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<div style = "align:left; background:#00ffff; font-size: 150%">
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If you
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use ideas, plots, text, code and other intellectual property developed by someone else
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in your `wikicoursenote' contribution , you have to cite the
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original source. If you copy a sentence or a paragraph from work done by someone
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else, in addition to citing the original source you have to use quotation marks to
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identify the scope of the copied material. Evidence of copying or plagiarism will
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cause a failing mark in the course.
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Example of citing the original source
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Assumptions Underlying Principal Component Analysis can be found here<ref>http://support.sas.com/publishing/pubcat/chaps/55129.pdf</ref>
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</div>
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==Important Notes==
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<span style="color:#ff0000;font-size: 200%"> To make distinction between the material covered in class and additional material that you have add to the course, use the following convention. For anything that is not covered in the lecture write:</span>
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<div style = "align:left; background:#F5F5DC; font-size: 120%">
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In the news recently was a story that captures some of the ideas behind PCA. Over the past two years, Scott Golder and Michael Macy, researchers from Cornell University, collected 509 million Twitter messages from 2.4 million users in 84 different countries. The data they used were words collected at various times of day and they classified the data into two different categories: positive emotion words and negative emotion words. Then, they were able to study this new data to evaluate subjects' moods at different times of day, while the subjects were in different parts of the world. They found that the subjects generally exhibited positive emotions in the mornings and late evenings, and negative emotions mid-day. They were able to "project their data onto a smaller dimensional space" using PCS. Their paper, "Diurnal and Seasonal Mood Vary with Work, Sleep, and Daylength Across Diverse Cultures," is available in the journal Science.<ref>http://www.pcworld.com/article/240831/twitter_analysis_reveals_global_human_moodiness.html</ref>.
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Assumptions Underlying Principal Component Analysis can be found here<ref>http://support.sas.com/publishing/pubcat/chaps/55129.pdf</ref>
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</div>
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== Introduction, Class 1 - Tuesday, May 7 ==
 
== Introduction, Class 1 - Tuesday, May 7 ==
  
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=== Four Fundamental Problems ===
 
=== Four Fundamental Problems ===
 
<!-- br tag for spacing-->
 
<!-- br tag for spacing-->
1. Classification: Given an input object X, we have a function which will take in this input X and identify which 'class (Y)' it belongs to (Discrete Case) <br />
+
1 Classification: Given input object X, we have a function which will take this input X and identify which 'class (Y)' it belongs to (Discrete Case) <br />
   i.e taking value from x, we could predict y.
+
   <font size="3">i.e taking value from x, we could predict y.</font>
 
(For example, if you have 40 images of oranges and 60 images of apples (represented by x), you can estimate a function that takes the images and states what type of fruit it is - note Y is discrete in this case.) <br />
 
(For example, if you have 40 images of oranges and 60 images of apples (represented by x), you can estimate a function that takes the images and states what type of fruit it is - note Y is discrete in this case.) <br />
2. Regression: Same as classification but in the continuous case except y is non discrete. (Example of stock prices) <br />
+
2 Regression: Same as classification but in the continuous case except y is non discrete. Results from regression are often used for prediction,forecasting and etc. (Example of stock prices, height, weight, etc.) <br />
 
(A simple practice might be investigating the hypothesis that higher levels of education cause higher levels of income.) <br />
 
(A simple practice might be investigating the hypothesis that higher levels of education cause higher levels of income.) <br />
3. Clustering: Use common features of objects in same class or group to form clusters.(in this case, x is given, y is unknown) <br />
+
3 Clustering: Use common features of objects in same class or group to form clusters.(in this case, x is given, y is unknown; For example, clustering by provinces to measure average height of Canadian men.) <br />
4. Dimensionality Reduction (aka Feature extraction, Manifold learning): Used when we have a variable in high dimension space and we want to reduce the dimension <br />
+
4 Dimensionality Reduction (also known as Feature extraction, Manifold learning): Used when we have a variable in high dimension space and we want to reduce the dimension <br />
  
 
=== Applications ===
 
=== Applications ===
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*Email all questions and concerns to UWStat340@gmail.com. Do not use your personal email address! Do not email instructor or TAs about the class directly to their personal accounts!
 
*Email all questions and concerns to UWStat340@gmail.com. Do not use your personal email address! Do not email instructor or TAs about the class directly to their personal accounts!
  
'''Wikicourse note (10% of final mark):'''
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'''Wikicourse note (complete at least 12 contributions to get 10% of final mark):'''
 
When applying for an account in the wikicourse note, please use the quest account as your login name while the uwaterloo email as the registered email. This is important as the quest id will be used to identify the students who make the contributions.
 
When applying for an account in the wikicourse note, please use the quest account as your login name while the uwaterloo email as the registered email. This is important as the quest id will be used to identify the students who make the contributions.
 
Example:<br/>
 
Example:<br/>
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Each time you make a contribution, check mark the table. Marks are calculated on an honour system, although there will be random verifications. If you are caught claiming to contribute but have not, you will not be credited.
 
Each time you make a contribution, check mark the table. Marks are calculated on an honour system, although there will be random verifications. If you are caught claiming to contribute but have not, you will not be credited.
  
'''Wikicoursenote contribution form''' : '''[https://docs.google.com/forms/d/1Sgq0uDztDvtcS5JoBMtWziwH96DrBz2JiURvHPNd-xs/viewform]'''
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'''Wikicoursenote contribution form''' : https://docs.google.com/forms/d/1Sgq0uDztDvtcS5JoBMtWziwH96DrBz2JiURvHPNd-xs/viewform
  
 
- you can submit your contributions multiple times.<br />
 
- you can submit your contributions multiple times.<br />
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- Variance reduction<br />
 
- Variance reduction<br />
 
- Markov Chain Monte Carlo
 
- Markov Chain Monte Carlo
 
=== Tentative Marking Scheme ===
 
{| class="wikitable"
 
|-
 
! Item
 
! Value
 
|-
 
| Assignments (~6)
 
| 30%
 
|-
 
| WikiCourseNote
 
| 10%
 
|-
 
| Midterm
 
| 20%
 
|-
 
| Final
 
| 40%
 
|}
 
 
 
'''The final exam is going to be closed book and only non-programmable calculators are allowed.''' <br>
 
'''A passing mark must be achieved in the final to pass the course'''
 
  
 
==Class 2 - Thursday, May 9==
 
==Class 2 - Thursday, May 9==
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Simulation is the imitation of a process or system over time. Computational power has introduced the possibility of using simulation study to analyze models used to describe a situation.
 
Simulation is the imitation of a process or system over time. Computational power has introduced the possibility of using simulation study to analyze models used to describe a situation.
  
In order to perform a simulation study, we must first:
+
In order to perform a simulation study, we should:
<br\> 1. Use a computer to generate (pseudo) random numbers.<br>
+
<br\> 1 Use a computer to generate (pseudo*) random numbers (rand in MATLAB).<br>
2. Use these numbers to generate values of random variable from distributions.<br>
+
2 Use these numbers to generate values of random variable from distributions: for example, set a variable in terms of uniform u ~ U(0,1).<br>
3. Using the concept of discrete events, we show how the random variables can be used to generate the behavior of a stochastic model over time. (Note: A stochastic model is the opposite of deterministic model, where there are several directions the process can evolve to)<br>
+
3 Using the concept of discrete events, we show how the random variables can be used to generate the behavior of a stochastic model over time. (Note: A stochastic model is the opposite of deterministic model, where there are several directions the process can evolve to)<br>
4. After continually generating the behavior of the system, we can obtain estimators and other quantities of interest.<br>
+
4 After continually generating the behavior of the system, we can obtain estimators and other quantities of interest.<br>
  
 
The building block of a simulation study is the ability to generate a random number. This random number is a value from a random variable distributed uniformly on (0,1). There are many different methods of generating a random number: <br>
 
The building block of a simulation study is the ability to generate a random number. This random number is a value from a random variable distributed uniformly on (0,1). There are many different methods of generating a random number: <br>
  <br>Physical Method: Roulette wheel, lottery balls, dice rolling, card shuffling etc. <br>
+
  <br><font size="3">Physical Method: Roulette wheel, lottery balls, dice rolling, card shuffling etc. <br>
  <br>Numerically/Arithmetically: Use of a computer to successively generate pseudorandom numbers. The <br />sequence of numbers can appear to be random; however they are deterministically calculated with an <br />equation which defines pseudorandom. <br>
+
  <br>Numerically/Arithmetically: Use of a computer to successively generate pseudorandom numbers. The <br />sequence of numbers can appear to be random; however they are deterministically calculated with an <br />equation which defines pseudorandom. <br></font>
  
 
(Source: Ross, Sheldon M., and Sheldon M. Ross. Simulation. San Diego: Academic, 1997. Print.)
 
(Source: Ross, Sheldon M., and Sheldon M. Ross. Simulation. San Diego: Academic, 1997. Print.)
 +
 +
*We use the prefix pseudo because computer generates random numbers based on algorithms, which suggests that generated numbers are not truly random. Therefore pseudo-random numbers is used.
  
 
In general, a deterministic model produces specific results given certain inputs by the model user, contrasting with a '''stochastic''' model which encapsulates randomness and probabilistic events.
 
In general, a deterministic model produces specific results given certain inputs by the model user, contrasting with a '''stochastic''' model which encapsulates randomness and probabilistic events.
 
[[File:Det_vs_sto.jpg]]
 
[[File:Det_vs_sto.jpg]]
<br>A computer cannot generate truly random numbers because computers can only run algorithms, which are deterministic in nature. They can, however, generate '''Pseudo Random Numbers'''; numbers that seem random but are actually deterministic. Although the pseudo random numbers are deterministic, these numbers have a sequence of value and all of them have the appearances of being independent uniform random variables. Being deterministic, pseudo random numbers are valuable and beneficial due to the ease to generate and manipulate.
+
<br>A computer cannot generate truly random numbers because computers can only run algorithms, which are deterministic in nature. They can, however, generate Pseudo Random Numbers<br>
  
When people do the test many times, the results will be the closed express values, which make the trial look deterministic, however for each trial, the result is random.
+
'''Pseudo Random Numbers''' are the numbers that seem random but are actually determined by a relative set of original values. It is a chain of numbers pre-set by a formula or an algorithm, and the value jump from one to the next, making it look like a series of independent random events. The flaw of this method is that, eventually the chain returns to its initial position and pattern starts to repeat, but if we make the number set large enough we can prevent the numbers from repeating too early. Although the pseudo random numbers are deterministic, these numbers have a sequence of value and all of them have the appearances of being independent uniform random variables. Being deterministic, pseudo random numbers are valuable and beneficial due to the ease to generate and manipulate.
So, it looks like pseudo random numbers.
+
 
 +
When people repeat the test many times, the results will be the closed express values, which make the trials look deterministic. However, for each trial, the result is random. So, it looks like pseudo random numbers.
  
 
==== Mod ====
 
==== Mod ====
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Generally, mod means taking the reminder after division by m.
 
Generally, mod means taking the reminder after division by m.
 
<br />
 
<br />
We say that n is congruent to r mod m if n = mq + r, where m is an integer. <br />
+
We say that n is congruent to r mod m if n = mq + r, where m is an integer.  
 +
Values are between 0 and m-1 <br />
 
if y = ax + b, then <math>b:=y \mod a</math>. <br />
 
if y = ax + b, then <math>b:=y \mod a</math>. <br />
  
For example:<br />
+
'''Example 1:'''<br />
30 = 4 * 7 + 2 <br />
 
2 = 30 mod 7<br />
 
  
25 = 8 * 3 + 1 <br />
+
<math>30 = 4 \cdot  7 + 2</math><br />
1 = 25 mod 3<br />
 
  
 +
<math>2 := 30\mod 7</math><br />
 +
<br />
 +
<math>25 = 8 \cdot  3 + 1</math><br />
  
 +
<math>1: = 25\mod 3</math><br />
 +
<br />
 +
<math>-3=5\cdot (-1)+2</math><br />
  
'''Note:''' <math>\mod</math> here is different from the modulo congruence relation in <math>\Z_m</math>, which is an equivalence relation instead of a function.
+
<math>2:=-3\mod 5</math><br />
  
The modulo operation is useful for determining if an integer divided by another integer produces a non-zero remainder. But both integers should satisfy n = mq + r, where m, r, q, and n are all integers, and r is smaller than m.
+
<br />
 +
'''Example 2:'''<br />
  
==== Mixed Congruential Algorithm ====
+
If <math>23 = 3 \cdot  6 + 5</math> <br />
We define the Linear Congruential Method to be <math>x_{k+1}=(ax_k + b) \mod m</math>, where <math>x_k, a, b, m \in \N, \;\text{with}\; a, m \neq 0</math>. Given a '''seed''' (i.e. an initial value <math>x_0 \in \N</math>), we can obtain values for <math>x_1, \, x_2, \, \cdots, x_n</math> inductively. The Multiplicative Congruential Method, invented by Berkeley professor D. H. Lehmer, may also refer to the special case where <math>b=0</math> and the Mixed Congruential Method is case where <math>b \neq 0</math> <br />
 
  
An interesting fact about '''Linear Congruential Method''' is that it is one of the oldest and best-known pseudorandom number generator algorithms. It is very fast and requires minimal memory to retain state. However, this method should not be used for applications where high-quality randomness is required. They should not be used for Monte Carlo simulation and cryptographic applications. (Monte Carlo simulation will consider possibilities for every choice of consideration, and it shows the extreme possibilities. This method is not precise enough.)<br />
+
Then equivalently, <math>5 := 23\mod 6</math><br />
 +
<br />
 +
If <math>31 = 31 \cdot  1</math> <br />
  
 +
Then equivalently, <math>0 := 31\mod 31</math><br />
 +
<br />
 +
If <math>-37 = 40\cdot (-1)+ 3</math> <br />
  
 +
Then equivalently, <math>3 := -37\mod 40</math><br />
  
'''First consider the following algorithm'''<br />
+
'''Example 3:'''<br />
<math>x_{k+1}=x_{k} \mod m</math>
+
<math>77 = 3 \cdot  25 + 2</math><br />
  
 +
<math>2 := 77\mod 3</math><br />
 +
<br />
 +
<math>25 = 25 \cdot  1 + 0</math><br />
  
'''Example'''<br />
+
<math>0: = 25\mod 25</math><br />
<math>\text{Let }x_{0}=10,\,m=3</math><br //>
+
<br />
  
