha_i</math> is the Lagrange multiplier, <math>\,\alpha_i \geq 0 \forall i</math>. Therefore, we have a new optimization problem:
 +
 +
: <math>\underset{\alpha}{\max} \sum_{i=1}^n{\alpha_i}- \,\frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\alpha_i\alpha_jy_iy_jx_i^Tx_j}} </math> , where
 +
 +
: <math>\,\alpha_i \ge 0 \forall i</math>&nbsp; and
 +
 +
: <math>\,\Sigma_i{\alpha_i y_i} = 0</math> .
 +
 +
This is a much simpler optimization problem and we can solve it by [http://en.wikipedia.org/wiki/Quadratic_programming]. Quadratic programming (QP) is a special type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a quadratic function of several variables subject to linear constraints on these variables.
 +
The general form of such a problem is minimize with respect to <math>\,x</math>
 +
: <math>f(x) = \frac{1}{2}x^TQx + c^Tx</math>
 +
subject to one or more constraints of the form <math>\,Ax\le b</math>, <math>\,Ex=d</math>.
 +
 +
See this [http://www.me.utexas.edu/~jensen/ORMM/supplements/methods/nlpmethod/S2_quadratic.pdf link] for a good description of general QP problem formulation and solution.
 +
 +
====Remark====
 +
Using Lagrangian Primal/Lagrangian Dual will give us two very different results when formulating this problem:
 +
-------------
 +
If use Lagrangian Primal:
 +
 +
:<math>\,min{L(\beta,\beta_0,\alpha)}</math>
 +
where
 +
:<math>\,L(\beta,\beta_0,\alpha) = \frac{1}{2}\|\beta\|^2 - \sum_{i=1}^n{\alpha_i\left(y_i(\beta^Tx_i+\beta_0)-1\right)}</math>
 +
such that
 +
:<math>\,\frac{\partial L}{\partial \beta} = 0</math> , <math>\,\frac{\partial L}{\partial \beta_0} = 0</math> and <math>\alpha_i\left(y_i(\beta^Tx_i+\beta_0)-1\right) = 0, \,\forall\, i</math>
 +
 +
-------------
 +
If use Lagrangian Dual:
 +
 +
:<math>\,max{L(\beta,\beta_0,\alpha)}</math>
 +
where
 +
:<math>\,L(\beta,\beta_0,\alpha) = \frac{1}{2}\|\beta\|^2 - \sum_{i=1}^n{\alpha_i\left(y_i(\beta^Tx_i+\beta_0)-1\right)}</math>
 +
such that
 +
:<math>\,\frac{\partial L}{\partial \beta} = 0</math> and <math>\,\frac{\partial L}{\partial \beta_0} = 0</math> (Note: no <math>\alpha_i\left(y_i(\beta^Tx_i+\beta_0)-1\right) = 0, \,\forall\, i</math> constraint anymore)
 +
 +
-------------
 +
We do not use Lagrangian Primal here because this formulation will generate quadratic constraints, which cannot be solved by 'quadprog.m'.
 +
Click [http://www.icom.nctu.edu.tw/CLSCM/download/Handouts/or706_week13.pdf here] for more details about Lagrangian Duality.
 +
 +
===Quadratic Programming Problem of SVMs and Dual Problem===
 +
 +
We have to find <math>\,\beta</math> and <math>\,\beta_0</math> such that <math>\,\frac{1}{2}\|\beta\|^2 </math> is minimized subject to <math> \,y_i (\beta^T x_i + \beta_0) \geq 1 \forall i </math>.
 +
Therefore, we need to optimize a quadratic function subject to linear constraints.
 +
Quadratic optimization problems are a well-known class of mathematical programming problems, and many (rather intricate) algorithms exist for solving them.
 +
The solution involves constructing a dual problem where a Lagrange multiplier <math>\,\alpha_i</math> is associated with every constraint in the primary problem.
 +
 +
The optimization goal is quadratic and the constraints are
 +
linear, it is a typical QP. Given such a constrained optimization problem, it is possible
 +
to construct another problem called the dual problem.
 +
We may now state the dual problem: given the training sample, find the
 +
Lagrange multipliers that maximize the objective function
 +
 +
: <math>\,Q(\alpha)= \underset{\alpha}{\max}  \sum_{i=1}^n{\alpha_i}- \,\frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\alpha_i\alpha_jy_iy_jx_i^Tx_j}} </math>
 +
 +
subject to the constraints
 +
 +
: <math>\,\alpha_i \ge 0 \forall i</math>&nbsp; and
 +
 +
: <math>\,\Sigma_i{\alpha_i y_i} = 0</math> .
 +
 +
=====Feasibility of the Primal and Dual Programming Problems=====
 +
 +
For Hard-Margin SVM, the primal and dual programming problems are feasible with any values of <math>\, y_i </math> and <math>\, x_i</math>.  Feasibility depends on satisfaction of the problem's constraints.  In the primal problem, the constraint <math> \,y_i (\beta^T x_i + \beta_0) \geq 1 \forall i </math> can be satisfied by setting <math>\, \beta = \underline{0} </math> and by setting <math>\, \beta_0 = \frac{1}{y_i} </math> for any values of <math>\, y_i </math> and <math>\, x_i</math>.  In the dual problem, the constrants <math>\,\alpha_i \ge 0 \forall i</math> and <math>\,\Sigma_i{\alpha_i y_i} = 0</math> can be satisfied by setting <math>\,\alpha_i = 0 \forall i</math> for any values of <math>\, y_i </math> and <math>\, x_i</math>.  The constaints can always be satisfied, so the programming problems are always feasible.
 +
 +
===Implementation===
 +
 +
The parameters of the maximum-margin hyperplane are derived by solving the optimization. There exist several specialized algorithms for quickly solving the QP problem that arises from SVMs, mostly reliant on heuristics for breaking the problem down into smaller, more-manageable chunks. A common method for solving the QP problem is the Sequential Minimal Optimization (SMO) algorithm by John Platt in 1998. A link to the original paper of which is available [http://research.microsoft.com/en-us/um/people/jplatt/smoTR.pdf here]. SMO which breaks the problem down into 2-dimensional sub-problems that may be solved analytically, eliminating the need for a numerical optimization algorithm.
 +
Another approach is to use an interior point method that uses Newton-like iterations to find a solution of the [http://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions Karush-Kuhn-Tucker conditions] of the primal and dual problems [10]. Instead of solving a sequence of broken down problems, this approach directly solves the problem as a whole. To avoid solving a linear system involving the large kernel matrix, a low rank approximation to the matrix is often used to use the kernel trick.Please refer to [http://www.mathworks.ch/help/toolbox/bioinfo/ref/svmtrain.html;jsessionid=q6MgMBHGsKf5hJrBv1H8pZsp4nLjsmnjFhvsGf5Ylnqzqh4fQMpn!2108730516] for code implementation of SVM.
 +
 +
=== Hard margin SVM Algorithm ===
 +
 +
[[image: H-SVM.png ]]
 +
 +
Source: John Shawe-Taylor and Nello Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, illustrated edition edition, June 2004.
 +
 +
===Multiclass SVM===
 +
 +
SVM is only directly applicable for two-class case. We want to generalize this algorithm to multi-class tasks. Multiclass SVM aims to assign labels to instances by using support vector machines, where the labels are drawn from a finite set of several elements. The dominating approach for doing so is to reduce the single multiclass problem into multiple binary problems. Each of the problems yields a binary classifier, which is assumed to produce an output function that gives relatively large values for examples from the positive class and relatively small values for examples belonging to the negative classes. Two common methods to build such binary classifiers are where each classifier distinguishes between (i) one of the labels to the rest (one-versus-all) or (ii) between every pair of classes (one-versus-one). Classification of new instances for one-versus-all case is done by a winner-takes-all strategy, in which the classifier with the highest output function assigns the class (it is important that the output functions be calibrated to produce comparable scores). For the one-versus-one approach, classification is done by a max-wins voting strategy, in which every classifier assigns the instance to one of the two classes, then the vote for the assigned class is increased by one vote, and finally the class with most votes determines the instance classification.
 +
 +
LIBSVM is an integrated software for support vector classification, regression and distribution estimation. It supports multi-class classification.
 +
[http://www.csie.ntu.edu.tw/~cjlin/libsvm/]
 +
 +
Koby Cramer and  Yoram Singer have proposed a direct way to learn multiclass SVMs. It is described here in their paper:
 +
[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.69.8716 On the Algorithmic Implementation of Multiclass Kernel-based Vector Machines]
 +
 +
 +
 +
==== Implements SVM multi-class ====
 +
 +
Spider is an object orientated environment for machine learning in MATLAB, for unsupervised, supervised or semi-supervised machine learning problems, and includes training, testing, model selection, cross-validation, and statistical tests. Implements SVM multi-class classification and regression. [http://www.kyb.tuebingen.mpg.de/bs/people/spider/ Spider]
 +
 +
===Support Vector Machines vs Artificial Neural Networks===
 +
 +
The development of ANNs followed a heuristic path, with applications and extensive experimentation preceding theory. In contrast, the development of SVMs involved sound theory first, then implementation and experiments. A significant advantage of SVMs is that whilst ANNs can suffer from multiple local minima, the solution to an SVM is global and unique. Two more advantages of SVMs are that they have a simple geometric interpretation and also a sparse solution. Unlike ANNs, the computational complexity of SVMs does not depend on the dimensionality of the input space. ANNs use empirical risk minimization, whilst SVMs use structural risk minimization. The reason that SVMs often outperform ANNs in practice is that they deal with the biggest problem with ANNs, SVMs are less prone to over-fitting since their solution is sparse. In contrast to neural networks SVMs automatically select their model size (by selecting the Support vectors)(Rychetsky (2001)).While the weight decay term is an important aspect for obtaining good generalization in the context of neural networks for regression, the gamma parameter (in soft-margin SVM) that is discussed below plays a somewhat similar role in classification problems.
 +
 +
===SVM packages===
 +
One of the popular Matlab toolboxes for SVM is [http://www.csie.ntu.edu.tw/~cjlin/libsvm/ LIBSVM], which has been developed in the department of Computer Science and Information Engineering, National Taiwan University, under supervision of Chih-Chung Chang and Chih-Jen Lin. In this page they have provided the society with many different interfaces for LIBSVM like Matlab, C++, Python, Perl, and many other languages, each one of those has been developed in different institutes and by variety of engineers and mathematicians. In this page you can also find a thorough introduction to the package and its various parameters.
 +
 +
A very helpful tool which you can find on the [http://www.csie.ntu.edu.tw/~cjlin/libsvm/ LIBSVM] page is a graphical interface for SVM; it is an applet by which we can draw points corresponding to each of the two classes of the classification problem and by adjusting the SVM parameters, observe the resulting solution.
 +
 +
If you found LIBSVM helpful and wanted to use it for your research, [http://www.csie.ntu.edu.tw/~cjlin/libsvm/faq.html#f203 please cite the toolbox].
 +
 +
A pretty long list of other SVM packages and comparison between all of them in terms of language, execution platform, multiclass and regression capabilities, is available [http://www.cs.ubc.ca/~murphyk/Software/svm.htm here].
 +
 +
The top 3 SVM software are:
 +
 +
1. LIBSVM
 +
 +
2. SVMlight
 +
 +
3. SVMTorch
 +
 +
Also, there are other two web pages introducing SVM software and their comparison: [http://www.svms.org/software.html] and [http://www.support-vector-machines.org/SVM_soft.html].
 +
 +
===References===
 +
 +
1. V. Vapnik and A. Chervonenkis, Theory of Pattern Recognition, Nauka, Moscow, 1974.
 +
 +
2. V. Vapnik, Estimation of Dependencies Based on Empirical Data, Nauka, Moscow, 1979.
 +
 +
3. V. Vapnik, The Nature of Statistical Learning Theory, Springer, 1995.
 +
 +
4. V. Vapnik, Statistical Learning Theory, Wiley-Interscience, New York, 1998.
 +
 +
5. P. H. Chen, C. J. Lin, and B. Schölkopf, A tutorial on ν-support vector machines, Appl. Stoch. Models. Bus. Ind. 21, 111-136, 2005.
