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=== Classification ===
 
=== Classification ===
  
'''Statistical classification''', usually simply known as classification, refers to the [http://en.wikipedia.org/wiki/Supervised_learning supervised learning] of the classes (also known as labels or groups) of novel data using models built using [http://en.wikipedia.org/wiki/Classifier_%28mathematics%29 classifiers] and labeled [http://en.wikipedia.org/wiki/Training_set training data]. Classification is an important task for people and society since the beginnings of history. The earliest application of classification in human society was probably done by prehistory peoples for recognizing which wild animals were beneficial to people and which one were harmful; and the earliest systematic use of classification was done by the famous Greek philosopher when he, for example, divided all living things into the two groups of plants and animals.[http://www.schools.utah.gov/curr/science/sciber00/7th/classify/sciber/history.htm] Classification is generally regarded as one of four major areas of statistics, with the other three major areas being [http://en.wikipedia.org/wiki/Regression_analysis regression regression], [http://en.wikipedia.or/wiki/Regression_analysis clustering], and [http://en.wikipedia.org/wiki/Regression_analysis dimensionality reduction] (also known as feature extraction or manifold learning).       
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'''Statistical classification''', usually 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 novel data to their labels (also known as classes or groups) by using knowledge of their features (also known as characteristics or attributes) that are derived from observation and/or measurement. In classification, this is done by first building a [http://en.wikipedia.org/wiki/Mathematical_model model] using labeled [http://en.wikipedia.org/wiki/Training_set training data], and then using a well-defined [http://en.wikipedia.org/wiki/Decision_rule decision rule] and the constructed model to predict the labels of new data.  
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Classification is an important task for people and society since the beginnings of history. The earliest application of classification in human society was probably done by prehistory peoples for recognizing which wild animals were beneficial to people and which one were harmful; and the earliest systematic use of classification was done by the famous Greek philosopher when he, for example, divided all living things into the two groups of plants and animals.[http://www.schools.utah.gov/curr/science/sciber00/7th/classify/sciber/history.htm] Classification is generally regarded as one of four major areas of statistics, with the other three major areas being [http://en.wikipedia.org/wiki/Regression_analysis regression regression], [http://en.wikipedia.or/wiki/Regression_analysis clustering], and [http://en.wikipedia.org/wiki/Regression_analysis dimensionality reduction] (also known as feature extraction or manifold learning).       
  
 
In '''classical statistics''', classification techniques were developed to learn useful information using small data sets where there is usually not enough of data. When '''machine learning''' was developed after the application of computers to statistics, classification techniques were developed to work with very large data sets where there is usually too many data. A major challenge facing data mining using machine learning is how to efficiently find useful patterns in very large amounts of data. A interesting quote that describes this problem quite well is the following one made by the retired Yale University Librarian Rutherford D. Rogers.
 
In '''classical statistics''', classification techniques were developed to learn useful information using small data sets where there is usually not enough of data. When '''machine learning''' was developed after the application of computers to statistics, classification techniques were developed to work with very large data sets where there is usually too many data. A major challenge facing data mining using machine learning is how to efficiently find useful patterns in very large amounts of data. A interesting quote that describes this problem quite well is the following one made by the retired Yale University Librarian Rutherford D. Rogers.

Revision as of 13:04, 25 September 2010

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Classfication-2010.09.21

Classification

Statistical classification, usually simply known as classification, is an area of supervised learning that addresses the problem of how to systematically assign novel data to their labels (also known as classes or groups) by using knowledge of their features (also known as characteristics or attributes) that are derived from observation and/or measurement. In classification, this is done by first building a model using labeled training data, and then using a well-defined decision rule and the constructed model to predict the labels of new data.

Classification is an important task for people and society since the beginnings of history. The earliest application of classification in human society was probably done by prehistory peoples for recognizing which wild animals were beneficial to people and which one were harmful; and the earliest systematic use of classification was done by the famous Greek philosopher when he, for example, divided all living things into the two groups of plants and animals.[1] Classification is generally regarded as one of four major areas of statistics, with the other three major areas being regression regression, clustering, and dimensionality reduction (also known as feature extraction or manifold learning).

In classical statistics, classification techniques were developed to learn useful information using small data sets where there is usually not enough of data. When machine learning was developed after the application of computers to statistics, classification techniques were developed to work with very large data sets where there is usually too many data. A major challenge facing data mining using machine learning is how to efficiently find useful patterns in very large amounts of data. A interesting quote that describes this problem quite well is the following one made by the retired Yale University Librarian Rutherford D. Rogers.

       "We are drowning in information and starving for knowledge."  
                                                         - Rutherford D. Rogers

In the Information Age, machine learning when it is combined with efficient classification techniques can be very useful for data mining using very large data sets. This is most useful when the structure of the data is not well understood but the data nevertheless exhibit strong statistical regularity. Areas in which machine learning and classification have been used together include search and recommendation (e.g. Google, Amazon),automatic speech recognition and speaker verification, medical diagnosis etc.

Error rate

Bayes Classifier

Bayesian vs. Frequentist

Linear and Quadratic Discriminant Analysis

Linear and Quadratic Discriminant Analysis cont'd - 2010.09.23

In the second lecture, Professor Ali Ghodsi recapitulates that by calculating the class posteriors [math]\Pr(Y=k|X=x)[/math] we have optimal classification. He also shows that by assuming that the classes have common covariance matrix [math]\Sigma_{k}=\Sigma \forall k [/math] the decision boundary between classes [math]k[/math] and [math]l[/math] is linear (LDA). However, if we do not assume same covariance between the two classes the decision boundary is quadratic function (QDA).

Some MATLAB samples are used to demonstrated LDA and QDA


LDA x QDA

Linear discriminant analysis[2] 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.

Quadratic Discriminant Analysis[3], on the other had, aims to find the quadratic combination of features. It is more general than Linear discriminant analysis. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical.

Summarizing LDA and QDA

We can summarize what we have learned so far into the following theorem.

Theorem:

Suppose that [math]\,Y \in \{1,\dots,k\}[/math], if [math]\,f_k(x) = Pr(X=x|Y=k)[/math] is Gaussian, the Bayes Classifier rule is

[math]\,h(X) = \arg\max_{k} \delta_k(x)[/math]

where

[math] \,\delta_k = - \frac{1}{2}log(|\Sigma_k|) - \frac{1}{2}(x-\mu_k)^\top\Sigma_k^{-1}(x-\mu_k) + log (\pi_k) [/math] (quadratic)
  • Note The decision boundary between classes [math]k[/math] and [math]l[/math] is quadratic in [math]x[/math].

If the covariance of the Gaussians are the same, this becomes

[math] \,\delta_k = x^\top\Sigma^{-1}\mu_k - \frac{1}{2}\mu_k^\top\Sigma^{-1}\mu_k + log (\pi_k) [/math] (linear)
  • Note [math]\,\arg\max_{k} \delta_k(x)[/math]returns the set of k for which [math]\,\delta_k(x)[/math] attains its largest value.

Reference

The Elements of Statistical Learning: Data Mining, Inference, and Prediction

The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition, February 2009 Trevor Hastie, Robert Tibshirani, Jerome Friedman

http://www-stat.stanford.edu/~tibs/ElemStatLearn/ (3rd Edition is available)