:<math>\begin{align}
 
  
x_{1} &{}= 10 &{}\mod{3} = 1 \\
 
  
x_{2} &{}= 1 &{}\mod{3} = 1 \\
 
  
x_{3} &{}= 1 &{}\mod{3} =1 \\
+
'''Note:''' <math>\mod</math> here is different from the modulo congruence relation in <math>\Z_m</math>, which is an equivalence relation instead of a function.
\end{align}</math>
 
<math>\ldots</math><br />
 
  
Excluding <math>x_{0}</math>, this example generates a series of ones. In general, excluding <math>x_{0}</math>, the algorithm above will always generate a series of the same number less than M. Hence, it has a period of 1. The '''period''' can be described as the length of a sequence before it repeats. We want a large period with a sequence that is random looking. We can modify this algorithm to form the Multiplicative Congruential Algorithm. <br />
+
The modulo operation is useful for determining if an integer divided by another integer produces a non-zero remainder. But both integers should satisfy <math>n = mq + r</math>, where <math>m</math>, <math>r</math>, <math>q</math>, and <math>n</math> are all integers, and <math>r</math> is smaller than <math>m</math>. The above rules also satisfy when any of <math>m</math>, <math>r</math>, <math>q</math>, and <math>n</math> is negative integer, see the third example.
 +
 
 +
==== Mixed Congruential Algorithm ====
 +
We define the Linear Congruential Method to be <math>x_{k+1}=(ax_k + b) \mod m</math>, where <math>x_k, a, b, m \in \N, \;\text{with}\; a, m \neq 0</math>. Given a '''seed''' (i.e. an initial value <math>x_0 \in \N</math>), we can obtain values for <math>x_1, \, x_2, \, \cdots, x_n</math> inductively. The Multiplicative Congruential Method, invented by Berkeley professor D. H. Lehmer, may also refer to the special case where <math>b=0</math> and the Mixed Congruential Method is case where <math>b \neq 0</math> <br />. Their title as "mixed" arises from the fact that it has both a multiplicative and additive term.
 +
 
 +
An interesting fact about '''Linear Congruential Method''' is that it is one of the oldest and best-known pseudo random number generator algorithms. It is very fast and requires minimal memory to retain state. However, this method should not be used for applications that require high randomness. They should not be used for Monte Carlo simulation and cryptographic applications. (Monte Carlo simulation will consider possibilities for every choice of consideration, and it shows the extreme possibilities. This method is not precise enough.)<br />
 +
 
 +
[[File:Linear_Congruential_Statment.png‎|600px]] "Source: STAT 340 Spring 2010 Course Notes"
 +
 
 +
'''First consider the following algorithm'''<br />
 +
<math>x_{k+1}=x_{k} \mod m</math> <br />
 +
 
 +
such that: if <math>x_{0}=5(mod 150)</math>, <math>x_{n}=3x_{n-1}</math>, find <math>x_{1},x_{8},x_{9}</math>. <br />
 +
<math>x_{n}=(3^n)*5(mod 150)</math> <br />
 +
<math>x_{1}=45,x_{8}=105,x_{9}=15</math> <br />
 +
 
 +
 
 +
 
 +
'''Example'''<br />
 +
<math>\text{Let }x_{0}=10,\,m=3</math><br //>
 +
 
 +
:<math>\begin{align}
 +
 
 +
x_{1} &{}= 10 &{}\mod{3} = 1 \\
 +
 
 +
x_{2} &{}= 1 &{}\mod{3} = 1 \\
 +
 
 +
x_{3} &{}= 1 &{}\mod{3} =1 \\
 +
\end{align}</math>
 +
<math>\ldots</math><br />
 +
 
 +
Excluding <math>x_{0}</math>, this example generates a series of ones. In general, excluding <math>x_{0}</math>, the algorithm above will always generate a series of the same number less than M. Hence, it has a period of 1. The '''period''' can be described as the length of a sequence before it repeats. We want a large period with a sequence that is random looking. We can modify this algorithm to form the Multiplicative Congruential Algorithm. <br />
  
  
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This example generates a sequence with a repeating cycle of two integers.<br />
 
This example generates a sequence with a repeating cycle of two integers.<br />
  
(If we choose the numbers properly, we could get a sequence of "random" numbers. However, how do we find the value of <math>a,b,</math> and <math>m</math>?  At the very least <math>m</math> should be a very '''large''', preferably prime number.  The larger <math>m</math> is, the higher possibility people get a sequence of "random" numbers.  This is easier to solve in Matlab. In Matlab, the command rand() generates random numbers which are uniformly distributed in the interval (0,1)). Matlab uses <math>a=7^5, b=0, m=2^{31}-1</math> – recommended in a 1988 paper, "Random Number Generators: Good Ones Are Hard To Find" by Stephen K. Park and Keith W. Miller (Important part is that <math>m</math> should be '''large and prime''')<br />  
+
(If we choose the numbers properly, we could get a sequence of "random" numbers. How do we find the value of <math>a,b,</math> and <math>m</math>?  At the very least <math>m</math> should be a very '''large''', preferably prime number.  The larger <math>m</math> is, the higher the possibility to get a sequence of "random" numbers.  This is easier to solve in Matlab. In Matlab, the command rand() generates random numbers which are uniformly distributed on the interval (0,1)). Matlab uses <math>a=7^5, b=0, m=2^{31}-1</math> – recommended in a 1988 paper, "Random Number Generators: Good Ones Are Hard To Find" by Stephen K. Park and Keith W. Miller (Important part is that <math>m</math> should be '''large and prime''')<br />  
  
 
Note: <math>\frac {x_{n+1}}{m-1}</math> is an approximation to the value of a U(0,1) random variable.<br />  
 
Note: <math>\frac {x_{n+1}}{m-1}</math> is an approximation to the value of a U(0,1) random variable.<br />  
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''(Note: <br />
 
''(Note: <br />
1. Keep repeating this command over and over again and you will seem to get random numbers – this is how the command rand works in a computer. <br />
+
1. Keep repeating this command over and over again and you will get random numbers – this is how the command rand works in a computer. <br />
2. There is a function in MATLAB called '''RAND''' to generate a number between 0 and 1. <br />
+
2. There is a function in MATLAB called '''RAND''' to generate a random number between 0 and 1. <br />
 
For example, in MATLAB, we can use '''rand(1,1000)''' to generate 1000's numbers between 0 and 1. This is essentially a vector with 1 row, 1000 columns, with each entry a random number between 0 and 1.<br />
 
For example, in MATLAB, we can use '''rand(1,1000)''' to generate 1000's numbers between 0 and 1. This is essentially a vector with 1 row, 1000 columns, with each entry a random number between 0 and 1.<br />
 
3. If we would like to generate 1000 or more numbers, we could use a '''for''' loop<br /><br />
 
3. If we would like to generate 1000 or more numbers, we could use a '''for''' loop<br /><br />
Line 267: Line 311:
 
2. close all: closes all figures.<br />
 
2. close all: closes all figures.<br />
 
3. who: displays all defined variables.<br />
 
3. who: displays all defined variables.<br />
4. clc: clears screen.<br /><br />
+
4. clc: clears screen.<br />
5. ; : prevents the results from printing.<br /><br />
+
5. ; : prevents the results from printing.<br />
 +
6. disstool: displays a graphing tool.<br /><br />
  
 
<pre style="font-size:16px">
 
<pre style="font-size:16px">
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x_{2} &{}= 3 \times 1 + 2 \mod{4} = 1 \\
 
x_{2} &{}= 3 \times 1 + 2 \mod{4} = 1 \\
 
\end{align}</math><br />
 
\end{align}</math><br />
 
+
Another Example, a =3, b =2, m = 5, x_0=1
 
etc.
 
etc.
 
<hr/>
 
<hr/>
 
<p style="color:red;font-size:16px;">FAQ:</P>
 
<p style="color:red;font-size:16px;">FAQ:</P>
1.Why in the example above is 1 to 30 not 0 to 30?<br>
+
1.Why is it 1 to 30 instead of 0 to 30 in the example above?<br>
 
''<math>b = 0</math> so in order to have <math>x_k</math> equal to 0, <math>x_{k-1}</math> must be 0 (since <math>a=13</math> is relatively prime to 31). However, the seed is 1. Hence, we will never observe 0 in the sequence.''<br>
 
''<math>b = 0</math> so in order to have <math>x_k</math> equal to 0, <math>x_{k-1}</math> must be 0 (since <math>a=13</math> is relatively prime to 31). However, the seed is 1. Hence, we will never observe 0 in the sequence.''<br>
 
Alternatively, {0} and {1,2,...,30} are two orbits of the left multiplication by 13 in the group <math>\Z_{31}</math>.<br>
 
Alternatively, {0} and {1,2,...,30} are two orbits of the left multiplication by 13 in the group <math>\Z_{31}</math>.<br>
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'''Examples:[From Textbook]'''<br />
 
'''Examples:[From Textbook]'''<br />
If <math>x_0=3</math> and <math>x_n=(5x_{n-1}+7)\mod 200</math>, find <math>x_1,\cdots,x_{10}</math>.<br />
+
<math>\text{If }x_0=3 \text{ and } x_n=(5x_{n-1}+7)\mod 200</math>, <math>\text{find }x_1,\cdots,x_{10}</math>.<br />
 
'''Solution:'''<br />
 
'''Solution:'''<br />
 
<math>\begin{align}
 
<math>\begin{align}
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'''Comments:'''<br />
 
'''Comments:'''<br />
 +
 +
Matlab code:
 +
a=5;
 +
b=7;
 +
m=200;
 +
x(1)=3;
 +
for ii=2:1000
 +
x(ii)=mod(a*x(ii-1)+b,m);
 +
end
 +
size(x);
 +
hist(x)
 +
 +
 +
 
Typically, it is good to choose <math>m</math> such that <math>m</math> is large, and <math>m</math> is prime. Careful selection of parameters '<math>a</math>' and '<math>b</math>' also helps generate relatively "random" output values, where it is harder to identify patterns. For example, when we used a composite (non prime) number such as 40 for <math>m</math>, our results were not satisfactory in producing an output resembling a uniform distribution.<br />
 
Typically, it is good to choose <math>m</math> such that <math>m</math> is large, and <math>m</math> is prime. Careful selection of parameters '<math>a</math>' and '<math>b</math>' also helps generate relatively "random" output values, where it is harder to identify patterns. For example, when we used a composite (non prime) number such as 40 for <math>m</math>, our results were not satisfactory in producing an output resembling a uniform distribution.<br />
  
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From the example shown above, if we want to create a large group of random numbers, it is better to have large, prime <math>m</math> so that the generated random values will not repeat after several iterations. Note: the period for this example is 8: from '<math>x_2</math>' to '<math>x_9</math>'.<br />
 
From the example shown above, if we want to create a large group of random numbers, it is better to have large, prime <math>m</math> so that the generated random values will not repeat after several iterations. Note: the period for this example is 8: from '<math>x_2</math>' to '<math>x_9</math>'.<br />
  
There has been a research about how to choose uniform sequence.  Many programs give you the options to choose the seed.  Sometimes the seed is chosen by CPU.<br />
+
There has been a research on how to choose uniform sequence.  Many programs give you the options to choose the seed.  Sometimes the seed is chosen by CPU.<br />
  
 
<span style="background:#F5F5DC">Theorem (extra knowledge)</span><br />
 
<span style="background:#F5F5DC">Theorem (extra knowledge)</span><br />
Line 364: Line 423:
 
(ii) (a-1) is divisible by all prime factor of m;<br />
 
(ii) (a-1) is divisible by all prime factor of m;<br />
 
(iii) if and only if m is divisible by 4, then a-1 is also divisible by 4.<br />
 
(iii) if and only if m is divisible by 4, then a-1 is also divisible by 4.<br />
 +
 +
We want our LCG to have a large cycle.
 +
We call a cycle with m element the maximal period.
 +
We can make it bigger by making m big and prime.
 +
Recall:any number you can think of can be broken into a factor of prime
 +
Define coprime:Two numbers X and Y, are coprime if they do not share any prime factors.
  
 
Example:<br />
 
Example:<br />
  Xn=(15Xn-1 + 4) mod 7<br />
+
  <font size="3">Xn=(15Xn-1 + 4) mod 7</font><br />
 
(i) m=7 c=4 -> coprime;<br />
 
(i) m=7 c=4 -> coprime;<br />
 
(ii) a-1=14 and a-1 is divisible by 7;<br />
 
(ii) a-1=14 and a-1 is divisible by 7;<br />
Line 374: Line 439:
  
  
In this part, I learned how to use R code to figure out the relationship between two integer
+
In this part, I learned how to use R code to figure out the relationship between two integers
 
division, and their remainder. And when we use R to calculate R with random variables for a range such as(1:1000),the graph of distribution is like uniform distribution.
 
division, and their remainder. And when we use R to calculate R with random variables for a range such as(1:1000),the graph of distribution is like uniform distribution.
 