 +
 +
6. S.-W. Lee and A. Verri (Eds.),Applications of Support Vector Machines for Pattern Recognition: SVM 2002, LNCS 2388, pp. 213-236, 2002.
 +
 +
7. V. D. Sanchez, Advanced support vector machines and kernel methods, Neurocomputing 55, 5-20, 2003.
 +
 +
8. C. Campbell, Kernel methods: a survey of current techniques, Neurocomputing, 48, 63-84, 2002.
 +
 +
9. K. R. Müller, S. Mika, G. Rätsch, K. Tsuda, and B. Schölkopf, An introduction to kernel-based learning algorithms, IEEE Trans.
 +
Neural Netw., 12, 181-201, 2001.
 +
 +
10. J. A. K. Suykens, Support vector machines: A nonlinear modelling and control perspective, Eur. J. Control, 7, 311-327, 2001.
 +
 +
11. V. N. Vapnik, An overview of statistical learning theory, IEEE Trans. Neural Netw., 10, 988-999, 1999.
 +
 +
12. B. Schölkopf, S. Mika, C. J. C. Burges, P. Knirsch, K. R. Muller, G. Ratsch, and A. J. Smola, Input space versus feature space in kernel-based methods, IEEE Trans. Neural Netw., 10, 1000-1017, 1999.
 +
 +
13. C. J. C. Burges, A tutorial on Support Vector Machines for pattern recognition, Data Min. Knowl. Discov., 2, 121-167, 1998.
 +
 +
14. A. J. Smola and B. Schölkopf, On a kernel-based method for pattern recognition, regression, approximation, and operator inversion, Algorithmica, 22, 211-231, 1998.
 +
 +
15. K. Jonsson, J. Kittler, and Y.P. Matas, Support vector machines for face authentication, Journal of Image and Vision Computing, vol. 20. pp. 369-375, 2002.
 +
 +
16. A. Tefas, C. Kotropoulos, and I. Pitas, Using support vector machines to enhance the performance of elastic graph matching for frontal face authentication, IEEE Transaction on Pattern  Analysis  and  Machine  Intelligence, vol. 23. No. 7, pp. 735-746, 2001.
 +
 +
17. E. Osuna, R. Freund, and F. Girosi, Training support machines: An application to face detection. In  Proceedings  of  IEEE  Conference  on  Computer  Vision  and Pattern Recognition, pp. 130-136, 1997.
 +
 +
18. Y. Wang, C.S. Chua, and Y.K, Ho. Facial feature detection and face recognition from 2D and 3D images, Pattern Recognition Letters, Feb., 2002.
 +
 +
19. Q. Tian, P. Hong, and T.S. Huang, Update relevant image weights for contentbased image retrieval using support vector machines, In Proceedings of IEEE Int. Conference on Multimedia and Expo, vol.2, pp. 1199-1202, 2000.
 +
 +
20. D. Gorgevik, D. Cakmakov, and V. Radevski, Handwritten digit recognition by combining support vector machines using rule-based reasoning, In Proceedings of 23rd Int. Conference on Information Technology Interfaces, pp. 139-144, 2001.
 +
 +
21. V. Wan and W.M. Campbell, Support vector machines for speaker verification and identification, In  Proceedings  of  IEEE  Workshop  on  Neural  Networks  for Signal Processing X, vol. 2, 2000.
 +
 +
22. A. Fan and M. Palaniswami, Selecting bankruptcy predictors using a support vector machine approach, vol. 6, pp. 354-359, 2000.
 +
 +
23. Joachims, T. Text categorization with support vector machines. Technical report, LS VIII Number 23, University of Dortmund, 1997. ftp://ftp-ai.informatik.uni-zortmund.de/pub/Reports/report23.ps.Z.
 +
 +
==''' Support Vector Machine Cont., Kernel Trick - November 11, 2010'''==
 +
 +
Recall in the previous lecture that instead of solving the primal problem of maximizing the margin, we can solve the dual problem without changing the solution as long as it subjects to the [http://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions Karush-Kuhn-Tucker] (KKT) conditions. KKT are the first-order conditions on the gradient for an optimal point. Leading to the following:
 +
 +
<math>\max_{\alpha} L(\alpha) = \sum_{i=1}^n{\alpha_i} - \frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\alpha_i\alpha_jy_iy_jx_i^Tx_j}}</math>
 +
:such that <math>\,\alpha_i \ge 0 \forall i</math>
 +
:and <math>\sum_{i=1}^n{\alpha_i y_i} = 0</math>
 +
 +
We are looking to maximize <math>\,\alpha</math>, which is our only unknown. Once we know <math>\,\alpha</math>, we can easily find <math>\,\beta</math> and <math>\,\beta_0</math> (see the Support Vector algorithm below for complete details).
 +
 +
If we examine the Lagrangian equation, we can see that <math>\,\alpha</math> is multiplied by itself; that is, the Lagrangian is quadratic with respect to <math>\,\alpha</math>. Our constraints are linear. This is therefore a problem that can be solved through [http://en.wikipedia.org/wiki/Quadratic_programming quadratic programming] techniques.
 +
 +
We can write the Lagrangian equation in matrix form:
 +
 +
<math>\max_{\alpha} L(\alpha) = \underline{\alpha}^T\underline{1} - \frac{1}{2}\underline{\alpha}^TS\underline{\alpha}</math>
 +
:such that <math>\underline{\alpha} \geq \underline{0}</math>
 +
:and <math>\underline{\alpha}^T\underline{y} = 0</math>
 +
 +
Where:
 +
* <math>\underline{\alpha}</math> denotes an <math>\,n \times 1</math> vector; <math>\underline{\alpha}^T = [\alpha_1, ..., \alpha_n]</math>
 +
* Matrix <math>S(i,j) = y_iy_jx_i^Tx_j = (y_ix_i)^T(y_jx_j)</math>
 +
* <math>\,\underline{0}</math> and <math>\,\underline{1}</math> are vectors containing all 0s or all 1s respectively
 +
 +
Using this matrix notation, we can use Matlab's built in quadratic programming routine, [http://www.mathworks.com/access/helpdesk/help/toolbox/optim/ug/quadprog.html quadprog].
 +
 +
'''Note'''
 +
 +
Matlab's <code>quadprog</code> function minimizes an equation of the following form:
 +
:<math>\min_x\frac{1}{2}x^THx+f^Tx</math>
 +
:such that: <math>\,A \cdot x \leq b</math>, <math>\,A_{eq} \cdot x = b_{eq}</math> and <math>\,lb \leq x \leq ub</math>
 +
 +
The function is called as such: <code>x = quadprog(H,f,A,b,Aeq,beq,lb,ub)</code>. The variables correspond to values in the equation above.
 +
 +
An example of running "quadprog" method in matlab:
 +
 +
:<math>\min_x\frac{1}{2}x_1^2+x_2^2-x_1x_2-2x_1-6x_2</math>
 +
:<math>\,s.t.  x_1+x_2 \leq 2</math>, <math>\,-x_1+2x_2 \leq 2</math>, <math>\,2x_1+x_2 \leq 3</math>, <math>\,0 \leq x_1</math>, <math>\,0 \leq x_2</math>
 +
 +
The matlab code is:
 +
 +
H = [1 -1; -1 2]
 +
f = [-2; -6]
 +
A = [1 1; -1 2; 2 1]
 +
b = [2; 2; 3]
 +
lb = zeros(2,1)
 +
[x,fval,exitflag,output,lambda] = quadprog(H,f,A,b,[],[],lb)
 +
 +
The result is:
 +
 +
x =
 +
 +
    0.6667
 +
    1.3333
 +
 +
 +
fval =
 +
 +
    -8.2222
 +
 +
 +
exitflag =
 +
 +
      1
 +
 +
 +
output =
 +
 +
          iterations: 3
 +
    constrviolation: 1.1102e-016
 +
          algorithm: 'medium-scale: active-set'
 +
      firstorderopt: []
 +
        cgiterations: []
 +
            message: 'Optimization terminated.'
 +
 +
 +
lambda =
 +
 +
      lower: [2x1 double]
 +
      upper: [2x1 double]
 +
      eqlin: [0x1 double]
 +
    ineqlin: [3x1 double]
 +
 +
=====Upper bound for Hard Margin SVM in MATLAB's Quadprog=====
 +
 +
The theoretical optimization problem for Hard Margin SVM provided at the beginning of this lecture shows that <math>\, \underline{\alpha}</math> is given no upper bound.  However, when this problem is implemented using MATLAB's <code>quadprog</code>, there are some cases where the correct solution is not produced. It is the experience of students who have taken this course that, for some Hard-Margin SVM problems, <code>quadprog</code> unexpectedly outputs a solution that has all elements of <math>\, \underline{\alpha}</math> set to zero.  Of course, some elements of <math>\, \underline{\alpha}</math> need to be non-zero in order for the support vectors to be determined, so this all-zero result is obviously incorrect.  One numerical trick that seems to correct this trouble with <code>quadprog</code> is to set the upper bound on the elements of <math>\, \underline{\alpha}</math> to a large number like <math>\, 10^6</math>. This can be done by setting the eighth parameter of <code>quadprog</code> to a vector with the same dimension of <math>\, \underline{\alpha}</math> and with all elements having a value of <math>\, 10^6</math>.
 +
 +
===Examining K.K.T. conditions===
 +
 +
[http://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions Karush-Kuhn-Tucker conditions] ([http://webrum.uni-mannheim.de/mokuhn/public/KarushKuhnTucker.pdf more info]) give us a closer look into the Lagrangian equation and the associated conditions.
 +
 +
Suppose we are looking to minimize <math>\,f(x)</math> such that <math>\,g_i(x) \geq 0, \forall{x}</math>. If <math>\,f</math> and <math>\,g</math> are differentiable, then the ''necessary'' conditions for <math>\hat{x}</math> to be a local minimum are:
 +
 +
# At the optimal point, <math>\frac{\partial L}{\partial \hat{x}} = 0</math>; i.e. <math>f'(\hat{x}) - \Sigma_i{\alpha_ig'(\hat{x})}=0</math>
 +
# <math>\alpha_i \ge 0 \forall{i}</math>. (Dual Feasibility)
 +
# <math>\alpha_ig_i({x}) = 0 \forall{i}</math> (Complementary Slackness)
 +
# <math>g_i(\hat{x}) \geq 0</math> (Primal Feasibility)
 +
 +
If any of these conditions are violated, then the problem is deemed not feasible.
 +
 +
These are all trivial except for condition 3. Let's examine it further in our support vector machine problem.
 +
 +
===Support Vectors===
 +
 +
Support vectors are the training points that determine the optimal separating hyperplane that we seek. Also, they are the most difficult points to classify and at the same time the most informative for classification.
 +
 +
In our case, the <math>g_i({x})</math> function is:
 +
:<math>\,g_i(x) = y_i(\beta^Tx_i+\beta_0)-1</math>
 +
 +
Substituting <math>\,g_i</math> into KKT condition 3, we get <math>\,\alpha_i[y_i(\beta^Tx_i+\beta_0)-1] = 0</math>. <br\>In order for this condition to be satisfied either <br/><math>\,\alpha_i= 0</math> or <br/><math>\,y_i(\beta^Tx_i+\beta_0)=1</math>
 +
 +
All points <math>\,x_i</math> will be either 1 or greater than 1 distance unit away from the hyperplane.
 +
 +
'''Case 1: a point away from the margin'''
 +
 +
If <math>\,y_i(\beta^Tx_i+\beta_0) > 1 \Rightarrow \alpha_i = 0</math>.
 +
 +
If point <math>\, x_i</math> is not on the margin, then the corresponding <math>\,\alpha_i=0</math>.
 +
 +
'''Case 2: a point on the margin'''
 +
 +
If  <math>\,y_i(\beta^Tx_i+\beta_0) = 1 \Rightarrow \alpha_i > 0 </math>.
 +
<br\>If point <math>\, x_i</math> is on the margin, then the corresponding <math>\,\alpha_i>0</math>.
 +
 +
 +
Points on the margin, with corresponding <math>\,\alpha_i > 0</math>, are called '''''support vectors'''''.
 +
 +
The optimal hyperplane is determined by only a few support vectors. Since it is impossible for us to know a priori which of the training data points would end up as the support vectors, it is necessary for us to work with the entire training set to find the optimal hyperplane.
 +
 +
===The support vector machine algorithm===
 +
 +
# Solve the quadratic programming problem:<math>\max_{\alpha} L(\alpha) = \sum_{i=1}^n{\alpha_i} - \frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\alpha_i\alpha_jy_iy_jx_i^Tx_j}}</math> such that <math>\alpha_i \geq 0 \forall{i}</math> and <math>\sum_{i=1}^n{\alpha_i y_i} = 0</math> <br/> (Use Matlab's quadprog to find the optimal <math>\,\underline{\alpha}</math>)
 +
# Find <math>\beta = \sum_{i=1}^n{\alpha_iy_i\underline{x_i}}</math>
 +
# Find <math>\,\beta_0</math> by choosing a support vector (a point with <math>\,\alpha_i > 0</math>) and solving <math>\,y_i(\beta^Tx_i+\beta_0) = 1</math>
 +
 +
===Advantages of Support Vector Machines===
 +
 +
* SVMs provide a good out-of-sample generalization. This means that, by choosing an appropriate generalization grade,
 +
SVMs can be robust, even when the training sample has some bias. This is mainly due to selection of optimal hyperplane.
 +
* SVMs deliver a unique solution, since the optimality problem is convex. This is an advantage compared
 +
to Neural Networks, which have multiple solutions associated with local minima and for this reason may
 +
not be robust over different samples.
 +
*State-of-the-art accuracy on many problems.
 +
*SVM can handle any data types by changing the kernel.
 +
*The support vector machine algorithm is insensitive to outliers. If <math>\,\alpha = 0</math>, then the cost function is also 0, and won't contribute to the solution of the SVM problem; only points on the margin — support vectors — contribute. Hence the model given by SVM is entirely defined by the support vectors, which is a very small subset of the entire training set. In this case we have a data-driven or 'nonparametric' model in which is the training set and algorithm will determine the support vectors, instead of fitting a set of parameters.
 +
 +
References:
 +
Wang, L, 2005. Support Vector Machines: Theory and Applications, Springer, 3
 +
 +
 +
===Disadvantages of Support Vector Machines [http://www.cse.unr.edu/~bebis/MathMethods/SVM/lecture.pdf]===
 +
 +
*Perhaps the biggest limitation of the support vector approach lies in choice of the kernel (Which we will study about in future).
 +
 +
*A second limitation is speed and size, both in training and testing (mostly in training - for large training sets, it typically selects a small number of support vectors, thereby minimizing the computational requirements during testing).
 +
 +
*Discrete data presents another problem, although with suitable rescaling excellent results have nevertheless been obtained.
 +
 +
*The optimal design for multiclass SVM classifiers is a further area for research.
 +
 +
*Although SVMs have good generalization performance, they can be abysmally slow in test phase.
 +
 +
*Besides the advantages of SVMs - from a practical point of view - they have some drawbacks. An important practical question that is not entirely solved, is the selection of the kernel function parameters - for Gaussian kernels the width parameter [sigma] - and the value of [epsilon] in the [epsilon]-insensitive loss function.
 +
 +
*However, from a practical point of view perhaps the most serious problem with SVMs is the high algorithmic complexity and extensive memory requirements of the required quadratic programming in large-scale tasks.
 +
 +
===Applications of Support Vector Machines===
 +
 +
The following papers describe some of the possible applications of support vector machines:
 +
 +
1- Training support vector machines: an application to face detection [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=609310 here]
 +
 +
2- Application of support vector machines in financial time series forecasting [http://svms.org/regression/TaCa01.pdf here]
 +
 +
3- Support vector machine active learning with applications to text classification [http://portal.acm.org/citation.cfm?id=944793&dl=GUIDE, here]
 +
 +
 +
 +
Note that SVMs start from the goal of separating the data with a hyperplane, and could be extended to non-linear decision boundaries using the kernel trick.
 +
 +
===Kernel Trick===
 +
{{Cleanup|date=November 2010|reason=It would be better to provide a link to exact proof of the fact that of we project data into high dimensional space then data will become linearly separable.}}
 +
 +
{{Cleanup|date=November 2010|reason=I dont know if such a proof exists, would this not depend on the data whether or not a high dimensional projection would make the data linearly seperable}}
 +
 +
Fig.1 shows how transforming the data into the higher dimension makes it linearly separable.
 +
 +
[[File:dimensoin.png|350px|thumb|right|Fig1.Transforming the data can make it linearly separable.]]
 +
 +
We talked about the [http://www.armyconference.org/ACAS00-02/ACAS02ShortCourse/ACASCourse10.pdf curse of dimensionality] at the beginning of this course. However, we now turn to the power of high dimensions in order to find a hyperplane between two classes of data points that can linearly separate the transformed (mapped) data in a space that has a higher dimension than the space in which the training data points reside. To understand this, imagine a two dimensional prison where a two dimensional person is constrained. Suppose magically we give the person a third dimension, then he can escape from the prison. In other words, the prison and the person are linearly separable now with respect to the third dimension. The intuition behind the [http://www.cs.berkeley.edu/~jordan/courses/281B-spring04/lectures/lec3.pdf kernel trick] is basically to map data to a higher dimension in which the mapped data are linearly separable by a hyperplane, even if the original data are not linearly separable.
 +
 +
[[File:Point_2d.png|200px|thumb|right|Imagine the point is a person.  They're stuck.]]
 +
[[File:Point_3d.png|200px|thumb|right|Escape through the third dimension!]]
 +
[[File:Unsep.png|200px|thumb|right|It's not possible to put a hyperplane through these points.]]
 +
[[File:Sep2.png|200px|thumb|right|After a simple transformation, a perfect classification plane can be found.]]
 +
 +
The original optimal hyperplane algorithm proposed by [http://en.wikipedia.org/wiki/Vladimir_Vapnik Vladimir Vapnik] in 1963 was a linear classifier. However, in 1992, Bernhard Boser, Isabelle Guyon and Vapnik suggested a way to create non-linear classifiers by applying the kernel trick to maximum-margin hyperplanes. The algorithm is very similar, except that every dot product is replaced by a non-linear kernel function as below. This allows the algorithm to fit the maximum-margin hyperplane in a transformed feature space. We have seen SVM as a linear classification problem that finds the maximum margin hyperplane in the given input space. However, for many real world problems a more complex decision boundary is required. The following simple method was devised in order to solve the same linear classification problem but in a higher dimensional space, a [http://en.wikipedia.org/wiki/Feature_space feature space], under which the maximum margin hyperplane is better suited.
 +
 +
Let <math>\,\phi</math> be a mapping,
 +
 +
<math>\phi:\mathbb{R}^d \rightarrow \mathbb{R}^D </math>, where <math>\,D > d</math>.<br /><br />
 +
 +
We wish to find a <math>\,\phi</math> such that our data will be suited for separation by a hyperplane. Given this function, we are led to solve the previous constrained quadratic optimization on the transformed dataset,<br /><br />
 +
 +
<math>\max_{\alpha} L(\alpha) = \sum_{i=1}^n{\alpha_i} - \frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\alpha_i\alpha_jy_iy_j\phi(x_i)^T\phi(x_j)}}</math> such that <math>\alpha_i \geq 0</math> and <math>\sum_{i=1}^n{\alpha_i y_i} = 0</math><br /><br />
 +
 +
The solution to this optimization problem is now well known; however a workable <math>\,\phi</math> must be determined. Possibly the largest drawback in this method is that we must compute the inner product of two vectors in the high dimensional space. As the number of dimensions in the initial data set increases, the inner product becomes computationally intensive or impossible.
 +
 +
However, we have a very useful result that says that there exists a class of functions, <math>\,\Phi</math>, which satisfy the above requirements and that for any function <math>\,\phi \in \Phi</math>,
 +
 +
<math>\,\phi(x_i)^T\phi(x_j) = K(x_i,x_j) </math><br /><br />
 +
 +
Where K is a ''kernel function'' in the input space satisfying [http://en.wikipedia.org/wiki/Mercer%27s_condition Mercer's condition] (to guarantee that it indeed corresponds to certain mapping function <math>\,\phi</math>).  As a result, if the objective function depends on inner products but not on coordinates, we can always use a kernel function to implicitly calculate in the feature space without storing the huge data. Not only does this solve the computation problems but it no longer requires us to explicitly determine a specific mapping function in order to use this method. In fact, it is now possible to use an infinite dimensional feature space (such as a [http://en.wikipedia.org/wiki/Hilbert_space Hilbert space] in SVM without even explicitly knowing the function <math>\,\phi</math>.
 +
 +
* one may look at <math>\,x_i^T x_j</math> as way of measuring similarity, where <math>\,K(\underline{x}_i,\underline{x}_j) </math> is another way of measuring similarity between <math>\,x_i </math> and <math>\,x_j</math>
 +
 +
Available [http://www.youtube.com/watch?v=3liCbRZPrZA here] is a a short but interesting and informative video by Udi Aharoni that illustrates how kernel SVM uses a kernel to map non-linearly-separable original data to a higher-dimensional space and then finding a hyperplane in that space that linearly separates the implicitly mapped data, and how this hyperplane ultimately translates to a non-linear decision boundary in the original space that classifies the original data.
 +
 +
 +
====Popular kernel choices for SVM====
 +
 +
There are many types of kernels that can be used in Support Vector Machines models. These include linear, polynomial and radial basis function (RBF).
 +
 +
linear: <math>\ K(\underline{x}_{i},\underline{x}_{j})= \underline{x}_{i}^T\underline{x}_{j}</math>,
 +
 +
polynomial: <math>\ K(\underline{x}_{i},\underline{x}_{j})= (\gamma\underline{x}_{i}^T\underline{x}_{j}+r)^{d}, \gamma > 0</math>,
 +
 +
radial Basis: <math>\ K(\underline{x}_{i},\underline{x}_{j})= exp(-\gamma \|\underline{x}_i - \underline{x}_j\|^{2}), \gamma > 0</math>,
 +
 +
Gaussian: <math>\ K(x_i,x_j)=exp(\frac{-||x_i-x_j||^2}{2\sigma^2 })</math>,
 +
 +
hyperbolic tangent: <math>\ K(x_i,x_j)=tanh(k_1\underline{x}_{i}^T\underline{x}_{j}+k_2)</math>,
 +
 +
The RBF kernel is by far the most popular choice of kernel types used in Support Vector Machines. This is mainly because of their localized and finite responses across the entire range of the real x-axis.The art of flexible modeling using basis expansions consists of picking an appropriate family of basis functions, and then controlling the complexity of the representation by selection, regularization, or both. Some of the families of basis functions have elements that are defined locally; for example, <math>\displaystyle B</math>-splines are defined locally in <math>\displaystyle R</math>. If more flexibility is desired in a particular region, then that region needs to be represented by more basis functions(which in the case of <math>\displaystyle B</math>-splines translates to more knots). Kernel methods achieve flexibility by fitting simple models in a region local to the target point <math>\displaystyle x_0</math>. Localization is achieved via a weighting kernel <math>\displaystyle K</math> and individual observations receive weights <math>\displaystyle K(x_0,x_i)</math>. The RBF kernel combines these ideas, by treating the kernel functions as basis functions.