<div style="border:1px solid #cccccc;border-radius:10px;box-shadow: 0 5px 15px 1px rgba(0, 0, 0, 0.6), 0 0 200px 1px rgba(255, 255, 255, 0.5);padding:20px;margin:20px;background:#FFFFAD;">
 
<div style="border:1px solid #cccccc;border-radius:10px;box-shadow: 0 5px 15px 1px rgba(0, 0, 0, 0.6), 0 0 200px 1px rgba(255, 255, 255, 0.5);padding:20px;margin:20px;background:#FFFFAD;">
Line 396: Line 461:
 
</pre>
 
</pre>
 
</div>
 
</div>
 +
Another algorithm for generating pseudo random numbers is the multiply with carry method. Its simplest form is similar to the linear congruential generator. They differs in that the parameter b changes in the MWC algorithm. It is as follows: <br>
 +
 +
1.) x<sub>k+1</sub> = ax<sub>k</sub> + b<sub>k</sub> mod m <br>
 +
2.) b<sub>k+1</sub> = floor((ax<sub>k</sub> + b<sub>k</sub>)/m) <br>
 +
3.) set k to k + 1 and go to step 1
 +
[http://www.javamex.com/tutorials/random_numbers/multiply_with_carry.shtml Source]
  
 
=== Inverse Transform Method ===
 
=== Inverse Transform Method ===
Line 409: Line 480:
 
'''Proof of the theorem:'''<br />
 
'''Proof of the theorem:'''<br />
 
The generalized inverse satisfies the following: <br />
 
The generalized inverse satisfies the following: <br />
<math>\begin{align}
+
 
\forall u \in \left[0,1\right], \, x \in \R, \\
+
:<math>P(X\leq x)</math> <br />
&{} F^{-1}\left(u\right) \leq x &{} \\
+
<math>= P(F^{-1}(U)\leq x)</math> (since <math>X= F^{-1}(U)</math> by the inverse method)<br />
\Rightarrow &{} F\Big(F^{-1}\left(u\right)\Big) \leq F\left(x\right) &&{} F \text{ is non-decreasing} \\
+
<math>= P((F(F^{-1}(U))\leq F(x))</math>  (since <math>F </math> is monotonically increasing) <br />
\Rightarrow &{} F\Big(\inf \{y \in \R | F(y)\geq u \}\Big) \leq F\left(x\right) &&{} \text{by definition of } F^{-1} \\
+
<math>= P(U\leq F(x)) </math> (since <math> P(U\leq a)= a</math> for <math>U \sim U(0,1), a \in [0,1]</math>,<br />
\Rightarrow &{} \inf \{F(y) \in [0,1] | F(y)\geq u \} \leq F\left(x\right) &&{} F \text{ is right continuous and non-decreasing} \\
+
<math>= F(x) , \text{ where } 0 \leq F(x) \leq 1 </math>  <br />
\Rightarrow &{} u \leq F\left(x\right) &&{} \text{by definition of } \inf \\
+
 
\Rightarrow &{} x \in \{y \in \R | F(y) \geq u\} &&{} \\
+
This is the c.d.f. of X.  <br />
\Rightarrow &{} x \geq \inf \{y \in \R | F(y)\geq u \}\Big) &&{} \text{by definition of } \inf \\
+
<br />
\Rightarrow &{} x \geq F^{-1}(u) &&{} \text{by definition of } F^{-1} \\
 
\end{align}</math>
 
  
 
That is <math>F^{-1}\left(u\right) \leq x \Leftrightarrow u \leq F\left(x\right)</math><br />
 
That is <math>F^{-1}\left(u\right) \leq x \Leftrightarrow u \leq F\left(x\right)</math><br />
Line 447: Line 516:
 
Step 2: <math>  x=\frac{-ln(U)}{\lambda} </math> <br /><br />
 
Step 2: <math>  x=\frac{-ln(U)}{\lambda} </math> <br /><br />
  
 +
 +
EXAMPLE 2 Normal distribution
 +
G(y)=P[Y<=y)
 +
      =P[-sqr (y) < z < sqr (y))
 +
      =integrate from -sqr(z) to Sqr(z) 1/sqr(2pi) e ^(-z^2/2) dz
 +
      = 2 integrate from 0 to sqr(y)  1/sqr(2pi) e ^(-z^2/2) dz
 +
its the cdf of Y=z^2
 +
 +
pdf g(y)= G'(y)
 +
pdf pf x^2 (1)
  
 
'''MatLab Code''':<br />
 
'''MatLab Code''':<br />
Line 452: Line 531:
 
<pre style="font-size:16px">
 
<pre style="font-size:16px">
 
>>u=rand(1,1000);
 
>>u=rand(1,1000);
>>hist(u)      #will generate a fairly uniform diagram
+
>>hist(u)      # this will generate a fairly uniform diagram
 
</pre>
 
</pre>
 
[[File:ITM_example_hist(u).jpg|300px]]
 
[[File:ITM_example_hist(u).jpg|300px]]
Line 488: Line 567:
 
Sol:  
 
Sol:  
 
Let <math>y=x^5</math>, solve for x: <math>x=y^\frac {1}{5}</math>. Therefore, <math>F^{-1} (x) = x^\frac {1}{5}</math><br />
 
Let <math>y=x^5</math>, solve for x: <math>x=y^\frac {1}{5}</math>. Therefore, <math>F^{-1} (x) = x^\frac {1}{5}</math><br />
Hence, to obtain a value of x from F(x), we first set u as an uniform distribution, then obtain the inverse function of F(x), and set
+
Hence, to obtain a value of x from F(x), we first set 'u' as an uniform distribution, then obtain the inverse function of F(x), and set
 
<math>x= u^\frac{1}{5}</math><br /><br />
 
<math>x= u^\frac{1}{5}</math><br /><br />
 +
 +
Algorithm:<br />
 +
Steps: <br />
 +
Step 1: Draw U ~ rand[0, 1];<br />
 +
Step 2: X=U^(1/5);<br />
  
 
'''Example 4 - BETA(1,β)''':<br/>
 
'''Example 4 - BETA(1,β)''':<br/>
Line 501: Line 585:
 
<math>x = 1-(1-u)^\frac {1}{\beta}</math><br />
 
<math>x = 1-(1-u)^\frac {1}{\beta}</math><br />
 
let β=3, use Matlab to construct N=1000 observations from Beta(1,3)<br />
 
let β=3, use Matlab to construct N=1000 observations from Beta(1,3)<br />
Matlab Code:<br />
+
'''MatLab Code''':<br />
>> u = rand(1,1000);<br />
+
 
  x = 1-(1-u)^(1/3);<br />
+
<pre style="font-size:16px">
>> hist(x,50)<br />
+
>> u = rand(1,1000);
>> mean(x)<br />
+
x = 1-(1-u)^(1/3);
 +
>> hist(x,50)
 +
>> mean(x)
 +
</pre>
  
 
'''Example 5 - Estimating <math>\pi</math>''':<br/>
 
'''Example 5 - Estimating <math>\pi</math>''':<br/>
Line 515: Line 602:
 
Thus <math>\pi= 4(\frac {N_c}{N})</math><br />
 
Thus <math>\pi= 4(\frac {N_c}{N})</math><br />
  
   For example, '''UNIF(a,b)'''<br />
+
   <font size="3">For example, '''UNIF(a,b)'''<br />
 
   <math>y = F(x) = (x - a)/ (b - a) </math>
 
   <math>y = F(x) = (x - a)/ (b - a) </math>
 
   <math>x = (b - a ) * y + a</math>
 
   <math>x = (b - a ) * y + a</math>
 
   <math>X = a + ( b - a) * U</math><br />
 
   <math>X = a + ( b - a) * U</math><br />
   where U is UNIF(0,1)
+
   where U is UNIF(0,1)</font>
  
 
'''Limitations:'''<br />
 
'''Limitations:'''<br />
Line 525: Line 612:
 
2. It may be impractical since some CDF's and/or integrals are not easy to compute such as Gaussian distribution.<br />
 
2. It may be impractical since some CDF's and/or integrals are not easy to compute such as Gaussian distribution.<br />
  
We learned how to prove the cdf transfer to inverse cdf,and use the uniform distribution to obtain a value of x from F(x).
+
We learned how to prove the transformation from cdf to inverse cdf,and use the uniform distribution to obtain a value of x from F(x).
We also can use uniform distribution in inverse method to determine other distribution.
+
We can also use uniform distribution in inverse method to determine other distributions.
The probability of getting a point for a circle over the triangle is a closed uniform distribution, each point in the circle and over the triangle is almost the same.
+
The probability of getting a point for a circle over the triangle is a closed uniform distribution, each point in the circle and over the triangle is almost the same. Then, we can look at the graph to determine what kind of distribution the graph resembles.
and we can look at the graph to determine what kind of distribution the graph belongs to.
 
  
 
==== Probability Distribution Function Tool in MATLAB ====
 
==== Probability Distribution Function Tool in MATLAB ====
Line 535: Line 621:
 
</pre>  
 
</pre>  
  
This command allows users to explore the effect of changes of parameters on the plot of either a CDF or PDF.  
+
This command allows users to explore different types of distribution and see how the changes affect the parameters on the plot of either a CDF or PDF.
 +
 
  
 
[[File:Disttool.jpg|450px]]
 
[[File:Disttool.jpg|450px]]
Line 542: Line 629:
 
== Class 3 - Tuesday, May 14 ==
 
== Class 3 - Tuesday, May 14 ==
 
=== Recall the Inverse Transform Method ===
 
=== Recall the Inverse Transform Method ===
 +
Let U~Unif(0,1),then the random variable  X = F<sup>-1</sup>(u) has distribution F.  <br />
 +
To sample X with CDF F(x), <br />
  
To sample X with CDF F(x), <br />
+
<math>1) U~ \sim~ Unif [0,1] </math>
 +
'''2) X = F<sup>-1</sup>(u)   '''<br />
  
'''1. Draw u~U(0,1) '''<br />
 
'''2. X = F<sup>-1</sup>(u)  '''<br />
 
  
  
'''Proof''' <br />
 
First note that
 
<math>P(U\leq a)=a, \forall a\in[0,1]</math> <br />
 
  
:<math>P(X\leq x)</math> <br />
 
<math>= P(F^{-1}(U)\leq x)</math> (since <math>X= F^{-1}(U)</math> by the inverse method)<br />
 
<math>= P((F(F^{-1}(U))\leq F(x))</math>  (since <math>F </math> is monotonically increasing) <br />
 
<math>= P(U\leq F(x)) </math> (since <math> P(U\leq a)= a</math> for <math>U \sim U(0,1), a \in [0,1]</math>, this is explained further below)<br />
 
<math>= F(x) , \text{ where } 0 \leq F(x) \leq 1 </math>  <br />
 
  
This is the c.d.f. of X.  <br />
 
 
<br />
 
<br />
  
'''Note''': that the CDF of a U(a,b) random variable is:
+
'''Note''': CDF of a U(a,b) random variable is:
 
:<math>
 
:<math>
 
   F(x)= \begin{cases}
 
   F(x)= \begin{cases}
Line 584: Line 663:
  
 
Note that on a single point there is no mass probability (i.e. <math>u</math> <= 0.5, is the same as <math> u </math> < 0.5)  
 
Note that on a single point there is no mass probability (i.e. <math>u</math> <= 0.5, is the same as <math> u </math> < 0.5)  
More formally, this is saying that <math> P(X = x) = F(x)- \lim_{s \to x^-}F(x)</math> which equals zero for any continuous random variable
+
More formally, this is saying that <math> P(X = x) = F(x)- \lim_{s \to x^-}F(x)</math> , which equals zero for any continuous random variable
  
 
====Limitations of the Inverse Transform Method====
 
====Limitations of the Inverse Transform Method====
Line 590: Line 669:
 
Though this method is very easy to use and apply,  it does have a major disadvantage/limitation:
 
Though this method is very easy to use and apply,  it does have a major disadvantage/limitation:
  
*  We need to find the inverse cdf <math> F^{-1}(\cdot) </math>. In some cases the inverse function does not exist, or is difficult to find.
+
*  We need to find the inverse cdf <math> F^{-1}(\cdot) </math>. In some cases the inverse function does not exist, or is difficult to find because it requires a closed form expression for F(x).
  
 
For example, it is too difficult to find the inverse cdf of the Gaussian distribution, so we must find another method to sample from the Gaussian distribution.
 
For example, it is too difficult to find the inverse cdf of the Gaussian distribution, so we must find another method to sample from the Gaussian distribution.
 +
 +
In conclusion, we need to find another way of sampling from more complicated distributions
  
 
=== Discrete Case ===
 
=== Discrete Case ===
Line 609: Line 690:
  
 
Note that after generating a random U, the value of X can be determined by finding the interval <math>[F(x_{j-1}),F(x_{j})]</math> in which U lies. <br />
 
Note that after generating a random U, the value of X can be determined by finding the interval <math>[F(x_{j-1}),F(x_{j})]</math> in which U lies. <br />
 +
 +
In summary:
 +
Generate a discrete r.v.x that has pmf:<br />
 +
  P(X=xi)=Pi,    x0<x1<x2<... <br />
 +
1. Draw U~U(0,1);<br />
 +
2. If F(x(i-1))<U<F(xi), x=xi.<br />
  
  
Line 644: Line 731:
 
We can define the U function so that:  
 
We can define the U function so that:  
  
If U <= 0.5, then X = 0
+
If <math>U\leq 0.5</math>, then X = 0
  
and if  0.5 < U <= 1, then X =1.  
+
and if  <math>0.5 < U\leq 1</math>, then X =1.  
  
 
This allows the probability of Heads occurring to be 0.5 and is a good generator of a random coin flip.
 
This allows the probability of Heads occurring to be 0.5 and is a good generator of a random coin flip.
Line 745: Line 832:
 
</pre>
 
</pre>
 
[[File:Discrete_example.jpg|300px]]
 
[[File:Discrete_example.jpg|300px]]
 +
 +
The algorithm above generates a vector (1,1000) containing 0's ,1's and 2's in differing proportions. Due to the criteria for accepting 0, 1 or 2 into the vector we get proportions of 0,1 &2 that correspond to their respective probabilities. So plotting the histogram (frequency of 0,1&2) doesn't give us the pmf but a frequency histogram that shows the proportions of each, which looks identical to the pmf.
  
 
'''Example 3.3''': Generating a random variable from pdf <br>
 
'''Example 3.3''': Generating a random variable from pdf <br>
Line 786: Line 875:
 
Step 5: Go to step 3<br>
 
Step 5: Go to step 3<br>
 
*Note: These steps can be found in Simulation 5th Ed. by Sheldon Ross.
 
*Note: These steps can be found in Simulation 5th Ed. by Sheldon Ross.
 +
*Note: Another method by seeing the Binomial as a sum of n independent Bernoulli random variables, U1, ..., Un. Then set X equal to the number of Ui that are less than or equal to p. To use this method, n random numbers are needed and n comparisons need to be done. On the other hand, the inverse transformation method is simpler because only one random variable needs to be generated and it makes 1 + np comparisons.<br>
 +
Step 1: Generate n uniform numbers U1 ... Un.<br>
 +
Step 2: X = <math>\sum U_i < = p</math> where P is the probability of success.
  
 
'''Example 3.6''': Generating a Poisson random variable <br>
 
'''Example 3.6''': Generating a Poisson random variable <br>
  
Let X ~ Poi(u). Write an algorithm to generate X.
+
"Let X ~ Poi(u). Write an algorithm to generate X.
 
The PDF of a poisson is:
 
The PDF of a poisson is:
 
:<math>\begin{align} f(x) = \frac {\, e^{-u} u^x}{x!} \end{align}</math>
 
:<math>\begin{align} f(x) = \frac {\, e^{-u} u^x}{x!} \end{align}</math>
Line 802: Line 894:
 
   <math>\begin{align} F = P(X = 0) = e^{-u}*u^0/{0!} = e^{-u} = p \end{align}</math>
 
   <math>\begin{align} F = P(X = 0) = e^{-u}*u^0/{0!} = e^{-u} = p \end{align}</math>
 
3) If U<F, output x <br>
 
3) If U<F, output x <br>
   Else, <math>\begin{align} p = (u/(x+1))^p \end{align}</math> <br>
+
   <font size="3">Else,</font> <math>\begin{align} p = (u/(x+1))^p \end{align}</math> <br>
 
         <math>\begin{align} F = F + p \end{align}</math> <br>
 
         <math>\begin{align} F = F + p \end{align}</math> <br>
 
         <math>\begin{align} x = x + 1 \end{align}</math> <br>
 
         <math>\begin{align} x = x + 1 \end{align}</math> <br>
4) Go to x <br>
+
4) Go to 1" <br>
 
   
 
   
Acknowledgements: This is from Stat 340 Winter 2013
+
Acknowledgements: This is an example from Stat 340 Winter 2013
  
  
Line 815: Line 907:
 
<math>P(X=x_i) = \, p (1-p)^{x_{i}-1}</math>
 
<math>P(X=x_i) = \, p (1-p)^{x_{i}-1}</math>
 
We have CDF:
 
We have CDF:
<math>F(x)=P(X \leq x)=1-P(X>x) = 1-(1-p)^x</math>, P(X>x) means we get at least x failures before observe the first success.
+
<math>F(x)=P(X \leq x)=1-P(X>x) = 1-(1-p)^x</math>, P(X>x) means we get at least x failures before we observe the first success.
 
Now consider the inverse transform:
 
Now consider the inverse transform:
 
:<math>
 
:<math>
Line 838: Line 930:
 
4. Else if <math>U \leq P_{0} + P_{1} + P_{2} </math> deliver <math>x = x_{2}</math><br />
 
4. Else if <math>U \leq P_{0} + P_{1} + P_{2} </math> deliver <math>x = x_{2}</math><br />
 
...  
 
...  
   Else if <math>U \leq P_{0} + ... + P_{k} </math> deliver <math>x = x_{k}</math><br />
+
   <font size="3">Else if</font> <math>U \leq P_{0} + ... + P_{k} </math> <font size="3">deliver</font> <math>x = x_{k}</math><br />
 +
 
 +
<br /'''>===Inverse Transform Algorithm for Generating a Binomial(n,p) Random Variable(from textbook)==='''
 +
<br />step 1: Generate a random number U
 +
<br />step 2: c=p/(1-p),i=0, pr=(1-p)<sup>n</sup>, F=pr.
 +
<br />step 3: If U<F, set X=i and stop.
 +
<br />step 4: pr =[c(n-i)/(i+1)]pr, F=F+pr, i=i+1.
 +
<br />step 5: Go to step 3.
 +
 
  
 
'''Problems'''<br />
 
'''Problems'''<br />
1. We have to find <math> F^{-1} </math>
+
Though this method is very easy to use and apply, it does have a major disadvantage/limitation:
 
+
We need to find the inverse cdf  F^{-1}(\cdot) . In some cases the inverse function does not exist, or is difficult to find because it requires a closed form expression for F(x).
2. For many distributions, such as Gaussian, it is too difficult to find the inverse of <math> F(x) ,</math>
+
For example, it is too difficult to find the inverse cdf of the Gaussian distribution, so we must find another method to sample from the Gaussian distribution.
flipping a coin is a discrete case of uniform distribution, and for the code it is randomly flipped 1000 times for the coin, and the result we can see is closed to the express value(0.5)
+
In conclusion, we need to find another way of sampling from more complicated distributions
and example 2 is another discrete distribution, it shows that we can discrete uniform for 3 part like ,0,1,2, and the probability of each part or each trial is the same.
+
Flipping a coin is a discrete case of uniform distribution, and the code below shows an example of flipping a coin 1000 times; the result is close to the expected value 0.5.<br>
Example 3 is use inverse method to figure out the probability range of each random varibles.
+
Example 2, as another discrete distribution, shows that we can sample from parts like 0,1 and 2, and the probability of each part or each trial is the same.<br>
 +
Example 3 uses inverse method to figure out the probability range of each random varible.
 
<div style="border:1px solid #cccccc;border-radius:10px;box-shadow: 0 5px 15px 1px rgba(0, 0, 0, 0.6), 0 0 200px 1px rgba(255, 255, 255, 0.5);padding:20px;margin:20px;background:#FFFFAD;">
 
<div style="border:1px solid #cccccc;border-radius:10px;box-shadow: 0 5px 15px 1px rgba(0, 0, 0, 0.6), 0 0 200px 1px rgba(255, 255, 255, 0.5);padding:20px;margin:20px;background:#FFFFAD;">
 
<h2 style="text-align:center;">Summary of Inverse Transform Method</h2>
 
<h2 style="text-align:center;">Summary of Inverse Transform Method</h2>
Line 885: Line 986:
 
</pre>
 
</pre>
 
</div>
 
</div>
 +
 +
=== Generalized Inverse-Transform Method ===
 +
 +
Valid for any CDF F(x): return X=min{x:F(x)<math>\leq</math> U}, where U~U(0,1)
 +
 +
1. Continues, possibly with flat spots (i.e. not strictly increasing)
 +
 +
2. Discrete
 +
 +
3. Mixed continues discrete
 +
 +
 +
'''Advantages of Inverse-Transform Method'''
 +
 +
Inverse transform method preserves monotonicity and correlation
 +
 +
which helps in
 +
 +
1. Variance reduction methods ...
 +
 +
2. Generating truncated distributions ...
 +
 +
3. Order statistics ...
  
 
===Acceptance-Rejection Method===
 
===Acceptance-Rejection Method===
Line 898: Line 1,022:
 
[[File:AR_Method.png]]
 
[[File:AR_Method.png]]
  
 
{{Cleanup|reason= Do not write <math>c*g(x)</math>. Instead write <math>c \times g(x)</math> or <math>\,c g(x)</math>
 
}}
 
  
 
The main logic behind the Acceptance-Rejection Method is that:<br>
 
The main logic behind the Acceptance-Rejection Method is that:<br>
Line 907: Line 1,028:
 
3. For each value of x, we accept and reject some points based on a probability, which will be discussed below.<br>
 
3. For each value of x, we accept and reject some points based on a probability, which will be discussed below.<br>
  
Note: If the red line was only g(x) as opposed to <math>\,c g(x)</math> (i.e. c=1), then <math>g(x) \geq f(x)</math> for all values of x if and only if g and f are the same functions. This is because the sum of pdf of g(x)=1 and the sum of pdf of f(x)=1, hence, <math>g(x) \ngeqq f(x)</math> &forall;x. <br>
+
Note: If the red line was only g(x) as opposed to <math>\,c g(x)</math> (i.e. c=1), then <math>g(x) \geq f(x)</math> for all values of x if and only if g and f are the same functions. This is because the sum of pdf of g(x)=1 and the sum of pdf of f(x)=1, hence, <math>g(x) \ngeqq f(x)</math> \,&forall;x. <br>
  
 
Also remember that <math>\,c g(x)</math> always generates higher probability than what we need. Thus we need an approach of getting the proper probabilities.<br><br>
 
Also remember that <math>\,c g(x)</math> always generates higher probability than what we need. Thus we need an approach of getting the proper probabilities.<br><br>
Line 917: Line 1,038:
 
3. Verify that <math>f(x)\leqslant c g(x)</math> at all the local maximums as well as the absolute maximums.<br>
 
3. Verify that <math>f(x)\leqslant c g(x)</math> at all the local maximums as well as the absolute maximums.<br>
 
4. Verify that <math>f(x)\leqslant c g(x)</math> at the tail ends by calculating <math>\lim_{x \to +\infty} \frac{f(x)}{\, c g(x)}</math> and <math>\lim_{x \to -\infty} \frac{f(x)}{\, c g(x)}</math> and seeing that they are both < 1. Use of L'Hopital's Rule should make this easy, since both f and g are p.d.f's, resulting in both of them approaching 0.<br>
 
4. Verify that <math>f(x)\leqslant c g(x)</math> at the tail ends by calculating <math>\lim_{x \to +\infty} \frac{f(x)}{\, c g(x)}</math> and <math>\lim_{x \to -\infty} \frac{f(x)}{\, c g(x)}</math> and seeing that they are both < 1. Use of L'Hopital's Rule should make this easy, since both f and g are p.d.f's, resulting in both of them approaching 0.<br>
5.Efficiency: the number of times N that steps 1 and 2 need to be called(also the number of iterations needed to successfully generate X) is a random variable and has a geometric distribution with success probability p=P(U<= f(Y)/(cg(Y))) , P(N=n)=(1-p^(n-1))p ,n>=1.Thus on average the number of iterations required is given by E(N)=1/p
+
5.Efficiency: the number of times N that steps 1 and 2 need to be called(also the number of iterations needed to successfully generate X) is a random variable and has a geometric distribution with success probability <math>p=P(U \leq f(Y)/(cg(Y)))</math> , <math>P(N=n)=(1-p(n-1))p ,n \geq 1</math>.Thus on average the number of iterations required is given by <math> E(N)=\frac{1} p</math>
  
 
c should be close to the maximum of f(x)/g(x), not just some arbitrarily picked large number. Otherwise, the Acceptance-Rejection method will have more rejections (since our probability <math>f(x)\leqslant c g(x)</math> will be close to zero). This will render our algorithm inefficient.  
 
c should be close to the maximum of f(x)/g(x), not just some arbitrarily picked large number. Otherwise, the Acceptance-Rejection method will have more rejections (since our probability <math>f(x)\leqslant c g(x)</math> will be close to zero). This will render our algorithm inefficient.  
Line 957: Line 1,078:
  
 
'''Some notes on the constant C'''<br>
 
'''Some notes on the constant C'''<br>
1. C is chosen such that <math> c g(y)\geq f(y)</math>, that is,<math> c X g(y)</math> will always dominate <math>f(y)</math>. Because of this,  
+
1. C is chosen such that <math> c g(y)\geq f(y)</math>, that is,<math> c g(y)</math> will always dominate <math>f(y)</math>. Because of this,  
C will always be greater than or equal to one and will only equal to one if and only if the proposal distribution and the target distribution are the same. It is normally best to choose C such that the absolute maxima of both <math> c X g(y)</math> and <math> f(y)</math> are the same.<br>
+
C will always be greater than or equal to one and will only equal to one if and only if the proposal distribution and the target distribution are the same. It is normally best to choose C such that the absolute maxima of both <math> c g(y)</math> and <math> f(y)</math> are the same.<br>
  
2. <math> \frac {1}{C} </math> is the area of <math> F(y)</math> over the area of  <math> c X G(y)</math> and is the acceptance rate of the points generated. For example, if <math> \frac {1}{C} = 0.7</math> then on average, 70 percent of all points generated are accepted.<br>
+
2. <math> \frac {1}{C} </math> is the area of <math> F(y)</math> over the area of  <math> c G(y)</math> and is the acceptance rate of the points generated. For example, if <math> \frac {1}{C} = 0.7</math> then on average, 70 percent of all points generated are accepted.<br>
  
 
3. C is the average number of times Y is generated from g .
 
3. C is the average number of times Y is generated from g .
Line 1,001: Line 1,122:
 
'''Comments:'''
 
'''Comments:'''
  
-Acceptance-Rejection Method is not good for all cases. The limitation with this method is that sometimes many points will be rejected. One obvious cons is that it could be very hard to pick the <math>g(y)</math> and the constant <math>c</math> in some cases. We have to pick the SMALLEST C such that <math>cg(x) \leq f(x)</math> else the the algorithm will not be efficient. This is because <math>f(x)/cg(x)</math> will become smaller and probability <math>u \leq f(x)/cg(x)</math> will go down and many points will be rejected making the algorithm inefficient.  
+
-Acceptance-Rejection Method is not good for all cases. The limitation with this method is that sometimes many points will be rejected. One obvious disadvantage is that it could be very hard to pick the <math>g(y)</math> and the constant <math>c</math> in some cases. We have to pick the SMALLEST C such that <math>cg(x) \leq f(x)</math> else the the algorithm will not be efficient. This is because <math>f(x)/cg(x)</math> will become smaller and probability <math>u \leq f(x)/cg(x)</math> will go down and many points will be rejected making the algorithm inefficient.  
  
 
-'''Note:''' When <math>f(y)</math> is very different than <math>g(y)</math>, it is less likely that the point will be accepted as the ratio above would be very small and it will be difficult for <math>U</math> to be less than this small value. <br/>An example would be when the target function (<math>f</math>) has a spike or several spikes in its domain - this would force the known distribution (<math>g</math>) to have density at least as large as the spikes, making the value of <math>c</math> larger than desired. As a result, the algorithm would be highly inefficient.
 