 +
 +
 +
Kernels can also be constructed from other kernels using the following rules:
 +
 +
<br>
 +
Let a(x,x') , b(x,x') both be kernel functions<br>
 +
<math> k(x,x') = ca(x,x') \forall c > 0  </math>
 +
<br>
 +
<math> k(x,x') = f(x)a(x,x')f(x') \forall</math> functions f(x)
 +
<br>
 +
<math> k(x,x') =  p(a(x,x')) \forall </math> polynomial functions p with non negative coefficients
 +
<br>
 +
<math>\, k(x,x') = e^{a(x,x')} </math>
 +
<br>
 +
<math>\, k(x,x') = a(x,x') + b(x,x') </math>
 +
<br>
 +
<math>\, k(x,x') = a(x,x')b(x,x') </math>
 +
<br>
 +
<math> k(x,x') = k3(\phi(x),\phi(x')) \forall </math> valid kernels k3 over the dimension of <math>\phi(x)</math>
 +
 +
<math> k(x,x') = x^{T}Ax' \forall A \succeq 0 </math>
 +
<br>
 +
<math>\, k(x,x') = k_c(x_c,x_d') + k_d(x_d,x_d') </math> where <math>\, x_c, x_d </math> are variables with <math>\, x = (x_c,x_d) </math> and where <math>\, k_c, k_d </math> are valid kernel functions
 +
<br>
 +
<math>\, k(x,x') = k_c(x_c,x_a')k_d(x_d,x_d') </math> where <math>\, x_c, x_d </math> are variables with <math>\, x = (x_c,x_d) </math> and where <math>\, k_c, k_d </math> are valid kernel functions
 +
<br>
 +
 +
Once we have chosen the Kernel function, we don't need to figure out what <math>\,\phi</math> is, just use <math>\,\phi(\underline{x}_i)^T\phi(\underline{x}_j) = K(\underline{x}_i,\underline{x}_j) </math> to replace <math>\,\underline{x}_i^T\underline{x}_j</math>.
 +
 +
Reference for rules: [[http://www.cedar.buffalo.edu/~srihari/CSE574/Chap6/Chap6.1-KernelMethods.pdf Rules]]
 +
 +
Since the transformation chosen is dependent on the shape of the data, the only automated way to choose an appropriate kernel is by trial and error. Otherwise it is chosen manually.
 +
 +
====Kernel Functions for Machine Learning Applications====
 +
Beyond the kernel functions we discussed in class (Linear Kernel, Polynomial Kernel and Gaussian Kernel), many more kernel functions can be used in the application of kernel methods for machine learning. Some examples of other kernels are: Exponential Kernel, Laplacian Kernel, ANOVA Kernel, Hyperbolic Tangent (Sigmoid) Kernel, Rational Quadratic Kernel, Multiquadric Kernel, Inverse Multiquadric Kernel, Circular Kernel, Spherical Kernel, Wave Kernel, Power Kernel, Log Kernel, Spline Kernel, B-Spline Kernel, Bessel Kernel, Cauchy Kernel, Chi-Square Kernel, Histogram Intersection Kernel, Generalized Histogram Intersection Kernel, Generalized T-Student Kernel, Bayesian Kernel, Wavelet Kernel, etc. For more details, see http://crsouza.blogspot.com/2010/03/kernel-functions-for-machine-learning.html#kernel_functions.
 +
 +
===Example in Matlab===
 +
 +
The following code, taken verbatim from the lecture, shows how to use Matlab built-in SVM routines (found in the Bioinformatics toolkit) to do classification through support vector machines.
 +
 +
Note: in Matlab R2009, the svmtrain function returns support vectors which should not be marked as such. This could be due to svmtrain centering and rescaling the data automatically. The issue has been fixed in later versions.
 +
 +
load 2_3;
 +
[U,Y] = princomp(X');
 +
data = Y(:,1:2);
 +
l = [-ones(1,200) ones(1,200)];
 +
[train,test] = crossvalind('holdOut',400);
 +
% Gives indices of train and test; so, train is a matrix of 0 or 1, 1 where the point should be used as part of the training set
 +
svmStruct = svmtrain(data(train,:), l(train), 'showPlot', true);
 +
 +
[[File:Svm1.png|frame|center|The plot produced by training on some of the 2_3 data's first two features.]]
 +
 +
yh = svmclassify(svmStruct, data(test,:), 'showPlot', true);
 +
 +
[[File:Svm2.png|frame|center|The plot produced by testing some of the 2_3 data.]]
 +
 +
% SVM kernel trick using rbf as the kernel
 +
yh = svmclassify(svmStruct, data(test,:), 'showPlot', true, 'Kernel_Function','rbf');
 +
 +
===Support Vector Machines as a Regression Technique===
 +
The idea of support vector machines has been also applied on regression problems, called [http://svms.org/regression/ support vector regression]. Still it contains all the main features that characterize maximum margin algorithm: a non-linear function is leaned by linear learning machine mapping into high dimensional kernel induced feature space. The capacity of the system is controlled by parameters that do not depend on the dimensionality of feature space. In the same way as with classification approach there is motivation to seek and optimize the generalization bounds given for regression. They relied on defining the loss function that ignores errors, which are situated within the certain distance of the true value. This type of function is often called – epsilon intensive – loss function. The figure below shows an example of one-dimensional linear regression function with – epsilon intensive – band. The variables measure the cost of the errors on the training points. These are zero for all points that are inside the band (you may want to continue reading this in [http://kernelsvm.tripod.com/ here]).
 +
 +
Here are some papers and works in this matter, by [http://svms.org/regression/SmSc98.pdf A. J. Smola, B. Scholkopf], and [http://www.cmlab.csie.ntu.edu.tw/~cyy/learning/papers/SVR_WellingsNote.pdf M. Welling].
 +
 +
=== 1-norm support vector regression ===
 +
 +
[[image: Norm 1.png]]
 +
 +
Pseudocode for 1-norm support vector regression   
 +
 +
Source: John Shawe-Taylor and Nello Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, illustrated edition edition, June 2004.
 +
 +
=== 2-norm support vector regression ===
 +
 +
[[image: Norm 2.png]]
 +
 +
Pseudocode for 2-norm support vector regression
 +
 +
Source: John Shawe-Taylor and Nello Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, illustrated edition edition, June 2004.
 +
 +
===Extension:Support Vector Machines===
 +
 +
==== Pattern Recognition ====
 +
[http://research.microsoft.com/en-us/um/people/cburges/papers/svmtutorial.pdf]
 +
This paper talks about linear Support Vector Machines for separable and non-separable data by working through a non-trivial example in detail, and also it describes a mechanical analog and when SVM solutions are unique and when they are global. From this paper we can know support vector training can be practically implemented, and the kernel mapping technique which is used to construct SVM solutions which are nonlinear in the data.
 +
 +
Results of some experiments which were inspired by these arguments are also presented.
 +
The writer gives numerous examples and proofs of most of the key theorems, he hopes the people can find old material is cast in a fresh light since the paper includes some new material.
 +
 +
==== Emotion Recognition ====
 +
Moreover, Linear Support Vector Machine (LSVM) is used in emotion recognition from facial expression and voice of subjects. In this approach, different emotional expressions of each subject are extracted. Then, LSVM is used to classify the extracted feature vectors into different emotion classes.[4]
 +
 +
=== Further reading ===
 +
The following are few papers in which different approaches and further explanation on support vector machines are made:
 +
 +
1- Least Squares Support Vector Machine Classifiers [http://www.springerlink.com/content/n75178640w32646j/ here]
 +
 +
2- Support vector machine classification and validation of cancer tissue samples using microarray expression data [http://bioinformatics.oxfordjournals.org/content/16/10/906.abstract here]
 +
 +
3- Support vector machine active learning for image retrieval [http://portal.acm.org/citation.cfm?id=500159 here]
 +
 +
4- Support vector machine learning for interdependent and structured output spaces [http://portal.acm.org/citation.cfm?id=1015341&dl=GUIDE, here]
 +
 +
===References===
 +
 +
1. The genetic kernel support vector machine: Description and evaluation
 +
[http://www.springerlink.com/content/yt25436213h77755/]
 +
 +
2. Improving support vector machine classifiers by modifying kernel functions
 +
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6T08-3XMHNK7-1&_user=10&_coverDate=07%2F31%2F1999&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1540531694&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=79edeef12d9593a6f37de4da09d725b7&searchtype=a]
 +
 +
3. Classification using intersection kernel support vector machines is efficient
 +
[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4587630]
 +
 +
4. Das, S.; Halder, A.; Bhowmik, P.; Chakraborty, A.; Konar, A.; Janarthanan, R.; ,[http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5393891&isnumber=5393306 A support vector machine classifier of emotion from voice and facial expression data],  Nature & Biologically Inspired Computing, 2009. NaBIC 2009. World Congress on , vol., no., pp.1010-1015, 9-11 Dec. 2009.
 +
 +
== ''' Support Vector Machine, Kernel Trick - Cont. Case II - November 16, 2010 ''' ==
 +
 +
==='''Case II: Non-separable data (Soft Margin)'''===
 +
 +
 +
We have seen how SVMs are able to find an optimally separating hyperplane of two separable classes of data, in which case the margin contains no data points. However, in the real world, data of different classes are usually mixed together at the boundary and it's hard to find a perfect boundary to totally separate them. In this, one may want to separate the training data set with the minimal number of errors . To address this problem, we slacken the classification rule to allow data cross the margin. Now each data point can have some error <math>\,\xi_i</math>. However, we only want data to cross the boundary when they have to and make the minimum sacrifice; thus, a penalty term is added correspondingly in the objective function to constrain the number of points that cross the margin. The optimization problem now becomes:
 +
[[File:non-separable.JPG|350px|thumb|right|Figure non-separable case]]
 +
 +
:<math>\min_{\alpha} \frac{1}{2}|\beta|^2+\gamma\sum_{i=1}^n{\xi_i}</math>
 +
:<math>\,s.t.</math>  <math>y_i(\beta^Tx+\beta_0) \geq 1-\xi_i</math>
 +
:<math>\xi_i \geq 0</math>
 +
 +
<br\>Note that <math>\,\xi_i</math> is not necessarily smaller than one, which means data can not only enter the margin but can also cross the separating hyperplane.
 +
 +
<br\>Minimizing the objective, one finds some minimal subset of errors. If these error data are excluded from the training data set, then one can separate the remaining part of training data without errors.
 +
 +
<br\>Soft-margin SVM is a generalization of hard-margin SVM. The hard-margin SVM was presented here before the soft-margin SVM because hard-margin SVM is more intuitive and it was historically conceived first. Note that as <math>\,\gamma \Rightarrow \infty </math>, all <math>\,\xi_i \Rightarrow 0</math>. Intuitively, we would set <math>\,\xi_i = 0</math> if it was known that the classes were separable, that is, if we knew a hard-margin SVM boundary could be used to separate the classes. As shown later in this lecture, the softmargin classifier with <math>\,\gamma = \infty </math> is, in fact, the same optimization problem as the hardmargin classifier mentioned in the previous lecture. By similar logic, it is wise to set higher <math>\,\gamma</math>, for sets that are more separable.
 +
 +
 +
With the formulation of the Primal form for non-separable case above, we can form the Lagrangian.
 +
 +
===Forming the Lagrangian===
 +
In this case we have have two constraints in the [http://en.wikipedia.org/wiki/Lagrangian Lagrangian] primal form and therefore we optimize with respect to two dual variables <math>\,\alpha</math> and <math>\,\lambda</math>,<br>
 +
:<math>L: \frac{1}{2} |\beta|^2 + \gamma \sum_{i} \xi_i - \sum_{i} \alpha_i[y_i(\beta^T x_i+\beta_0)-1+\xi_i]-\sum_{i} \lambda_i \xi_i</math>
 +
:<math>\alpha_i \geq 0, \lambda_i \geq 0</math>
 +
 +
Now we apply KKT conditions, and come up with a new function to optimize. As we will see, the equation that we will attempt to optimize in the SVM algorithm for non-separable data sets is the same as the optimization for the separable case, with slightly different conditions.