-'''Note:''' When <math>f(y)</math> is very different than <math>g(y)</math>, it is less likely that the point will be accepted as the ratio above would be very small and it will be difficult for <math>U</math> to be less than this small value. <br/>An example would be when the target function (<math>f</math>) has a spike or several spikes in its domain - this would force the known distribution (<math>g</math>) to have density at least as large as the spikes, making the value of <math>c</math> larger than desired. As a result, the algorithm would be highly inefficient.
Line 1,009: Line 1,130:
 
We wish to generate X~Bi(2,0.5), assuming that we cannot generate this directly.<br/>
 
We wish to generate X~Bi(2,0.5), assuming that we cannot generate this directly.<br/>
 
We use a discrete distribution DU[0,2] to approximate this.<br/>
 
We use a discrete distribution DU[0,2] to approximate this.<br/>
<math>f(x)=Pr(X=x)=2Cx*(0.5)^2</math><br/>
+
<math>f(x)=Pr(X=x)=2Cx×(0.5)^2\,</math><br/>
  
 
{| class=wikitable  align=left
 
{| class=wikitable  align=left
Line 1,030: Line 1,151:
 
1. Generate <math>u,v~U(0,1)</math><br/>
 
1. Generate <math>u,v~U(0,1)</math><br/>
 
2. Set <math>y= \lfloor 3*u \rfloor</math> (This is using uniform distribution to generate DU[0,2]<br/>
 
2. Set <math>y= \lfloor 3*u \rfloor</math> (This is using uniform distribution to generate DU[0,2]<br/>
3. If <math>(y=0)</math> and <math>(v<1/2), output=0</math> <br/>
+
3. If <math>(y=0)</math> and <math>(v<\tfrac{1}{2}), output=0</math> <br/>
If <math>(y=2) </math> and <math>(v<1/2), output=2 </math><br/>
+
If <math>(y=2) </math> and <math>(v<\tfrac{1}{2}), output=2 </math><br/>
 
Else if <math>y=1, output=1</math><br/>
 
Else if <math>y=1, output=1</math><br/>
  
  
 
An elaboration of “c”<br/>
 
An elaboration of “c”<br/>
c is the expected number of times the code runs to output 1 random variable.  Remember that when <math>u < f(x)/(cg(x))</math> is not satisfied, we need to go over the code again.<br/>
+
c is the expected number of times the code runs to output 1 random variable.  Remember that when <math>u < \tfrac{f(x)}{cg(x)}</math> is not satisfied, we need to go over the code again.<br/>
  
 
Proof<br/>
 
Proof<br/>
Line 1,057: Line 1,178:
 
=== Example of Acceptance-Rejection Method===
 
=== Example of Acceptance-Rejection Method===
  
Generating a random variable having p.d.f.  
+
Generating a random variable having p.d.f. <br />
                                <math>f(x) = 20x(1 - x)^3,        0< x <1  </math>
+
<math>\displaystyle f(x) = 20x(1 - x)^3,        0< x <1  </math><br />
Since this random variable (which is beta with parameters 2, 4) is concentrated in the interval (0, 1), let us consider the acceptance-rejection method with
+
Since this random variable (which is beta with parameters (2,4)) is concentrated in the interval (0, 1), let us consider the acceptance-rejection method with<br />
                                    g(x) = 1,           0 < x < 1
+
<math>\displaystyle g(x) = 1,0<x<1</math><br />
To determine the constant c such that f(x)/g(x) <= c, we use calculus to determine the maximum value of
+
To determine the constant c such that f(x)/g(x) <= c, we use calculus to determine the maximum value of<br />
                                  <math> f(x)/g(x) = 20x(1 - x)^3 </math>
+
<math>\displaystyle f(x)/g(x) = 20x(1 - x)^3 </math><br />
Differentiation of this quantity yields                              
+
Differentiation of this quantity yields <br />                             
                                  <math>d/dx[f(x)/g(x)]=20*[(1-x)^3-3x(1-x)^2]</math>
+
<math>\displaystyle d/dx[f(x)/g(x)]=20*[(1-x)^3-3x(1-x)^2]</math><br />
 
Setting this equal to  0  shows that the maximal value is attained when x = 1/4,  
 
Setting this equal to  0  shows that the maximal value is attained when x = 1/4,  
and thus,                            
+
and thus, <br />
                                  <math>f(x)/g(x)<= 20*(1/4)*(3/4)^3=135/64=c </math>                                  
+
<math>\displaystyle f(x)/g(x)<= 20*(1/4)*(3/4)^3=135/64=c </math><br />
Hence,
+
Hence,<br />
                                  <math>f(x)/cg(x)=(256/27)*(x*(1-x)^3)</math>                            
+
<math>\displaystyle f(x)/cg(x)=(256/27)*(x*(1-x)^3)</math><br />
 
and thus the simulation procedure is as follows:
 
and thus the simulation procedure is as follows:
  
Line 1,086: Line 1,207:
 
===Another Example of Acceptance-Rejection Method===
 
===Another Example of Acceptance-Rejection Method===
 
Generate a random variable from:<br />  
 
Generate a random variable from:<br />  
  <math>f(x)=3*x^2</math>, 0< x <1<br />
+
<math>\displaystyle f(x)=3*x^2, 0<x<1 </math><br />
 
Assume g(x) to be uniform over interval (0,1), where 0< x <1<br />
 
Assume g(x) to be uniform over interval (0,1), where 0< x <1<br />
 
Therefore:<br />
 
Therefore:<br />
  <math>c = max(f(x)/(g(x)))= 3</math><br />   
+
<math>\displaystyle c = max(f(x)/(g(x)))= 3</math><br />   
  
 
the best constant c is the max(f(x)/(cg(x))) and the c make the area above the f(x) and below the g(x) to be small.
 
the best constant c is the max(f(x)/(cg(x))) and the c make the area above the f(x) and below the g(x) to be small.
because g(.) is uniform so the g(x) is 1. max(g(x)) is 1
+
because g(.) is uniform so the g(x) is 1. max(g(x)) is 1<br />
  <math>f(x)/(cg(x))= x^2</math><br />
+
<math>\displaystyle f(x)/(cg(x))= x^2</math><br />
 
Acknowledgement: this is example 1 from http://www.cs.bgu.ac.il/~mps042/acceptance.htm
 
Acknowledgement: this is example 1 from http://www.cs.bgu.ac.il/~mps042/acceptance.htm
  
 
== Class 4 - Thursday, May 16 ==  
 
== Class 4 - Thursday, May 16 ==  
*When we want to find a target distribution, denoted as <math>f(x)</math>, we need to first find a proposal distribution <math>g(x)</math>  that is easy to sample from. <br>  
+
 
*The relationship between the proposal distribution and target distribution is: <math> c \cdot g(x) \geq f(x) </math>, where c is a constant. The means that the area of f(x) is under the area of <math> c \cdot g(x)</math>. <br>
+
'''Goals'''<br>
 +
*When we want to find target distribution <math>f(x)</math>, we need to first find a proposal distribution <math>g(x)</math>  that is easy to sample from. <br>  
 +
*Relationship between the proposal distribution and target distribution is: <math> c \cdot g(x) \geq f(x) </math>, where c is constant. This means that the area of f(x) is under the area of <math> c \cdot g(x)</math>. <br>
 
*Chance of acceptance is less if the distance between <math>f(x)</math> and <math> c \cdot g(x)</math> is big, and vice-versa, we use <math> c </math> to keep <math> \frac {f(x)}{c \cdot g(x)} </math> below 1 (so <math>f(x) \leq c \cdot g(x)</math>). Therefore, we must find the constant <math> C </math> to achieve this.<br />
 
*Chance of acceptance is less if the distance between <math>f(x)</math> and <math> c \cdot g(x)</math> is big, and vice-versa, we use <math> c </math> to keep <math> \frac {f(x)}{c \cdot g(x)} </math> below 1 (so <math>f(x) \leq c \cdot g(x)</math>). Therefore, we must find the constant <math> C </math> to achieve this.<br />
*In other words, a <math>C</math> is chosen to make sure  <math> c \cdot g(x) \geq f(x) </math>. However, it will not make sense if <math>C</math> is simply chosen to be arbitrarily large. We need to choose <math>C</math> such that <math>c \cdot g(x)</math> fits <math>f(x)</math> as tightly as possible.<br />
+
*In other words, <math>C</math> is chosen to make sure  <math> c \cdot g(x) \geq f(x) </math>. However, it will not make sense if <math>C</math> is simply chosen to be arbitrarily large. We need to choose <math>C</math> such that <math>c \cdot g(x)</math> fits <math>f(x)</math> as tightly as possible. This means that we must find the minimum c such that the area of f(x) is under the area of c*g(x). <br />
*The constant c can not be negative number.<br />
+
*The constant c cannot be a negative number.<br />
 +
 
  
 
'''How to find C''':<br />
 
'''How to find C''':<br />
 +
 
<math>\begin{align}
 
<math>\begin{align}
 
&c \cdot g(x) \geq f(x)\\
 
&c \cdot g(x) \geq f(x)\\
Line 1,109: Line 1,234:
 
&c= \max \left(\frac{f(x)}{g(x)}\right)  
 
&c= \max \left(\frac{f(x)}{g(x)}\right)  
 
\end{align}</math><br>
 
\end{align}</math><br>
 +
 
If <math>f</math> and <math> g </math> are continuous, we can find the extremum by taking the derivative and solve for <math>x_0</math> such that:<br/>
 
If <math>f</math> and <math> g </math> are continuous, we can find the extremum by taking the derivative and solve for <math>x_0</math> such that:<br/>
 
<math> 0=\frac{d}{dx}\frac{f(x)}{g(x)}|_{x=x_0}</math> <br/>
 
<math> 0=\frac{d}{dx}\frac{f(x)}{g(x)}|_{x=x_0}</math> <br/>
 +
 
Thus <math> c = \frac{f(x_0)}{g(x_0)} </math><br/>
 
Thus <math> c = \frac{f(x_0)}{g(x_0)} </math><br/>
  
*The logic behind this:
+
Note: This procedure is called the Acceptance-Rejection Method.<br>  
The Acceptance-Rejection method involves finding a distribution that we know how to sample from (g(x)) and multiplying g(x) by a constant c so that <math>c \cdot g(x)</math> is always greater than or equal to f(x). Mathematically, we want <math> c \cdot g(x) \geq f(x) </math>.
 
And it means c has to be greater or equal to <math>\frac{f(x)}{g(x)}</math>. So the smallest possible c that satisfies the condition is the maximum value of <math>\frac{f(x)}{g(x)}</math> <br />. If c is made to be too large, the chance of acceptance of generated values will be small, and the algorithm will lose its purpose since the acceptance will be very small. Therefore, it is best to get the smallest posisble c such that <math> c g(x) \geq f(x)</math>. <br>
 
  
*For this method to be efficient, the constant c must be selected so that the rejection rate is low.(The efficiency for this method is<math>\left ( \frac{1}{c} \right )</math>)<br>
+
'''The Acceptance-Rejection method''' involves finding a distribution that we know how to sample from, g(x), and multiplying g(x) by a constant c so that <math>c \cdot g(x)</math> is always greater than or equal to f(x). Mathematically, we want <math> c \cdot g(x) \geq f(x) </math>.
*It is easy to show that the expected number of trials for an acceptance is <math>total number of trials / c</math>. Thus, the smaller the c is, the lower the rejection rate, and the better the algorithm:<br>
+
And it means, c has to be greater or equal to <math>\frac{f(x)}{g(x)}</math>. So the smallest possible c that satisfies the condition is the maximum value of <math>\frac{f(x)}{g(x)}</math><br/>.
*recall the acceptance rate is 1/c.(not rejection rate)  
+
But in case of c being too large, the chance of acceptance of generated values will be small, thereby losing efficiency of the algorithm. Therefore, it is best to get the smallest possible c such that <math> c g(x) \geq f(x)</math>. <br>
 +
 
 +
'''Important points:'''<br>
 +
 
 +
*For this method to be efficient, the constant c must be selected so that the rejection rate is low. (The efficiency for this method is <math>\left ( \frac{1}{c} \right )</math>)<br>
 +
*It is easy to show that the expected number of trials for an acceptance is <math> \frac{Total Number of Trials} {C} </math>. <br>
 +
*recall the '''acceptance rate is 1/c'''. (Not rejection rate)  
 
:Let <math>X</math> be the number of trials for an acceptance, <math> X \sim~ Geo(\frac{1}{c})</math><br>
 
:Let <math>X</math> be the number of trials for an acceptance, <math> X \sim~ Geo(\frac{1}{c})</math><br>
 
:<math>\mathbb{E}[X] = \frac{1}{\frac{1}{c}} = c </math>
 
:<math>\mathbb{E}[X] = \frac{1}{\frac{1}{c}} = c </math>
 
*The number of trials needed to generate a sample size of <math>N</math> follows a negative binomial distribution. The expected number of trials needed is then <math>cN</math>.<br>
 
*The number of trials needed to generate a sample size of <math>N</math> follows a negative binomial distribution. The expected number of trials needed is then <math>cN</math>.<br>
 
*So far, the only distribution we know how to sample from is the '''UNIFORM''' distribution. <br>
 
*So far, the only distribution we know how to sample from is the '''UNIFORM''' distribution. <br>
 +
  
 
'''Procedure''': <br>
 
'''Procedure''': <br>
 +
 
1. Choose <math>g(x)</math> (simple density function that we know how to sample, i.e. Uniform so far) <br>
 
1. Choose <math>g(x)</math> (simple density function that we know how to sample, i.e. Uniform so far) <br>
The easiest case is UNIF(0,1). However, in other cases we need to generate UNIF(a,b). We may need to perform a linear transformation on the UNIF(0,1) variable. <br>
+
The easiest case is <math>U~ \sim~ Unif [0,1] </math>. However, in other cases we need to generate UNIF(a,b). We may need to perform a linear transformation on the <math>U~ \sim~ Unif [0,1] </math> variable. <br>
 
2. Find a constant c such that :<math> c \cdot g(x) \geq f(x) </math>, otherwise return to step 1.
 
2. Find a constant c such that :<math> c \cdot g(x) \geq f(x) </math>, otherwise return to step 1.
  
Line 1,135: Line 1,268:
 
#If <math>U \leq \frac{f(Y)}{c \cdot g(Y)}</math> then X=Y; else return to step 1 (This is not the way to find C. This is the general procedure.)
 
#If <math>U \leq \frac{f(Y)}{c \cdot g(Y)}</math> then X=Y; else return to step 1 (This is not the way to find C. This is the general procedure.)
  
<hr><b>Example: Generate a random variable from the pdf</b><br>
+
<hr><b>Example: <br>
 +
 
 +
Generate a random variable from the pdf</b><br>
 
<math> f(x) =  
 
<math> f(x) =  
 
\begin{cases}  
 
\begin{cases}  
Line 1,145: Line 1,280:
 
<math>beta(a,b)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{(a-1)}(1-x)^{(b-1)}</math><br>
 
<math>beta(a,b)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{(a-1)}(1-x)^{(b-1)}</math><br>
  
Where &Gamma; (n)=(n-1)! if n is positive integer
+
Where &Gamma; (n) = (n - 1)! if n is positive integer
  
 
<math>Gamma(z)=\int _{0}^{\infty }t^{z-1}e^{-t}dt</math>
 
<math>Gamma(z)=\int _{0}^{\infty }t^{z-1}e^{-t}dt</math>
Line 1,166: Line 1,301:
 
Taking x = 1 gives the highest possible c, which is c=2
 
Taking x = 1 gives the highest possible c, which is c=2
 
<br />Note that c is a scalar greater than 1.
 