 +
 +
===Applying KKT conditions[http://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions]===
 +
# <math>\frac{\partial L}{\partial p} = 0</math> at an optimal solution <math>\, \hat p</math>, for each primal variable <math>\,p = \{\beta, \beta_0, \xi\}</math><br><math>\frac{\partial L}{\partial \beta}=\beta - \sum_{i} \alpha_i y_i x_i = 0 \Rightarrow \beta=\sum_{i}\alpha_i y_i x_i</math> <br\><math>\frac{\partial L}{\partial \beta_0}=-\sum_{i} \alpha_i y_i =0 \Rightarrow \sum_{i} \alpha_i y_i =0</math> since the sign does not make a difference<br><math>\frac{\partial L}{\partial \xi_i}=\gamma - \alpha_i - \lambda_i \Rightarrow \gamma = \alpha_i+\lambda_i</math>.  This is the only new condition added here
 +
#<math>\,\alpha_i \geq 0, \lambda_i \geq 0</math>, dual feasibility
 +
#<math>\,\alpha_i[y_i(\beta^T x_i+\beta_0)-1+\xi_i]=0</math> and <math>\,\lambda_i\xi_i=0</math>
 +
#<math>\,y_i( \beta^T x_i+ \beta_0)-1+ \xi_i \geq 0</math>
 +
 +
=== Objective Function ===
 +
With our KKT conditions and the Lagrangian equation, <math>\,\alpha</math> could be estimated by Quadratic programming.
 +
 +
<br\> Similar to what we did for the separable case after applying KKT conditions, replace the primal variables in terms of dual variables into the Lagrangian equations and simplify as follows:
 +
 +
:<math>L =  \frac{1}{2} |\beta|^2 + \gamma \sum_{i} \xi_i - \beta^T \sum_{i} \alpha_i y_i x_i - \beta_0 \sum_{i} \alpha_i
 +
 +
y_i
 +
 +
+ \sum_{i} \alpha_i - \sum_{i} \alpha_i \xi_i - \sum_{i} \lambda_i \xi_i</math>
 +
 +
From KKT conditions:
 +
:<math> \beta = \sum_{i} \alpha_i y_i x_i \Rightarrow \beta^T\beta = |\beta|^2</math> and
 +
:<math> \displaystyle \sum_{i} \alpha_i y_i = 0</math>
 +
 +
Rewriting the above equation we have:
 +
 +
:<math>L =  \frac{1}{2} |\beta|^2 - |\beta|^2 + \gamma \sum_{i} \xi_i + \sum_{i} \alpha_i - \sum_{i} \alpha_i \xi_i -
 +
 +
\sum_{i} \lambda_i \xi_i</math>
 +
 +
We know that <math>\frac{1}{2} |\beta|^2 - |\beta|^2 = -\frac{1}{2} |\beta|^2 = - \frac{1}{2} \sum_{i} \sum_{j} \alpha_i
 +
 +
\alpha_j  y_i y_j x_i^T x_i </math>
 +
 +
 +
:<math>\Rightarrow L = - \frac{1}{2} \sum_{i} \sum_{j} \alpha_i\alpha_j  y_i y_j x_i^T x_j + \sum_{i} \alpha_i + \sum_{i}
 +
 +
\gamma \xi_i - \sum_{i} \alpha_i \xi_i - \sum_{i} \lambda_i \xi_i</math>
 +
 +
:<math>\Rightarrow L = - \frac{1}{2} \sum_{i} \sum_{j} \alpha_i\alpha_j  y_i y_j x_i^T x_j + \sum_{i} \alpha_i + \sum_{i}
 +
 +
(\gamma - \alpha_i - \lambda_i) \xi_i</math>
 +
 +
We know that by KKT condition <math>\displaystyle \gamma - \alpha_i - \lambda_i = 0 </math>
 +
 +
Finally we have the simplest form of Lagrangian for non-separable case:
 +
 +
:<math>L = \sum_{i} \alpha_i - \frac{1}{2} \sum_{i} \sum_{j} \alpha_i\alpha_j  y_i y_j x_i^T x_j </math>
 +
 +
 +
You can see that there is no difference in objective function of Hard & Soft Margin. Now let's see the constraints for above objective function.
 +
 +
=== Constraints ===
 +
Following will be the constraints of above objective function:
 +
 +
:<math>\,\alpha_i \geq 0 \forall i</math>
 +
:<math>\lambda_i \geq 0 \forall i</math>
 +
:<math>\displaystyle \sum_{i} \alpha_i y_i = 0</math><br />
 +
 +
From the KKT conditions above, we have:<br />
 +
<math>\frac{\partial L}{\partial \xi_i}=\gamma - \alpha_i - \lambda_i \Rightarrow \gamma = \alpha_i+\lambda_i</math><br />
 +
 +
Therefore, If <math>\displaystyle \lambda_i \ge 0 \,\Rightarrow \, \alpha_i \le \gamma</math>, hence, <math>\,\lambda_i \geq 0 </math> constraint can be replaced by <math>\displaystyle \alpha_i \le \gamma</math>.
 +
 +
===Dual Problem or Quadratic Programming Problem===
 +
 +
We have formalized the Dual Problem which is as follows:
 +
 +
:<math>\displaystyle \max_{\alpha_i} \sum_{i}{\alpha_i} - \frac{1}{2}\sum_{i}{\sum_{j}{\alpha_i \alpha_j y_i y_j x_i^T x_j}}</math>
 +
 +
subject to the constraints
 +
:<math> \displaystyle 0 \le \alpha_i \le \gamma </math> and
 +
:<math>\displaystyle \sum_{i}{\alpha_i y_i} = 0</math>
 +
 +
You can see that the only difference in the Hard and Soft Margin is the upper bound of <math>\displaystyle \alpha</math> i.e. <math>\displaystyle \alpha \le \gamma</math>.
 +
 +
As <math>\displaystyle \gamma \rightarrow \infty </math>  soft margin  <math>\displaystyle \rightarrow</math> Hard margin.
 +
 +
=== Recovery of Hyperplane ===
 +
 +
We can easily recover the hyperplane <math>\displaystyle \underline \beta^T \underline x + \beta_0 = 0</math> by finding the values of <math>\displaystyle \underline \beta</math> and <math>\displaystyle \beta_0</math>.
 +
 +
* <math>\displaystyle \underline \beta</math> can be calculated from first KKT condition i.e. <math>\displaystyle \underline \beta = \sum_{i} \alpha_i y_i \underline x_i</math>
 +
 +
* <math>\displaystyle \beta_0</math> can be calculated by choosing a point that satisfy <math> \displaystyle 0 < \alpha_i \le \gamma </math>, then third KKT condition becomes
 +
:: <math>\displaystyle y_i( \underline \beta^T \underline x_i+ \beta_0)=1</math> which can be solved for <math>\displaystyle \beta_0</math>
 +
 +
===SVM algorithm for non-separable data sets===
 +
 +
The algorithm, for non-separable data sets is:
 +
 +
# Use <code>quadprog</code> (or another quadratic programming technique) to solve the above optimization and find <math>\,\alpha</math>
 +
# Find <math>\,\underline{\beta}</math> by solving <math>\,\underline{\beta} = \sum_{i}{\alpha_i y_i \underline x_i}</math>
 +
# Find <math>\,\beta_0</math> by choosing a point where <math>\,0 < \alpha_i \le \gamma</math> and then solving <math>\,y_i(\underline{\beta}^T \underline x_i + \beta_0) - 1 = 0</math>
 +
 +
=== Support Vectors ===
 +
 +
Kernel-based techniques (such as support vector machines, Bayes point
 +
machines, kernel principal component analysis, and Gaussian processes) represent
 +
a major development in machine learning algorithms. Support vector
 +
machines (SVM) are a group of supervised learning methods that can be
 +
applied to classification or regression.<ref name="cccc"> Ovidiu Ivanciuc, Review: Applications of Support Vector Machines in Chemistry, Rev. Comput. Chem. 2007, 23, 291-400</ref>Support vectors are the training points that determine the optimal separating hyperplane that we seek. Also, they are the most difficult points to classify and at the same time the most informative for classification.
 +
 +
For the non-separable case, the third KKT condition yields: if <math>\displaystyle \alpha_i > 0 \Rightarrow y_i(\underline \beta^T \underline x_i+\beta_0)-1+\xi_i=0</math>. These above points are called support vectors.
 +
 +
* Case 1: Support Vectors are on the Margin
 +
::If <math>\displaystyle \lambda_i > 0 \Rightarrow \xi_i = 0 </math>, then this support vector is on the margin.
 +
 +
* Case 2: Support Vectors are inside the Margin
 +
::If <math>\displaystyle \alpha = \gamma</math>, then this support vectors is inside the margin.
 +
 +
===  Support Vectors Machine Demo Tool ===
 +
 +
[[image:SVM_Demo.png]]
 +
 +
This demo tool shows the linear boundary found by SVM and illustrates its behaviour on some 2D data. This is an interactive demonstration to provide insight into how SVM finds the classification boundary.[http://www.mathworks.com/matlabcentral/fileexchange/28302-svm-demo File]
 +
 +
 +
=== Relevance Vector Machines ===
 +
Support vector machines have been used in a variety of classification and regression applications. Nevertheless, they suffer from a number of limitations, several of which have been highlighted already in earlier sessions. In particular, the outputs of an SVM represent decisions rather than posterior probabilities. Also, the SVM was originally formulated for two classes, and the extension toK > 2 classes is problematic.
 +
There is a complexity parameter C, or ν (as well as a parameter epsilon in the case of regression), that must be found using a hold-out method such as cross-validation. Finally, predictions are expressed as linear combinations of kernel functions that are centred on training data points and that are required to be positive definite.
 +
 +
The relevance vector machine or RVM (Tipping, 2001) is a Bayesian sparse kernel technique for regression and classification that shares many of the characteristics of the SVM whilst avoiding its principal limitations. Additionally, it typically leads to much sparser models resulting in correspondingly faster performance on test data whilst maintaining comparable generalization error.
 +
 +
Borrowed from Bishop's great book, Pattern Recognition and Machine Learning [http://research.microsoft.com/en-us/um/people/cmbishop/prml/]. For more details on RVM, you may refer to this book.
 +
 +
===  Further reading on the Kernel Trick ===
 +
1- The kernel trick for distances [http://74.125.155.132/scholar?q=cache:AfKdFY6a1cMJ:scholar.google.com/&hl=en&as_sdt=2000 here]
 +
 +
2- Exploiting the kernel trick to correlate fragment ions for peptide identification via tandem mass spectrometry [http://bioinformatics.oxfordjournals.org/content/20/12/1948.short here]
 +
 +
3- Kernel-based methods and function approximation [http://ieeexplore.ieee.org/xpl/freeabs_all.jsparnumber=939539 here]
 +
 +
4- SVM-KNN: Discriminative Nearest Neighbor Classification for Visual Category Recognition [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1641014 here]
 +
 +
5- SVM application list[http://www.clopinet.com/isabelle/Projects/SVM/applist.html]
 +
 +
6- Some readings about SVM  and the kernel trick [http://www.cs.cmu.edu/~guestrin/Class/10701-S07/Slides/kernels.pdf] and [http://www.cs.cmu.edu/~tom/10601_sp08/slides/svm3-26.ppt]
 +
 +
7- General overview of SVM and Kernel Methods. Easy to understand presentation. [http://www.support-vector.net/icml-tutorial.pdf]
 +
 +
== ''' Naive Bayes,  K Nearest Neighbours, Boosting, Bagging and Decision Trees,  - November 18, 2010 ''' ==
 +
 +
 +
Now that we've covered a number of more advanced classification algorithms, we can look at some of the simpler classification algorithms that are usually discussed at the beginning of a discussion on classification.