<br />Note that c is a scalar greater than 1.
 +
<br />cg(x) is proposal dist, and f(x) is target dist.
  
 
[[File:Beta(2,1)_example.jpg|750x750px]]
 
[[File:Beta(2,1)_example.jpg|750x750px]]
  
Note: g follows uniform distribution, it only covers half of the graph which runs from 0 to 1 on y-axis. Thus we need to multiply by c to ensure that <math>c\cdot g</math> can cover entire f(x) area. In this case, c=2, so that makes g runs from 0 to 2 on y-axis which covers f(x).
+
'''Note:''' g follows uniform distribution, it only covers half of the graph which runs from 0 to 1 on y-axis. Thus we need to multiply by c to ensure that <math>c\cdot g</math> can cover entire f(x) area. In this case, c=2, so that makes g run from 0 to 2 on y-axis which covers f(x).
  
Comment:
+
'''Comment:'''<br>
From the picture above, we could observe that the area under f(x)=2x is a half of the area under the pdf of UNIF(0,1). This is why in order to sample 1000 points of f(x) we need to sample approximately 2000 points in UNIF(0,1).
+
From the picture above, we could observe that the area under f(x)=2x is a half of the area under the pdf of UNIF(0,1). This is why in order to sample 1000 points of f(x), we need to sample approximately 2000 points in UNIF(0,1).
 
And in general, if we want to sample n points from a distritubion with pdf f(x), we need to scan approximately <math>n\cdot c</math> points from the proposal distribution (g(x)) in total. <br>
 
And in general, if we want to sample n points from a distritubion with pdf f(x), we need to scan approximately <math>n\cdot c</math> points from the proposal distribution (g(x)) in total. <br>
 
<b>Step</b>
 
<b>Step</b>
Line 1,182: Line 1,318:
 
</ol>
 
</ol>
  
Note: In the above example, we sample 2 numbers. If second number (u) is less than or equal to first number (y), then accept x=y, if not then start all over.
+
'''Note:''' In the above example, we sample 2 numbers. If second number (u) is less than or equal to first number (y), then accept x=y, if not then start all over.
  
 
<span style="font-weight:bold;color:green;">Matlab Code</span>
 
<span style="font-weight:bold;color:green;">Matlab Code</span>
Line 1,199: Line 1,335:
 
     end
 
     end
 
   end
 
   end
>>hist(x)
+
>>hist(x)         # It is a histogram
 
>>jj
 
>>jj
 
   jj = 2024        # should be around 2000
 
   jj = 2024        # should be around 2000
Line 1,205: Line 1,341:
 
[[File:ARM_Example.jpg|300px]]
 
[[File:ARM_Example.jpg|300px]]
  
:'''*Note:''' The reason that a for loop is not used is that we need continue the looping until we get 1000 successful samples. We will reject some samples during the process and therefore do not know the number of y we are going to generate.  
+
:'''*Note:''' The reason that a for loop is not used is that we need to continue the looping until we get 1000 successful samples. We will reject some samples during the process and therefore do not know the number of y we are going to generate.  
  
 
:'''*Note2:''' In this example, we used c=2, which means we accept half of the points we generate on average. Generally speaking, 1/c would be the probability of acceptance, and an indicator of the efficiency of your chosen proposal distribution and algorithm.  
 
:'''*Note2:''' In this example, we used c=2, which means we accept half of the points we generate on average. Generally speaking, 1/c would be the probability of acceptance, and an indicator of the efficiency of your chosen proposal distribution and algorithm.  
Line 1,232: Line 1,368:
  
 
<span style="font-weight:bold;colour:green;">Matlab Tip:</span>
 
<span style="font-weight:bold;colour:green;">Matlab Tip:</span>
Periods, ".",meaning "element-wise", are used to describe the operation you want performed on each element of a vector. In the above example, to take the square root of every element in U, the notation U.^0.5 is used. However if you want to take the Square root of the entire matrix U the period, "*.*" would be excluded. i.e. Let matrix B=U^0.5, then <math>B^T*B=U</math>. For example if we have a two 1 X 3 matrices and we want to find out their product; using "." in the code will give us their product; however, if we don't use "." it will just give us an error. For example, a =[1 2 3] b=[2 3 4] are vectors, a.*b=[2 6 12], but a*b does not work since matrix dimensions must agree.
+
Periods, ".",meaning "element-wise", are used to describe the operation you want performed on each element of a vector. In the above example, to take the square root of every element in U, the notation U.^0.5 is used. However if you want to take the square root of the entire matrix U the period, "." would be excluded. i.e. Let matrix B=U^0.5, then <math>B^T*B=U</math>. For example if we have a two 1 X 3 matrices and we want to find out their product; using "." in the code will give us their product. However, if we don't use ".", it will just give us an error. For example, a =[1 2 3] b=[2 3 4] are vectors, a.*b=[2 6 12], but a*b does not work since the matrix dimensions must agree.
  
 
'''
 
'''
Line 1,247: Line 1,383:
 
<math> cg(x)\geq f(x),
 
<math> cg(x)\geq f(x),
 
c\frac{1}{2} \geq \frac{3}{4} (1-x^2) /1,  
 
c\frac{1}{2} \geq \frac{3}{4} (1-x^2) /1,  
c=max 2*\frac{3}{4} (1-x^2) = 3/2 </math>
+
c=max 2\cdot\frac{3}{4} (1-x^2) = 3/2 </math>
  
 
The process:
 
The process:
Line 1,262: Line 1,398:
 
=====Example of Acceptance-Rejection Method=====
 
=====Example of Acceptance-Rejection Method=====
  
<math> f(x) = 3x^2,  0<x<1 </math>
+
<math>\begin{align}
<math>g(x)=1,  0<x<1</math>
+
& f(x) = 3x^2,  0<x<1 \\
 +
\end{align}</math><br\>
 +
 
 +
<math>\begin{align}
 +
& g(x)=1,  0<x<1 \\
 +
\end{align}</math><br\>
  
 
<math>c = \max \frac{f(x)}{g(x)} = \max \frac{3x^2}{1} = 3 </math><br>
 
<math>c = \max \frac{f(x)}{g(x)} = \max \frac{3x^2}{1} = 3 </math><br>
Line 1,269: Line 1,410:
  
 
1. Generate two uniform numbers in the unit interval <math>U_1, U_2 \sim~ U(0,1)</math><br>
 
1. Generate two uniform numbers in the unit interval <math>U_1, U_2 \sim~ U(0,1)</math><br>
2. If <math>U_2 \leqslant {U_1}^2</math>, accept <math>U_1</math> as the random variable with pdf <math>f</math>, if not return to Step 1
+
2. If <math>U_2 \leqslant {U_1}^2</math>, accept <math>\begin{align}U_1\end{align}</math> as the random variable with pdf <math>\begin{align}f\end{align}</math>, if not return to Step 1
  
We can also use <math>g(x)=2x</math> for a more efficient algorithm
+
We can also use <math>\begin{align}g(x)=2x\end{align}</math> for a more efficient algorithm
  
 
<math>c = \max \frac{f(x)}{g(x)} = \max \frac {3x^2}{2x} = \frac {3x}{2}  </math>.
 
<math>c = \max \frac{f(x)}{g(x)} = \max \frac {3x^2}{2x} = \frac {3x}{2}  </math>.
Use the inverse method to sample from <math>g(x)</math>
+
Use the inverse method to sample from <math>\begin{align}g(x)\end{align}</math>
<math>G(x)=x^2</math>.
+
<math>\begin{align}G(x)=x^2\end{align}</math>.
Generate <math>U</math> from <math>U(0,1)</math> and set <math>x=sqrt(u)</math>
+
Generate <math>\begin{align}U\end{align}</math> from <math>\begin{align}U(0,1)\end{align}</math> and set <math>\begin{align}x=sqrt(u)\end{align}</math>
  
 
1. Generate two uniform numbers in the unit interval <math>U_1, U_2 \sim~ U(0,1)</math><br>
 
1. Generate two uniform numbers in the unit interval <math>U_1, U_2 \sim~ U(0,1)</math><br>
 
2. If <math>U_2 \leq \frac{3\sqrt{U_1}}{2}</math>, accept <math>U_1</math> as the random variable with pdf <math>f</math>, if not return to Step 1
 
2. If <math>U_2 \leq \frac{3\sqrt{U_1}}{2}</math>, accept <math>U_1</math> as the random variable with pdf <math>f</math>, if not return to Step 1
  
 +
*Note :the function <math>\begin{align}q(x) = c * g(x)\end{align}</math> is called an envelop or majoring function.<br>
 +
To obtain a better proposing function <math>\begin{align}g(x)\end{align}</math>, we can first assume a new <math>\begin{align}q(x)\end{align}</math> and then solve for the normalizing constant by integrating.<br>
 +
In the previous example, we first assume <math>\begin{align}q(x) = 3x\end{align}</math>. To find the normalizing constant, we need to solve  <math>k *\sum 3x = 1</math> which gives us k = 2/3. So,<math>\begin{align}g(x) = k*q(x) = 2x\end{align}</math>.
  
 +
*Source: http://www.cs.bgu.ac.il/~mps042/acceptance.htm*       
  
 
'''Possible Limitations'''
 
'''Possible Limitations'''
Line 1,361: Line 1,506:
 
'''One more example about AR method''' <br/>
 
'''One more example about AR method''' <br/>
 
(In this example, we will see how to determine the value of c when c is a function with unknown parameters instead of a value)
 
(In this example, we will see how to determine the value of c when c is a function with unknown parameters instead of a value)
Let <math>f(x)=x*e^{-x}, x>0 </math> <br/>
+
Let <math>f(x)=x×e^{-x}, x > 0 </math> <br/>
Use <math>g(x)=a*e^{-a*x}</math>to generate random variable <br/>
+
Use <math>g(x)=a×e^{-a×x}</math> to generate random variable <br/>
 
<br/>
 
<br/>
 
Solution: First of all, we need to find c<br/>
 
Solution: First of all, we need to find c<br/>
Line 1,405: Line 1,550:
  
 
'''Summary of when to use the Accept Rejection Method''' <br/>
 
'''Summary of when to use the Accept Rejection Method''' <br/>
1) When the calculation of inverse cdf cannot to be computed or too difficult to compute. <br/>
+
1) When the calculation of inverse cdf cannot to be computed or is too difficult to compute. <br/>
 
2) When f(x) can be evaluated to at least one of the normalizing constant. <br/>
 
2) When f(x) can be evaluated to at least one of the normalizing constant. <br/>
 
3) A constant c where <math>f(x)\leq c\cdot g(x)</math><br/>
 
3) A constant c where <math>f(x)\leq c\cdot g(x)</math><br/>
 
4) A uniform draw<br/>
 
4) A uniform draw<br/>
  
----
+
==== Interpretation of 'C' ====
 +
We can use the value of c to calculate the acceptance rate by <math>\tfrac{1}{c}</math>.
 +
 
 +
For instance, assume c=1.5, then we can tell that 66.7% of the points will be accepted (<math>\tfrac{1}{1.5} = 0.667</math>). We can also call the efficiency of the method is 66.7%.
 +
 
 +
Likewise, if the minimum value of possible values for C is <math>\tfrac{4}{3}</math>, <math>1/ \tfrac{4}{3}</math> of the generated random variables will be accepted. Thus the efficient of the algorithm is 75%.
  
==== Interpretation of 'C' ====
+
In order to ensure the algorithm is as efficient as possible, the 'C' value should be as close to one as possible, such that <math>\tfrac{1}{c}</math> approaches 1 => 100% acceptance rate.
We can use the value of c to calculate the acceptance rate by '1/c'.
 
  
For instance, assume c=1.5, then we can tell that 66.7% of the points will be accepted (1/1.5=0.667). We can also call the efficiency of the method is 66.7%.
+
 
 +
>> close All
 +
>> clear All
 +
>> i=1
 +
>> j=0;
 +
>> while ii<1000
 +
y=rand
 +
u=rand
 +
if u<=y;
 +
x(ii)=y
 +
ii=ii+1
 +
end
 +
end
  
 
== Class 5 - Tuesday, May 21 ==
 
== Class 5 - Tuesday, May 21 ==
Line 1,438: Line 1,599:
 
   end
 
   end
 
>>hist(x,20)                  # 20 is the number of bars
 
>>hist(x,20)                  # 20 is the number of bars
 +
 +
>>hist(x,30)                #30 is the number of bars
 
</pre>
 
</pre>
  
 +
calculate process:
 +
<math>u_{1} <= \sqrt (1-(2u-1)^2) </math> <br>
 +
<math>(u_{1})^2 <=(1-(2u-1)^2) </math> <br>
 +
<math>(u_{1})^2 -1 <=(-(2u-1)^2) </math> <br>
 +
<math>1-(u_{1})^2 >=((2u-1)^2-1) </math> <br>
  
  
MATLAB tips: hist(x,y) where y is the number of bars in the graph.
+
MATLAB tips: hist(x,y) plots a histogram of variable x, where y is the number of bars in the graph.
  
 
[[File:ARM_cont_example.jpg|300px]]
 
[[File:ARM_cont_example.jpg|300px]]
  
a histogram to show variable x, and the bars number is y.
 
 
=== Discrete Examples ===
 
=== Discrete Examples ===
 
* '''Example 1''' <br>
 
* '''Example 1''' <br>
Line 1,462: Line 1,629:
 
The following algorithm then yields our X:
 
The following algorithm then yields our X:
  
Step 1. Draw discrete uniform distribution of 1, 2, 3, 4 and 5, <math>Y \sim~ g</math>.<br/>
+
Step 1 Draw discrete uniform distribution of 1, 2, 3, 4 and 5, <math>Y \sim~ g</math>.<br/>
Step 2. Draw <math>U \sim~ U(0,1)</math>.<br/>
+
Step 2 Draw <math>U \sim~ U(0,1)</math>.<br/>
Step 3. If <math>U \leq \frac{f(Y)}{c \cdot g(Y)}</math>, then <b> X = Y </b>;<br/>
+
Step 3 If <math>U \leq \frac{f(Y)}{c \cdot g(Y)}</math>, then <b> X = Y </b>;<br/>
 
Else return to Step 1.<br/>
 
Else return to Step 1.<br/>
  
Line 1,472: Line 1,639:
 
:<math>\frac{p(x)}{cg(x)} =  \frac{p(x)}{1.5*0.2} = \frac{p(x)}{0.3} </math><br>
 
:<math>\frac{p(x)}{cg(x)} =  \frac{p(x)}{1.5*0.2} = \frac{p(x)}{0.3} </math><br>
 
Note: The U is independent from y in Step 2 and 3 above.
 