 +
 +
=== [http://en.wikipedia.org/wiki/Naive_Bayes_classifier Naive Bayes Classifiers] ===
 +
 +
Recall that one of the major drawbacks of the Bayes classifier was the difficulty in estimating a joint density in a multidimensional space.  Naive Bayes classifiers are one possible solution to the problem.  They are especially popular for problems with high-dimensional features.
 +
 +
 +
A naive Bayes classifier applies a strong independence assumption to the conditional probability <math>\ P(X|Y) = P(x_1,x_2,...,x_d |Y)</math>. It assumes that inputs within each class are conditionally independent.  In other words, it assumes the dimensions of the inputs in each class are independent. The Naive Bayes classifier does this by reducing the number of parameters to be estimated dramatically when modeling <math>\ P(X|Y)</math>.
 +
 +
Under the conditional independence assumption:
 +
 +
<math>\ P(X|Y) = P(x_1,x_2,...,x_d |Y) =\prod_{i=1}^{d}P(X = x_i | Y)</math>.
 +
 +
====Naive Bayes is Equivalent to a Generalized Additive Model====
 +
 +
Continuing with the notation used above, consider a binary classification problem <math>\, y \in \{0,1\}</math>. Beginning with the corresponding Bayes classifier, we have
 +
 +
:<math>\, h^*(x)= \left\{\begin{matrix}
 +
1 &\text{if } P(Y=1|X=x)>P(Y=0|X=x)  \\
 +
0 &\mathrm{otherwise}  \end{matrix}\right.</math>
 +
 +
:<math>\, = \left\{\begin{matrix}
 +
1 &\text{if } \log\left(\frac{P(Y=1|X=x)}{P(Y=0|X=x)}\right)>0  \\
 +
0 &\mathrm{otherwise}  \end{matrix}\right.</math>
 +
 +
:<math>\, = \frac{1}{2}\text{sign}\left[\log\left(\frac{P(Y=1|X=x)}{P(Y=0|X=x)}\right)\right] + \frac{1}{2}</math>
 +
 +
:<math>\, = \frac{1}{2}\text{sign}\left[\log\left(\frac{P(Y=1)P(X=x|Y=1)}{P(Y=0)P(X=x|Y=0)}\right)\right] + \frac{1}{2}</math>
 +
 +
then, by the conditional independence assumption provided by Naive Bayes,
 +
 +
:<math>\, = \frac{1}{2}\text{sign}\left[\log\left(\frac{P(Y=1)}{P(Y=0)} \prod_{i=1}^d \frac{P(X_i=x_i|Y=1)}{P(X_i=x_i|Y=0)}\right)\right] + \frac{1}{2}</math>
 +
 +
:<math>\, = \frac{1}{2}\text{sign}\left[\log\left(\frac{P(Y=1)}{P(Y=0)}\right) + \sum_{i=1}^d \log\left(\frac{P(X_i=x_i|Y=1)}{P(X_i=x_i|Y=0)}\right)\right] + \frac{1}{2}</math>
 +
 +
which is a generalized additive model.
 +
 +
==== Naive Bayes for Continuous Input ====
 +
 +
A naive Bayes classifier applies a strong independence assumption to the class density <math>\,f_{k}(x)</math>. 
 +
 +
Recall that the Bayes rule is :
 +
 +
<math>\ h(x) = \arg\max_{ k}  \pi_{k}f_{k}(x). </math>
 +
 +
Although the Bayes classifier is the best classifier, in practice, it is difficult to give an estimate for the multi-variable prior probabilities which are required to determine the classification. Therefore, by assuming independence between the features, we can transform an n-variable distribution into n independent one-variable distributions which are easier to handle, and then apply the Bayes classification.
 +
 +
The density function of inputs can be written as below under the independence assumption :
 +
 +
<math>\ f_{k}(x) = f_{k}(x_1 ,x_2,...,x_d) = \prod_{j=1}^d f_{kj}(x_{j})</math>
 +
 +
Each of the <math>\,d</math> marginal densities can be estimated separately using one-dimensional density estimates.  If one of the components <math>\,x_{j}</math> is discrete then its density can be estimated using a histogram.  We can thus mix discrete and continuous variables in a naive Bayes classifier.
 +
 +
Naive Bayes classifiers often perform extremely well in practice despite these 'naive' and seemingly optimistic assumptions.  This is because while individual class density estimates could be biased, the bias does not carry through to the posterior probabilities. 
 +
 +
It is also possible to train naive Bayes classifiers using maximum likelihood estimation.
 +
 +
An interesting example by Jose M. Vidal that shows how the naive Bayes classifier can be used to solve a real-world classification task is available [http://jmvidal.cse.sc.edu/talks/bayesianlearning/nbex.xml here].
 +
 +
==== Naive Bayes for Discrete Inputs ====
 +
 +
Naive Bayes with discrete inputs is very similar to that of continuous inputs. From examples researched, the major difference is that instead of using a probability distribution to characterize the likelihood, we use feature frequencies, or (in English) the proportion of time cases in which variables X fall under class C vs. total number of cases that fall under class C. The following example shows how this would work:
 +
 +
You are running a scientific study meant to find the optimal features under which a girl you encounter will wear her glasses. The data you collect represent the setting of your encounter (library, park, bar), whether she is a student or not (yes, no), and what her eye colour is (blue, green, brown).
 +
 +
{|
 +
|-
 +
! scope="col" | Case
 +
! scope="col" | Setting
 +
! scope="col" | Student
 +
! scope="col" | Eye colour
 +
! scope="col" | Wears glasses?
 +
|-
 +
! scope="row" | 1
 +
| Bar || yes || Blue || no
 +
|-
 +
! scope="row" | 2
 +
| Park || yes || Brown || yes
 +
|-
 +
! scope="row" | 3
 +
| Library || no || Green || yes
 +
|-
 +
! scope="row" | 4
 +
| Library || no || Blue || no
 +
|-
 +
! scope="row" | 5
 +
| Bar || no || Brown || yes
 +
|-
 +
! scope="row" | 6
 +
| Park || yes || Green || yes
 +
|-
 +
! scope="row" | 7
 +
| Bar || no || Brown || yes
 +
|-
 +
! scope="row" | 8
 +
| Library || yes || Brown || yes
 +
|-
 +
! scope="row" | 9
 +
| Bar || yes || Green || no
 +
|-
 +
! scope="row" | 10
 +
| Park || yes || Blue || no
 +
|}
 +
 +
 +
From this, we extract the following feature frequencies:
 +
 +
{|
 +
|-
 +
! scope="col" | Eye Colour
 +
! scope="col" | Wearing glasses
 +
! scope="col" | Not wearing glasses
 +
|-
 +
! scope="row" | Blue
 +
| 0 || 3
 +
|-
 +
! scope="row" | Brown
 +
| 4 || 0
 +
|-
 +
! scope="row" | Green
 +
| 2 || 1
 +
|}
 +
 +
{|
 +
|-
 +
! scope="col" | Student?
 +
! scope="col" | Wearing glasses
 +
! scope="col" | Not wearing glasses
 +
|-
 +
! scope="row" | Not a student
 +
| 3 || 1
 +
|-
 +
! scope="row" | Student
 +
| 3 || 3
 +
|}
 +
 +
{|
 +
|-
 +
! scope="col" | Setting
 +
! scope="col" | Wearing glasses
 +
! scope="col" | Not wearing glasses
 +
|-
 +
! scope="row" | Bar
 +
| 2 || 2
 +
|-
 +
! scope="row" | Library
 +
| 2 || 1
 +
|-
 +
! scope="row" | Park
 +
| 2 || 1
 +
|}
 +
 +
You also note that of the 10 girls you saw, 6 were wearing their glasses and 4 weren't. Therefore, given the new case of a green eyed student in a bar, we calculate the probabilities of her wearing vs. not wearing her glasses as such:
 +
 +
P(Wearing glasses | green eyed student in a bar) = P(Wearing glasses | student)*P(Wearing glasses | green eyed)*P(Wearing glasses | in a bar) = 3/6 * 2/6 * 2/6 = 0.0556<br>
 +
P(Not wearing glasses | green eyed student in a bar) = P(Not wearing glasses | student)*P(Not wearing glasses | green eyed)*P(Not wearing glasses | in a bar) = 3/4 * 1/4 * 2/4 = 0.09375<br>
 +
 +
Since P(Wearing glasses | green eyed student in a bar) < P(Not wearing glasses | green eyed student in a bar), it is not likely that a green eyed student will be wearing her glasses in a bar.
 +
 +
==== Further reading Naive Bayes  ====
 +
 +
The following are some papers to show how Naive Bayes is used in different aspects of classifications.
 +
 +
1- An empirical study of the naive Bayes classifier [http://www.cc.gatech.edu/home/isbell/classes/reading/papers/Rish.pdf here]
 +
 +
2- Naive (Bayes) at forty: The independence assumption in information retrieval [http://www.springerlink.com/content/wu3g458834583125/ here]
 +
 +
3- Emotion Recognition Using a Cauchy Naive Bayes Classifier [http://www.computer.org/portal/web/csdl/doi/10.1109/ICPR.2002.1044578 here]
 +
 +
===  References ===
 +
 +
1. Scaling up the accuracy of naive-Bayes classifiers: A decision-tree hybrid
 +
[http://www.cs.ust.hk/~qyang/537/Papers/kohavi96scaling.pdf]
 +
 +
2. A comparative study of discretization methods for naive-bayes classifiers
 +
[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.15.298&rep=rep1&type=pdf]
 +
 +
3. Semi-naive Bayesian classifier
 +
[http://www.springerlink.com/content/m4p7863g61502515/]
 +
 +
=== [http://en.wikipedia.org/wiki/K-nearest_neighbor_algorithm K-Nearest Neighbors Classification] ===
 +
 +
 +
<math>\,K</math>-Nearest Neighbors is a very simple algorithm that classifies points based on a majority vote of the <math>\ k</math> nearest points in the feature space, with the object being assigned to the class most common among its <math>\ k</math> nearest neighbors. <math>\ k</math> is a positive integer, typically small which can be chosen using cross validation. If <math>\ k=1</math>, then the object is simply assigned to the class of its nearest neighbor.
 +
 +
1. Ties are broken at random.
 +
 +
2. If we assume the features are real, we can use the Euclidean distance in feature space. More complex distance measures such as an adaptive [http://en.wikipedia.org/wiki/Mahalanobis_distance Mahalanobis distance] that is detailed in Verdier ''et al.'''s [http://www.emse.fr/~verdier/ENSMSE%20CMP%20WP2009_14.pdf paper] can be used as well.
 +
 +
3. Since the features are measured in different units, we can standardize the features to have mean zero and variance 1.
 +
 +
4. K can be chosen by cross-validation.
 +
 +
====  Advantage:====
 +
Robust to noisy training data (especially if we use inverse square of weighted distance as the “distance”)[http://people.revoledu.com/kardi/tutorial/KNN/What-is-K-Nearest-Neighbor-Algorithm.html]
 +
 +
Effective if the training data is large.[http://people.revoledu.com/kardi/tutorial/KNN/What-is-K-Nearest-Neighbor-Algorithm.html]
 +
 +
====Disadvantage:====
 +
 +
Need to determine value of parameter K (number of nearest neighbors)[http://people.revoledu.com/kardi/tutorial/KNN/What-is-K-Nearest-Neighbor-Algorithm.html]
 +
 +
Distance based learning is not clear which type of distance to use and which attribute to use to produce the best results.[http://people.revoledu.com/kardi/tutorial/KNN/What-is-K-Nearest-Neighbor-Algorithm.html]
 +
 +
Misclassification rate is large when training data is small.
 +
 +
A major drawback is that if  the frequency of one class is greater than the other ones significantly , the samples in this class with the largest frequency tend to dominate the prediction of a new point . An approach to overcome it is  attaching weights to the samples ,for instance ,add  larger weights to the neighbors which are closer to the new points than those that are further away.
 +
 +
====Note of Interest:====
 +
In K-nearest neighbours over fitting may occur when a small k is used. In contrast to other methods, in K-nearest neighbours k=1 is the most complex case.
 +
 +
====Property[http://en.wikipedia.org/wiki/K-nearest_neighbor_algorithm#Properties]====
 +
 +
K-nearest neighbor algorithm has some good and strong results. As the number of data points goes to infinity, the algorithm is guaranteed to yield an error rate no worse than twice the Bayes error rate (the minimum achievable error rate given the distribution of the data). K-nearest neighbor is guaranteed to approach the Bayes error rate, for some value of k (where k increases as a function of the number of data points).
 +
See ''Nearest Neighbour Pattern Classification'', T.M. Cover and P.E. Hart, for interesting theoretical results about the algorithm, including proof of the above properties.
 +
 +
Effect of K on Bias Variance Trade-off for KNN
 +
 +
Large K high bias , low variance
 +
Samll K low bias , high variance
 +
 +
==== Algorithm ====
 +
Here is step by step on how to compute K-nearest neighbors KNN algorithm:
 +
 +
1. Determine number of nearest neighbors (K-parameter).
 +
 +
2. Calculate the distance between the query-instance and all the training samples.
 +
 +
3. Sort the distance and determine nearest neighbors based on the 'K-th' minimum distance.
 +
 +
4. Gather the category of the nearest neighbors.
 +
 +
5. Use simple majority of the category of nearest neighbors as the prediction value of
 +
the query instance. A random tie-break is used if each class results in the same number of neighbors.
 +
 +
==== Working Example ====
 +
 +
We have data from examination laboratory and the objective testing with two attributes (having a flu and his temperature is high) to classify whether a person has a flu or not. Next table shows the four training samples we have:
 +
 +
{| class="wikitable"
 +
|-
 +
! X1 = having Flu
 +
! X2= having high temperature
 +
! Y = Classification
 +
|-
 +
| 7
 +
| 7
 +
| Bad - Condition
 +
|-
 +
| 7
 +
| 4
 +
| Bad - Condition
 +
|-
 +
| 3
 +
| 4
 +
| Good - Condition
 +
|-
 +
| 1
 +
| 4
 +
| Good - Condition
 +
|}
 +
 +
 +
Now we have a new patient that pass laboratory test with X1 = 3 and X2 = 7. Without another expensive survey, can we guess what the condition (classification) of this new patient is?
 +
 +
==== Applying K-NN ====
 +
 +
1. Determine parameter K = number of nearest neighbors, Let us assume that K = 3.
 +
 +
2. Calculate the distance between the query-instance and all the training samples:
 +
Coordinate of query instance is (3, 7), instead of calculating the distance we compute square distance which is faster to calculate (without square root)
 +
 +
{| class="wikitable"
 +
|-
 +
! X1
 +
! X2
 +
! Square Distance to query instance (3, 7)
 +
! Rank minimum distance
 +
! Is it included in 3-Nearest neighbors?
 +
|-
 +
| 7
 +
| 7
 +
| (7-3).^2+(7-7).^2=16
 +
| 3
 +
| Yes
 +
|-
 +
| 7
 +
| 4
 +
| (7-3).^2+(4-7).^2=25
 +
| 4
 +
| No
 +
|-
 +
| 3
 +
| 4
 +
| (3-3).^2+(4-7).^2=9
 +
| 1
 +
| Yes
 +
|-
 +
| 1
 +
| 4
 +
| (1-3).^2+(4-7).^2=13
 +
| 2
 +
| Yes
 +
|}
 +
 +
 +
4. Gather the category  of the nearest neighbors. Notice in the second row last column that the category of nearest neighbor (Y) is not included because the rank of this data is more than 3 (=K).
 +
{| class="wikitable"
 +
|-
 +
! X1
 +
! X2
 +
! Square Distance to query instance (3, 7)
 +
! Rank minimum distance
 +
! Is it included in 3-Nearest neighbors?
 +
! Y = Category of nearest Neighbor
 +
|-
 +
| 7
 +
| 7
 +
| (7-3).^2+(7-7).^2=16
 +
| 3
 +
| Yes
 +
| Bad
 +
|-
 +
| 7
 +
| 4
 +
| (7-3).^2+(4-7).^2=25
 +
| 4
 +
| No
 +
| -
 +
|-
 +
| 3
 +
| 4
 +
| (3-3).^2+(4-7).^2=9
 +
| 1
 +
| Yes
 +
| Good
 +
|-
 +
| 1
 +
| 4
 +
| (1-3).^2+(4-7).^2=13
 +
| 2
 +
| Yes
 +
| Good
 +
|}
 +
 +
 +
5. Use simple majority of the category of nearest neighbors as the prediction value of the query instance.
 +
 +
We have 2 good and 1 bad, since 2>1 then we conclude that a new patient that pass laboratory test with X1 = 3 and X2 = 7 is included in Good Condition category.
 +
 +
====Example in Matlab====
 +
 +
sample = [.9 .8;.1 .3;.2 .6]
 +
training=[0 0;.5 .5;1 1]
 +
group = [1;2;3]
 +
class = knnclassify(sample, training, group)
 +
 +
=== Boosting ===
 +
 +
[http://en.wikipedia.org/wiki/Boosting Boosting] algorithms are a class of machine learning meta-algorithms that can improve weak classifiers.The idea is to incorporate unequal weights in learning process given higher weights to misclassified points . If we have different weak classifiers which slightly do better than random classification, then by assigning larger weights to points which are misclassified and  minimizing the new cost function by choosing an optimal weak classifier,we can update the weights in a way related to  the minimum value of the new cost function. This procedure can be repeated for a finite number of times and then a new classifier which is a weighed aggregation of the generated classifiers will be used as the boosted classifier. The better each generated classifier is the more its weight is in the final classifier.
 +
 +
[http://www.site.uottawa.ca/~stan/csi5387/boost-tut-ppr.pdf Paper about Boosting]:
 +
Boosting is a general method for improving the accuracy of any given learning algorithm.
 +
This paper introduces the boosting algorithm AdaBoost, and explains the underlying theory of boosting, including an explanation of why boosting often does not suffer
 +
from overfitting as well as boosting’s relationship to support-vector machines. Finally, this paper gives some examples of recent applications of boosting.
 +
 +
Boosting is a general method of producing a very accurate prediction rule by combining rough and moderately inaccurate "rules of thumb." Much recent work has been on the "AdaBoost" boosting algorithm and its extensions.
 +
[http://www.cs.princeton.edu/~schapire/boost.html]
 +
 +
==== [http://en.wikipedia.org/wiki/AdaBoost AdaBoost] ====
 +
AdaBoost (which is short for adaptive boosting) is a linear classifier with all its desirable properties. Its output converges to the logarithm of likelihood ratio.
 +
It has good generalization properties and is a feature selector with a principled strategy (minimization of upper
 +
bound on empirical error).
 +
AdaBoost produces a sequence of gradually more complex classifiers).
 +
 +
Advantages
 +
 +
*Very simple to implement
 +
*Feature selection on very large sets of features
 +
*Fairly good generalization
 +
 +
Disadvantages
 +
 +
*Suboptimal solution for <math>\,\Rightarrow\alpha</math>
 +
*Can overfit in presence of noise
 +
 +
[[File:1111.JPG|200px|thumb|right|Fig1.j=1]]
 +
[[File:2222.JPG|200px|thumb|right|Fig2.j=2]]
 +
[[File:3333.JPG|200px|thumb|right|Fig3.j=3]]
 +
[[File:4444.JPG|200px|thumb|right|Fig4.j=4]]
 +
[[File:5555.JPG|200px|thumb|right|Fig5.j=5]]
 +
[[File:6666.JPG|200px|thumb|right|Fig6.j=6]]
 +
[[File:7777.JPG|200px|thumb|right|Fig7.j=7]]
 +
[[File:8888.JPG|200px|thumb|right|Fig8.j=J]]
 +
 +
==== AdaBoost Algorithm ====
 +
 +
 +
 +
Let's first look at the adaptive boosting algorithm:
 +
#Set all the weights of all points equal <math>w_i\leftarrow \frac{1}{n}</math> where we have <math>\,n</math> points.
 +
#For <math>j=1,\dots, J</math>
 +
## Find <math>h_j:X\rightarrow \{-1,+1\}</math> that minimizes the weighted error <math>\,L_j</math><br><math>h_j=\mbox{argmin}_{h_j \in H} L_j </math> where <math>L_j=\frac{\sum_{i=1}^n w_i I[y_i\neq h_j(x_i)]}{\sum_{i=1}^n w_i} </math>.<br /><math>\ H </math> is a set of classifiers which need to be improved and <math>I</math> is:<br />
 +
:<math>\, I= \left\{\begin{matrix}
 +
1 &  for \quad  y_i\neq h_j(x_i) \\
 +
0 &  for \quad y_i = h_j(x_i)  \end{matrix}\right.</math><br />
 +
## Let <math>\alpha_j\leftarrow\log(\frac{1-L_j}{L_j})</math>
 +
## Update the weights: <math>w_i\leftarrow w_i e^{a_j I[y_j\neq h_j(x_i)]}</math>
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#The final hypothesis is <math>h(x)=\mbox{sign}\left(\sum_{j=1}^J \alpha_j h_j(x)\right)</math><br />
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The final hypothesis <math>h(x)</math> can be completely nonlinear. <br />
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* If we have a classifier that is random <math> {L_j} = 0 \Rightarrow \alpha_j = 0</math>, where else if the classifier is a little bit better than chance <math> \alpha_j\ \geq 0 </math>
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* If we have a good classifier and incorrectly misclassified <math>{x_i}</math>, then <math>{w_i}</math> is increased heavily
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When applying Adaboosting to different classifiers, the first step in 2 may be different since we can define the most proper misclassification error according to the problem. However, the major idea is to give higher weight to misclassified examples, which does not change across classifiers.
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AdaBoosting works very well in practice, and there are a lot of research and published works on why it has a good performance. One possible explanation is that it actually maximizes the margin of classifiers.
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We can see that in AdaBoost if training points are accurately classified, then their weights of being used in the next classifier is kept unchanged, while if points are not accurately classified, their weights of being used again is raised.  At a result easier examples get classified in the very first few classifiers and hard examples are learned later with increasing emphasis. Finally, all the classifiers are combined through a majority vote, which is also weighted by their accuracy, taking consideration of both the easy and hard points. In other words, the Boost focuses on the more informative or difficult points.
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A short but interesting video by Kai O. Arras that shows how AdaBoost can create a strong classifier of a toy problem is available [http://www.youtube.com/watch?v=k4G2VCuOMMg here].
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==== Training and Test Error of Boosting.====
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The most basic theretical property of AdaBoost concerns its ability to reduce the training error.Suppose that the cost function <math>\ L_j = \frac{1}{2}- \gamma_{j}, \gamma_{j}>0 </math> .Freund and Schapire[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.8918] prove that training error of the final hypothesis h is at most
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<math>\ \prod_{j} 2 \sqrt{L_j(1-L_j)}= \prod \sqrt{1-4 \gamma_j^2} \leq
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e^{-2 \Sigma_{j} \gamma_j^2} </math> .
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Thus , if each weak classifier is slightly better than random which means <math>\ \gamma_j > 0 </math>, the training error drops exponentially fast .