Note: The U is independent from y in Step 2 and 3 above.
~The constant c is a indicator of rejection rate or efficiency of the algorithm.
+
~The constant c is a indicator of rejection rate or efficiency of the algorithm. It can represent the average number of trials of the algorithm. Thus, a higher c would mean that the algorithm is comparatively inefficient.
  
the acceptance-rejection method of pmf, the uniform probability is the same for all variables, and there 5 parameters(1,2,3,4,5), so g(x) is 0.2
+
the acceptance-rejection method of pmf, the uniform probability is the same for all variables, and there are 5 parameters(1,2,3,4,5), so g(x) is 0.2
  
 
Remember that we always want to choose <math> cg </math> to be equal to or greater than <math> f </math>, but as close as possible.
 
Remember that we always want to choose <math> cg </math> to be equal to or greater than <math> f </math>, but as close as possible.
 +
<br />limitations: If the form of the proposal dist g is very different from target dist f, then c is very large and the algorithm is not computatively efficient.
  
 
* '''Code for example 1'''<br />
 
* '''Code for example 1'''<br />
Line 1,509: Line 1,677:
 
Let g be the uniform distribution of 1, 2, or 3<br />
 
Let g be the uniform distribution of 1, 2, or 3<br />
 
g(x)= 1/3<br />
 
g(x)= 1/3<br />
<math>c=max(p_{x}/g(x))=0.6/(1/3)=1.8</math><br />
+
<math>c=max(\tfrac{p_{x}}{g(x)})=0.6/(\tfrac{1}{3})=1.8</math><br />
Hence p(x)/cg(x) = p(x)/(1.8*1/3)= p(x)/0.6
+
Hence <math>\tfrac{p(x)}{cg(x)} = p(x)/(1.8 (\tfrac{1}{3}))= \tfrac{p(x)}{0.6}</math>
  
 
1,y~g<br />
 
1,y~g<br />
Line 1,520: Line 1,688:
 
>>close all
 
>>close all
 
>>clear all
 
>>clear all
>>p=[.1 .3 .6];  
+
>>p=[.1 .3 .6];     %This a vector holding the values 
 
>>ii=1;
 
>>ii=1;
 
>>while ii < 1000
 
>>while ii < 1000
     y=unidrnd(3);
+
     y=unidrnd(3);   %generates random numbers for the discrete uniform distribution with maximum 3
     u=rand;
+
     u=rand;          
 
     if u<= p(y)/0.6
 
     if u<= p(y)/0.6
       x(ii)=y;
+
       x(ii)=y;    
       ii=ii+1;
+
       ii=ii+1;     %else ii=ii+1
 
     end
 
     end
 
   end
 
   end
Line 1,535: Line 1,703:
  
 
* '''Example 3'''<br>
 
* '''Example 3'''<br>
<math>p_{x}=e^{-3}3^{x}/x! , x>=0</math><br>(poisson distribution)
 
Try the first few p_{x}'s:  .0498 .149 .224 .224 .168 .101 .0504 .0216 .0081 .0027<br>
 
  
Use the geometric distribution for <math>g(x)</math>;<br>
+
Suppose <math>\begin{align}p_{x} = e^{-3}3^{x}/x! , x\geq 0\end{align}</math> (Poisson distribution)
<math>g(x)=p(1-p)^{x}</math>, choose p=0.25<br>
 
Look at <math>p_{x}/g(x)</math> for the first few numbers: .199 .797 1.59 2.12 2.12 1.70 1.13 .647 .324 .144<br>
 
We want <math>c=max(p_{x}/g(x))</math> which is approximately 2.12<br>
 
  
1. Generate <math>U_{1} \sim~ U(0,1); U_{2} \sim~ U(0,1)</math><br>
+
'''First:''' Try the first few <math>\begin{align}p_{x}'s\end{align}</math>:  0.0498, 0.149, 0.224, 0.224, 0.168, 0.101, 0.0504, 0.0216, 0.0081, 0.0027 for <math>\begin{align} x = 0,1,2,3,4,5,6,7,8,9 \end{align}</math><br>
2. <math>j = \lfloor \frac{ln(U_{1})}{ln(.75)} \rfloor+1;</math><br>
 
3. if <math>U_{2} < \frac{p_{j}}{cg(j)}</math>, set X = x<sub>j</sub>, else go to step 1.
 
  
Note: In this case, f(x)/g(x) is extremely difficult to differentiate so we were required to test points. If the function is easily differentiable, we can calculate the max as if it were a continuous function then check the two surrounding points for which is the highest discrete value.
+
'''Proposed distribution:''' Use the geometric distribution for <math>\begin{align}g(x)\end{align}</math>;<br>
 +
 
 +
<math>\begin{align}g(x)=p(1-p)^{x}\end{align}</math>, choose <math>\begin{align}p=0.25\end{align}</math><br>
 +
 
 +
Look at <math>\begin{align}p_{x}/g(x)\end{align}</math> for the first few numbers: 0.199 0.797 1.59 2.12 2.12 1.70 1.13 0.647 0.324 0.144 for <math>\begin{align} x = 0,1,2,3,4,5,6,7,8,9 \end{align}</math><br>
 +
 
 +
We want <math>\begin{align}c=max(p_{x}/g(x))\end{align}</math> which is approximately 2.12<br>
 +
 
 +
'''The general procedures to generate <math>\begin{align}p(x)\end{align}</math> is as follows:'''
 +
 
 +
1. Generate <math>\begin{align}U_{1} \sim~ U(0,1); U_{2} \sim~ U(0,1)\end{align}</math><br>
 +
 
 +
2. <math>\begin{align}j = \lfloor \frac{ln(U_{1})}{ln(.75)} \rfloor+1;\end{align}</math><br>
 +
 
 +
3. if <math>U_{2} < \frac{p_{j}}{cg(j)}</math>, set <math>\begin{align}X = x_{j}\end{align}</math>, else go to step 1.
 +
 
 +
Note: In this case, <math>\begin{align}f(x)/g(x)\end{align}</math> is extremely difficult to differentiate so we were required to test points. If the function is very easy to differentiate, we can calculate the max as if it were a continuous function then check the two surrounding points for which is the highest discrete value.
 +
 
 +
* Source: http://www.math.wsu.edu/faculty/genz/416/lect/l04-46.pdf*
  
 
*'''Example 4''' (Hypergeometric & Binomial)<br>  
 
*'''Example 4''' (Hypergeometric & Binomial)<br>  
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<math> F(x) = \int_0^{x} \frac{e^{-y}y^{t-1}}{(t-1)!} \mathrm{d}y, \; \forall x \in (0,+\infty)</math>, where <math>t \in \N^+ \text{ and } \lambda \in (0,+\infty)</math>.<br>
 
<math> F(x) = \int_0^{x} \frac{e^{-y}y^{t-1}}{(t-1)!} \mathrm{d}y, \; \forall x \in (0,+\infty)</math>, where <math>t \in \N^+ \text{ and } \lambda \in (0,+\infty)</math>.<br>
  
 +
Note that the CDF of the Gamma distribution does not have a closed form.
  
Neither Inverse Transformation nor Acceptance/Rejection Method can be easily applied to Gamma distribution.
+
The gamma distribution is often used to model waiting times between a certain number of events. It can also be expressed as the sum of infinitely many independent and identically distributed exponential distributions. This distribution has two parameters: the number of exponential terms n, and the rate parameter <math>\lambda</math>. In this distribution there is the Gamma function, <math>\Gamma </math> which has some very useful properties. "Source: STAT 340 Spring 2010 Course Notes" <br/>
 +
 
 +
Neither Inverse Transformation nor Acceptance-Rejection Method can be easily applied to Gamma distribution.
 
However, we can use additive property of Gamma distribution to generate random variables.
 
However, we can use additive property of Gamma distribution to generate random variables.
  
Line 1,640: Line 1,822:
 
If we want to sample from the Gamma distribution, we can consider sampling from <math>t</math> independent exponential distributions using the Inverse Method for each <math> X_i</math> and add them up. Note that this only works the specific set of gamma distributions where t is a positive integer.
 
If we want to sample from the Gamma distribution, we can consider sampling from <math>t</math> independent exponential distributions using the Inverse Method for each <math> X_i</math> and add them up. Note that this only works the specific set of gamma distributions where t is a positive integer.
  
According to this property, a random variable that follows Gamma distribution is the sum of i.i.d (independent and identically distributed) exponential random variables. Now we want to generate 1000 values of <math>Gamma(20,10)</math> random variables, so we need to obtain the value of each one by adding 20 values of <math>X_i \sim~ Exp(10)</math>. To achieve this, we generate a 20-by-1000 matrix whose entries follow <math>Exp(10)</math> and add the rows together.
+
According to this property, a random variable that follows Gamma distribution is the sum of i.i.d (independent and identically distributed) exponential random variables. Now we want to generate 1000 values of <math>Gamma(20,10)</math> random variables, so we need to obtain the value of each one by adding 20 values of <math>X_i \sim~ Exp(10)</math>. To achieve this, we generate a 20-by-1000 matrix whose entries follow <math>Exp(10)</math> and add the rows together.<br />
<math> x_1 </math>~Exp(<math>\lambda </math>)
+
<math> x_1 \sim~Exp(\lambda)</math><br />
<math>x_2 </math>~Exp(<math> \lambda </math>)
+
<math>x_2 \sim~Exp(\lambda)</math><br />
...
+
...<br />
<math>x_t </math>~Exp(<math> \lambda </math>)
+
<math>x_t \sim~Exp(\lambda)</math><br />
<math>x_1+x_2+...+x_t</math>
+
<math>x_1+x_2+...+x_t~</math>
  
 
<pre style="font-size:16px">
 
<pre style="font-size:16px">
Line 1,679: Line 1,861:
 
                             all the elements are generated by rand
 
                             all the elements are generated by rand
 
>>x = (-1/lambda)*log(1-u);      Note: log(1-u) is essentially the same as log(u) only if u~U(0,1)  
 
>>x = (-1/lambda)*log(1-u);      Note: log(1-u) is essentially the same as log(u) only if u~U(0,1)  
>>xx = sum(x)                     Note: sum(x) will sum all elements in the same column.  
+
>>xx = sum(x)                   Note: sum(x) will sum all elements in the same column.  
 
                                                 size(xx) can help you to verify
 
                                                 size(xx) can help you to verify
 
>>size(sum(x))                  Note: see the size of x if we forget it
 
>>size(sum(x))                  Note: see the size of x if we forget it
Line 1,704: Line 1,886:
 
</pre>
 
</pre>
  
in the matrix rand(20,1000) means 20 row with 1000 numbers for each.
+
In the matrix rand(20,1000) means 20 row with 1000 numbers for each.
 
use the code to show the generalize the distributions for multidimensional purposes in different cases, such as sum xi (each xi not equal xj), and they are independent, or matrix. Finally, we can see the conclusion is shown by the histogram.
 
use the code to show the generalize the distributions for multidimensional purposes in different cases, such as sum xi (each xi not equal xj), and they are independent, or matrix. Finally, we can see the conclusion is shown by the histogram.
  
Line 1,712: Line 1,894:
 
<math> R=\sqrt{x_{1}^2+x_{2}^2}= x_{2}/sin(\theta)= x_{1}/cos(\theta)</math> <br />
 
<math> R=\sqrt{x_{1}^2+x_{2}^2}= x_{2}/sin(\theta)= x_{1}/cos(\theta)</math> <br />
 
<math> tan(\theta)=x_{2}/x_{1} \rightarrow \theta=tan^{-1}(x_{2}/x_{1})</math> <br />
 
<math> tan(\theta)=x_{2}/x_{1} \rightarrow \theta=tan^{-1}(x_{2}/x_{1})</math> <br />
 +
 +
*Box-Muller Transformation:<br>
 +
It is a transformation that consumes two continuous uniform random variables <math> X \sim U(0,1), Y \sim U(0,1) </math> and outputs a bivariate normal random variable with <math> Z_1\sim N(0,1), Z_2\sim N(0,1). </math>
  
 
=== '''Matlab''' ===
 
=== '''Matlab''' ===
Line 1,734: Line 1,919:
 
:<math>f(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}</math>
 
:<math>f(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}</math>
  
*Warning : the General Normal distribution is  
+
*Warning : the General Normal distribution is:
:
 
 
<table>
 
<table>
 
<tr>
 
<tr>
Line 1,787: Line 1,971:
  
 
Let <math> \theta </math> and R denote the Polar coordinate of the vector (X, Y)  
 
Let <math> \theta </math> and R denote the Polar coordinate of the vector (X, Y)  
 +
where <math> X = R \cdot \sin\theta </math> and <math> Y = R \cdot \cos \theta </math>
  
 
[[File:rtheta.jpg]]
 
[[File:rtheta.jpg]]
Line 1,803: Line 1,988:
 
We know that  
 
We know that  
  
:R<sup>2</sup>= X<sup>2</sup>+Y<sup>2</sup> and <math> \tan(\theta) = \frac{y}{x} </math> where X and Y are two independent standard normal
+
<math>R^{2}= X^{2}+Y^{2}</math> and <math> \tan(\theta) = \frac{y}{x} </math> where X and Y are two independent standard normal
 
:<math>f(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}</math>
 
:<math>f(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}</math>
 
:<math>f(y) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} y^2}</math>
 
:<math>f(y) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} y^2}</math>
:<math>f(x,y) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2} * \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} y^2}=\frac{1}{2\pi}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} (x^2+y^2)} </math><br /> - Since for independent distributions, their joint probability function is the multiplication of two independent probability functions
+
:<math>f(x,y) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2} * \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} y^2}=\frac{1}{2\pi}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} (x^2+y^2)} </math><br /> - Since for independent distributions, their joint probability function is the multiplication of two independent probability functions. It can also be shown using 1-1 transformation that the joint distribution of R and θ is given by, 1-1 transformation:<br />
It can also be shown using 1-1 transformation that the joint distribution of R and θ is given by,
+
 