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Freund and Schapire[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.8918] show that the true error, with high probability , is at most
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<math>\ \hat{Pr}[H(x) \neq y]+ \tilde{O} (\sqrt{\frac{m}{TD}}) </math>
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where <math>\ T </math> is the number of boosting rounds and <math>\ \hat{Pr} [.] </math>
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denotes the empirical probability on training sample.
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This bounds suggests that AdaBoost will overfit if run too many rounds. In fact , this sometimes happen. However, in early experiments, several author abserved empirically that boost often does not overfit even run for thousands of times .Moreover, it was abserved that Adaboost would sometimes continue to drive down the true error after the training error had reached zero.
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Therefor Boosting often does not suffer from overfitting .[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.107.3285]
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==== AnyBoost ====
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Many boosting algorithms belong to a class called AnyBoost which are gradient descent algorithms for choosing linear combinations of elements of an inner product space in order to minimize some cost function.
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We are primarily interested in weighted combinations of classifiers <math>H(x) = sgn(\sum_{j=1}^J \alpha_j h_j(x))</math>
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We want to find H such that the cost functional <math>C(F) = \frac{1}{m}\sum_{i=1}^m c(y_i F(x_i))</math> is minimized for a suitable cost function <math>c</math>
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<math>h_j:X\rightarrow \{-1,+1\}</math> are weak base classifiers from some class <math>\ H</math> and <math> \alpha_j</math> are classifier weights.  The margin of an example <math>(x_i,y_i)</math> is defined by <math>y_i H(x_i)</math>.
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The base hypotheses h and their linear combinations H can be considered to be elements of an inner product function space <math>(S,\langle,\rangle)</math>.
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We define the inner product as <math>\langle F,G \rangle = \frac{1}{m}\sum_{i=1}^m F(x_i) G(x_i)</math> but the AnyBoost algorithm is valid for any cost function and inner product.  We have a function <math>H</math> as a linear combination of base classifiers and wish to add a base classifier h to H so that cost <math>\ C(H + \epsilon h)</math> decreases for arbitrarily small <math> \epsilon</math>.  The direction we seek is found by maximizing <math>-\langle\nabla C(H),h\rangle</math>
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AnyBoost algorithm:
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#<math>\ H_0(x) = 0</math>
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#For <math>j=0,\dots, J</math>
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## Find <math>h_{j+1}:X\rightarrow \{-1,+1\}</math> that maximizes the inner product <math>-\langle\nabla C(H),h_{j+1}\rangle</math>
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## If <math>-\langle\nabla C(H),h_{j+1}\rangle \leq 0 </math> then
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### Return <math>\ H_j</math>
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## Choose step size <math>\ \alpha_{j+1}</math>
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## <math>\ H_{j+1} = H_j + \alpha_{j+1} h_{j+1}</math>
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#The final classifier is <math>\ H_{J+1}</math>
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Figures 1 to 8 show how Anyboost algorithm works to classify the data.
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Other voting methods, including AdaBoost, can be viewed as special cases of this algorithm.
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====Connection between Boost and  Support Vector Machine====
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There are some relationships between Boost and Support Vector Machines. Freund and Schapire[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.107.3285]show that Adaboost and SVMs can be described in a way that they have a similar goal of maximizing a minimal margin while with different norms.
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Combination with boost and SVM  is proved to be beneficial[http://www.springerlink.com/content/bg1xcjbn86349y2e/] .One method is to boost the SVMs with different norms such as <math>\ l_1 </math> norm , <math>\ l_{\infty} </math>. While the <math>\ l_2 </math> norm SVMs is widely used , other norms are useful in some special cases .Here is some papers which provide some methods to combine boost and SVM :
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A Method to Boost Support Vector Machines.[http://www.springerlink.com/content/bg1xcjbn86349y2e/ here]
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Adaptive Boosting of Support Vector Machine Component Classifiers Applied in Face Detection.[http://www.ece.rice.edu/~sv4/papers/EBC_86_607.pdf here]
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===Boosting k-Nearest Neighbor Classifier===
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As the author stated, although the k-nearest neighbours classifier is one of the most widely used methods of classification due to several interesting features, no successful method has been reported so far to apply boosting to k-NN. As boosting methods have proved very effective in improving the generalization capabilities of many classification algorithms, proposing an appropriate application of boosting to k-nearest neighbours is of great interest. In the article, http://cib.uco.es/documents/TR-2008-03.pdf,  Nicolas Garcıa Pedrajas gave more details about how to combine the boosting methods into KNN method, also the brief summary of related work on KNN and boosting methods is presented. Finally, the comparison of evaluation on methods is given under an experimental data.
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===  Reference ===
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The Elements of Statistical Learning, Second Edition. Trevor Hastie,Robert Tibshirani,Jerome Friedman.
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K-Nearest Neighbors Tutorial.[http://people.revoledu.com/kardi/tutorial/KNN/What-is-K-Nearest-Neighbor-Algorithm.html]
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A Method to Boost Support Vector Machines.[http://www.springerlink.com/content/bg1xcjbn86349y2e/]
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=== Bagging ===
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==== History ====
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Bagging ('''B'''ootstrap '''agg'''regat'''ing''') was proposed by [[Leo Breiman]] in 1994 to improve the classification by combining classifications of randomly generated training sets. See Breiman, 1994. Technical Report No. 421.
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Bagging, or [http://en.wikipedia.org/wiki/Bootstrap_aggregating bootstrap aggregating], is another technique used to reduce the variance of classifiers with high variability.  It exploits the fact that a bootstrap mean is approximately equal to the posterior average.  It is most effective for highly nonlinear classifiers such as decision trees. In particular because of the highly unstable nature of these classifiers, they stand most likely to benefit from bagging.
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Bagging is one of the most effective computationally intensive procedures to improve on unstable estimators or classifiers, useful especially for high dimensional data set problems. Hard decisions create instability, and bagging is shown to smooth such hard decisions, yielding smaller variance and mean squared error.
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==== Bagging Classifier ====
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The idea is to train classifiers <math>\ h_{1}(x)</math> to <math>\ h_{B}(x)</math> using B bootstrap samples from the data set.  The final classification is obtained using an average or 'plurality vote' of the B classifiers as follows:
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:<math>\, h(x)= \left\{\begin{matrix}
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1 &  \frac{1}{B} \sum_{i=1}^{B} h_{b}(x) \geq \frac{1}{2}  \\
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0 &  \mathrm{otherwise}  \end{matrix}\right.</math>
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Many classifiers, such as trees, already have underlying functions that estimate the class probabilities at <math>\,x</math>.  An alternative strategy is to average these class probabilities instead of the final classifiers.  This approach can produce bagged estimates with lower variance and usually better performance.
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==== Example: Ozone data ====
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This example illustrates the basic principles of bagging.[http://en.wikipedia.org/wiki/Bootstrap_aggregating Ozone Data]
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=== Boosting vs. Bagging ===
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• Bagging doesn’t work so well with stable models.Boosting might still help.
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• Boosting might hurt performance on noisy datasets. Bagging doesn’t have this problem.
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• In practice bagging almost always helps.
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• On average, boosting usually helps more than bagging, but it is also more common for boosting to hurt performance.
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• The weights grow exponentially.
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• Bagging is easier to parallelize.
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==== Reference ====
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1. CS578 Computer Science Dept., Cornell University, Fall 2004
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2. An empirical comparison of voting classification algorithms: Bagging, boosting, and variants
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[http://www.springerlink.com/content/l006m1614w023752/]
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3. Bagging predictors
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[http://www.springerlink.com/content/l4780124w2874025/]
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====Example====
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An example given by comparison of the bagging and the boosting methods http://www.doiserbia.nb.rs/ft.aspx?id=1820-02140602057M
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===Decision Trees===
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A "decision tree" is used as a visual and analytical decision support tool, where the expected values of competing alternatives are calculated. It uses principle of divide and conquer for classification. Decision trees have traditionally been created manually. Trees can be used for classification, regression, or both.  Trees map features of a decision problem onto a conclusion, or label.
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We fit a tree model by minimizing some measure of impurity.  For a single covariate <math>\,X_{1}</math> we choose a point t on the real line that splits the real line into two sets R1 = <math>(-\infty,t]</math>, R2 = <math>[t,\infty)</math> in a way that minimizes impurity.
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We denote by <math> \hat p_{s}(j) </math> the proportion of observations in <math>\ R_{s}</math> that <math>\ Y_{i} = j</math>.
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<math> \hat p_{s}(j) = \frac{\sum_{i = 1}^{n} I(Y_{i} = j,X_{i} \in R_{s})}{\sum_{i = 1}^{n} I(X_{i} \in R_{s})}</math>
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==== CART ====
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Classification and regression trees (CART) is a non-parametric Decision tree learning technique that produces either classification or regression trees, depending on whether the dependent variable is categorical or numeric, respectively. (Wikipedia)
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Classification and Regression Trees is a classification method which uses historical data to construct so-called decision trees. Decision trees are then used to classify new data. In order to use CART we need to know number of classes a priori. ([http://edoc.hu-berlin.de/master/timofeev-roman-2004-12-20/PDF/timofeev.pdf])
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CART methodology was developed in 80s by Breiman, Freidman, Olshen, Stone in their paper ”Classification and Regression Trees” (1984). For building decision trees, CART uses so-called learning sample - a set of historical data with pre-assigned classes for all observations. For example, learning sample for credit scoring system would be fundamental information about previous borrows (variables) matched with actual
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payoff results (classes). ([http://edoc.hu-berlin.de/master/timofeev-roman-2004-12-20/PDF/timofeev.pdf])
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Official Statistics Toolbox of Matlab provides CART. Here is a simple code for training and evaluation of a CART.
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% Tree Construction - Learning Phase - Statistics Toolbox Built-in Function
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tree = classregtree(data_train,labels_train,'method','classification');
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% Tree in Action - Recalling Phase - Statistics Toolbox Built-in Function
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labels_test_hat = tree.eval(data_test));
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% Confusion Matrix Estimation - Statistics Toolbox Built-in Function
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C = confusionmat(labels_test,labels_test_hat);
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CCR = sum(diag(C))/sum(sum(C));
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These are some pros and cons of CART (from here: [http://edoc.hu-berlin.de/master/timofeev-roman-2004-12-20/PDF/timofeev.pdf])
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1. CART is nonparametric. Therefore this method does not require specification of any functional form.
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2. CART does not require variables to be selected in advance. CART algorithm will itself identify the most significant variables and eliminate non-significant ones.
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3. CART results are invariant to monotone transformations of its independent variables. Changing one or several variables to its logarithm or square root will not change the structure of the tree. Only the splitting values (but not variables) in the questions will be different.
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4. CART can easily handle outliers. Outliers can negatively affect the results of some statistical models, like Principal Component Analysis (PCA) and linear regression. But the splitting algorithm of CART will easily handle noisy data: CART will isolate the outliers in a separate node. This property is very important, because financial data very often have outliers due to financial crisises or defaults.
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==== Examples====
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[[image:Decision_trees.GIF]]
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In this classification tree above ,we classify the samples by two features <math>\ x_1 </math> and <math>\ x_2 </math>. First , we classify the data according to the <math>\ x_1 </math> features . Then we make more accurate classification by <math>\ x_{2} </math> feature.
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[[image:Decision_Square.GIF]]
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A classification tree can also be viewed as squares as above . The classification rules can be more and more complex to make the training error rate reach to zero .
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Extension:
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[http://www.mindtools.com/dectree.html Decision Tree Analysis Decision Trees from Mind Tools]
 +
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''useful link'':
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Algorithm, Overfitting, Examples:[http://www.cs.cmu.edu/afs/cs.cmu.edu/project/theo-20/www/mlbook/ch3.pdf],[http://robotics.stanford.edu/people/nilsson/MLDraftBook/ch6-ml.pdf],[http://www.autonlab.org/tutorials/dtree18.pdf]
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