1-1 transformation:<br />
+
 
Let <math>d=R^2</math><br />
+
'''Let <math>d=R^2</math>'''<br />
 +
 
 
  <math>x= \sqrt {d}\cos \theta </math>
 
  <math>x= \sqrt {d}\cos \theta </math>
 
  <math>y= \sqrt {d}\sin \theta </math>
 
  <math>y= \sqrt {d}\sin \theta </math>
 
then  
 
then  
 
<math>\left| J\right| = \left| \dfrac {1} {2}d^{-\frac {1} {2}}\cos \theta d^{\frac{1}{2}}\cos \theta +\sqrt {d}\sin \theta \dfrac {1} {2}d^{-\frac{1}{2}}\sin \theta \right| = \dfrac {1} {2}</math>
 
<math>\left| J\right| = \left| \dfrac {1} {2}d^{-\frac {1} {2}}\cos \theta d^{\frac{1}{2}}\cos \theta +\sqrt {d}\sin \theta \dfrac {1} {2}d^{-\frac{1}{2}}\sin \theta \right| = \dfrac {1} {2}</math>
It can be shown that the pdf of <math> d </math> and <math> \theta </math> is:
+
It can be shown that the joint density of <math> d /R^2</math> and <math> \theta </math> is:
 
:<math>\begin{matrix}  f(d,\theta) = \frac{1}{2}e^{-\frac{d}{2}}*\frac{1}{2\pi},\quad  d = R^2 \end{matrix},\quad for\quad 0\leq d<\infty\ and\quad 0\leq \theta\leq 2\pi </math>
 
:<math>\begin{matrix}  f(d,\theta) = \frac{1}{2}e^{-\frac{d}{2}}*\frac{1}{2\pi},\quad  d = R^2 \end{matrix},\quad for\quad 0\leq d<\infty\ and\quad 0\leq \theta\leq 2\pi </math>
  
Line 1,821: Line 2,007:
 
Note that <math> \begin{matrix}f(r,\theta)\end{matrix}</math> consists of two density functions, Exponential and Uniform, so assuming that r and <math>\theta</math> are independent
 
Note that <math> \begin{matrix}f(r,\theta)\end{matrix}</math> consists of two density functions, Exponential and Uniform, so assuming that r and <math>\theta</math> are independent
 
<math> \begin{matrix} \Rightarrow d \sim~ Exp(1/2),  \theta \sim~ Unif[0,2\pi] \end{matrix} </math>
 
<math> \begin{matrix} \Rightarrow d \sim~ Exp(1/2),  \theta \sim~ Unif[0,2\pi] \end{matrix} </math>
::* <math> \begin{align} R^2 = x^2 + y^2 \end{align} </math>
+
::* <math> \begin{align} R^2 = d = x^2 + y^2 \end{align} </math>
 
::* <math> \tan(\theta) = \frac{y}{x} </math>
 
::* <math> \tan(\theta) = \frac{y}{x} </math>
 
<math>\begin{align} f(d) = Exp(1/2)=\frac{1}{2}e^{-\frac{d}{2}}\ \end{align}</math>  
 
<math>\begin{align} f(d) = Exp(1/2)=\frac{1}{2}e^{-\frac{d}{2}}\ \end{align}</math>  
Line 1,827: Line 2,013:
 
<math>\begin{align} f(\theta) =\frac{1}{2\pi}\ \end{align}</math>
 
<math>\begin{align} f(\theta) =\frac{1}{2\pi}\ \end{align}</math>
 
<br>
 
<br>
 +
 
To sample from the normal distribution, we can generate a pair of independent standard normal X and Y by:<br />
 
To sample from the normal distribution, we can generate a pair of independent standard normal X and Y by:<br />
 +
 
1) Generating their polar coordinates<br />
 
1) Generating their polar coordinates<br />
 
2) Transforming back to rectangular (Cartesian) coordinates.<br />
 
2) Transforming back to rectangular (Cartesian) coordinates.<br />
  
Alternative Method of Generating Standard Normal Random Variables 
 
  
Step 1: Generate <math>u1~Unif(0,1)</math>
+
'''Alternative Method of Generating Standard Normal Random Variables'''<br />
Step 2: Generate <math>Y1~Exp(1),Y2~Exp(2)</math>
 
Step 3: If <math>Y2 \geq(Y1-1)^2/2</math>,set <math>V=Y1</math>,otherwise,go to step 1
 
Step 4: If <math>u1 \leq 1/2</math>,then <math>X=-V</math>
 
  
==== Expectation of a Standard Normal distribution ====
+
Step 1: Generate <math>u_{1}</math> ~<math>Unif(0,1)</math><br />
The expectation of a standard normal distribution is 0
+
Step 2: Generate <math>Y_{1}</math> ~<math>Exp(1)</math>,<math>Y_{2}</math>~<math>Exp(2)</math><br />
:Below is the proof:
+
Step 3: If <math>Y_{2} \geq(Y_{1}-1)^2/2</math>,set <math>V=Y1</math>,otherwise,go to step 1<br />
 +
Step 4: If <math>u_{1} \leq 1/2</math>,then <math>X=-V</math><br />
 +
 
 +
===Expectation of a Standard Normal distribution===<br />
 +
 
 +
The expectation of a standard normal distribution is 0<br />
 +
 
 +
'''Proof:''' <br />
  
 
:<math>\operatorname{E}[X]= \;\int_{-\infty}^{\infty} x \frac{1}{\sqrt{2\pi}}  e^{-x^2/2} \, dx.</math>
 
:<math>\operatorname{E}[X]= \;\int_{-\infty}^{\infty} x \frac{1}{\sqrt{2\pi}}  e^{-x^2/2} \, dx.</math>
Line 1,849: Line 2,040:
 
:<math>= - \left[\phi(x)\right]_{-\infty}^{\infty}</math>
 
:<math>= - \left[\phi(x)\right]_{-\infty}^{\infty}</math>
 
:<math>= 0</math><br />
 
:<math>= 0</math><br />
More intuitively, because x is an odd function (f(x)+f(-x)=0). Taking integral of x will give <math>x^2/2 </math> which is an even function (f(x)=f(-x)). If support is from negative infinity to infinity, then the integral will return 0.<br />
 
  
* '''Procedure (Box-Muller Transformation Method):''' <br />
+
'''Note,''' more intuitively, because x is an odd function (f(x)+f(-x)=0). Taking integral of x will give <math>x^2/2 </math> which is an even function (f(x)=f(-x)). This is in relation to the symmetrical properties of the standard normal distribution. If support is from negative infinity to infinity, then the integral will return 0.<br />
 +
 
 +
 
 +
'''Procedure (Box-Muller Transformation Method):''' <br />
 +
 
 
Pseudorandom approaches to generating normal random variables used to be limited. Inefficient methods such as inverse Gaussian function, sum of uniform random variables, and acceptance-rejection were used. In 1958, a new method was proposed by George Box and Mervin Muller of Princeton University. This new technique was easy to use and also had the accuracy to the inverse transform sampling method that it grew more valuable as computers became more computationally astute. <br>
 
Pseudorandom approaches to generating normal random variables used to be limited. Inefficient methods such as inverse Gaussian function, sum of uniform random variables, and acceptance-rejection were used. In 1958, a new method was proposed by George Box and Mervin Muller of Princeton University. This new technique was easy to use and also had the accuracy to the inverse transform sampling method that it grew more valuable as computers became more computationally astute. <br>
 
The Box-Muller method takes a sample from a bivariate independent standard normal distribution, each component of which is thus a univariate standard normal. The algorithm is based on the following two properties of the bivariate independent standard normal distribution: <br>
 
The Box-Muller method takes a sample from a bivariate independent standard normal distribution, each component of which is thus a univariate standard normal. The algorithm is based on the following two properties of the bivariate independent standard normal distribution: <br>
 
if <math>Z = (Z_{1}, Z_{2}</math>) has this distribution, then <br>
 
if <math>Z = (Z_{1}, Z_{2}</math>) has this distribution, then <br>
 +
 
1.<math>R^2=Z_{1}^2+Z_{2}^2</math> is exponentially distributed with mean 2, i.e. <br>
 
1.<math>R^2=Z_{1}^2+Z_{2}^2</math> is exponentially distributed with mean 2, i.e. <br>
 
<math>P(R^2 \leq x) = 1-e^{-x/2}</math>. <br>
 
<math>P(R^2 \leq x) = 1-e^{-x/2}</math>. <br>
 
2.Given <math>R^2</math>, the point <math>(Z_{1},Z_{2}</math>) is uniformly distributed on the circle of radius R centered at the origin. <br>
 
2.Given <math>R^2</math>, the point <math>(Z_{1},Z_{2}</math>) is uniformly distributed on the circle of radius R centered at the origin. <br>
 
We can use these properties to build the algorithm: <br>
 
We can use these properties to build the algorithm: <br>
 +
  
 
1) Generate random number <math> \begin{align} U_1,U_2 \sim~ \mathrm{Unif}(0, 1) \end{align} </math> <br />
 
1) Generate random number <math> \begin{align} U_1,U_2 \sim~ \mathrm{Unif}(0, 1) \end{align} </math> <br />
Line 1,877: Line 2,073:
  
  
Note: In steps 2 and 3, we are using a similar technique as that used in the inverse transform method. <br />
+
'''Note:''' In steps 2 and 3, we are using a similar technique as that used in the inverse transform method. <br />
 
The Box-Muller Transformation Method generates a pair of independent Standard Normal distributions, X and Y (Using the transformation of polar coordinates). <br />
 
The Box-Muller Transformation Method generates a pair of independent Standard Normal distributions, X and Y (Using the transformation of polar coordinates). <br />
 +
 
If you want to generate a number of independent standard normal distributed numbers (more than two), you can run the Box-Muller method several times.<br/>
 
If you want to generate a number of independent standard normal distributed numbers (more than two), you can run the Box-Muller method several times.<br/>
 
For example: <br />
 
For example: <br />
Line 1,885: Line 2,082:
  
  
* '''Code'''<br />
+
'''Matlab Code'''<br />
 +
 
 
<pre style="font-size:16px">
 
<pre style="font-size:16px">
 
>>close all
 
>>close all
Line 1,900: Line 2,098:
 
>>hist(y)
 
>>hist(y)
 
</pre>
 
</pre>
 +
<br>
 +
'''Remember''': For the above code to work the "." needs to be after the d to ensure that each element of d is raised to the power of 0.5.<br /> Otherwise matlab will raise the entire matrix to the power of 0.5."<br>
  
"''Remember'': For the above code to work the "." needs to be after the d to ensure that each element of d is raised to the power of 0.5.<br /> Otherwise matlab will raise the entire matrix to the power of 0.5."
+
'''Note:'''<br>the first graph is hist(tet) and it is a uniform distribution.<br>The second one is hist(d) and it is a exponential distribution.<br>The third one is hist(x) and it is a normal distribution.<br>The last one is hist(y) and it is also a normal distribution.
 
 
Note:<br>the first graph is hist(tet) and it is a uniform distribution.<br>The second one is hist(d) and it is a exponential distribution.<br>The third one is hist(x) and it is a normal distribution.<br>The last one is hist(y) and it is also a normal distribution.
 
  
 
Attention:There is a "dot" between sqrt(d) and "*". It is because d and tet are vectors. <br>
 
Attention:There is a "dot" between sqrt(d) and "*". It is because d and tet are vectors. <br>
Line 1,920: Line 2,118:
 
>>hist(x)
 
>>hist(x)
 
>>hist(x+2)
 
>>hist(x+2)
>>hist(x*2+2)
+
>>hist(x*2+2)<br>
 
</pre>
 
</pre>
 
+
<br>
Note: randn is random sample from a standard normal distribution.<br />
+
'''Note:'''<br>
Note: hist(x+2) will be centered at 2 instead of at 0. <br />
+
1. randn is random sample from a standard normal distribution.<br />
      hist(x*3+2) is also centered at 2. The mean doesn't change, but the variance of x*3+2 becomes nine times (3^2) the variance of x.<br />
+
2. hist(x+2) will be centered at 2 instead of at 0. <br />
 +
3. hist(x*3+2) is also centered at 2. The mean doesn't change, but the variance of x*3+2 becomes nine times (3^2) the variance of x.<br />
 
[[File:Normal_x.jpg|300x300px]][[File:Normal_x+2.jpg|300x300px]][[File:Normal(2x+2).jpg|300px]]
 
[[File:Normal_x.jpg|300x300px]][[File:Normal_x+2.jpg|300x300px]][[File:Normal(2x+2).jpg|300px]]
 
<br />
 
<br />
  
<b>Comment</b>: Box-Muller transformations are not computationally efficient. The reason for this is the need to compute sine and cosine functions. A way to get around this time-consuming difficulty is by an indirect computation of the sine and cosine of  a random angle (as opposed to a direct computation which generates  U  and then computes the sine and cosine of 2πU. <br />
+
<b>Comment</b>:<br />
 +
Box-Muller transformations are not computationally efficient. The reason for this is the need to compute sine and cosine functions. A way to get around this time-consuming difficulty is by an indirect computation of the sine and cosine of  a random angle (as opposed to a direct computation which generates  U  and then computes the sine and cosine of 2πU. <br />
 +
 
 +
 
  
 
'''Alternative Methods of generating normal distribution'''<br />
 
'''Alternative Methods of generating normal distribution'''<br />
 +
 
1. Even though we cannot use inverse transform method, we can approximate this inverse using different functions.One method would be '''rational approximation'''.<br />
 
1. Even though we cannot use inverse transform method, we can approximate this inverse using different functions.One method would be '''rational approximation'''.<br />
 
2.'''Central limit theorem''' : If we sum 12 independent U(0,1) distribution and subtract 6 (which is E(ui)*12)we will approximately get a standard normal distribution.<br />
 
2.'''Central limit theorem''' : If we sum 12 independent U(0,1) distribution and subtract 6 (which is E(ui)*12)we will approximately get a standard normal distribution.<br />
Line 1,945: Line 2,148:
 
=== Proof of Box Muller Transformation ===
 
=== Proof of Box Muller Transformation ===
  
Definition:
+
'''Definition:'''<br />
 
A transformation which transforms from a '''two-dimensional continuous uniform''' distribution to a '''two-dimensional bivariate normal''' distribution (or complex normal distribution).
 
A transformation which transforms from a '''two-dimensional continuous uniform''' distribution to a '''two-dimensional bivariate normal''' distribution (or complex normal distribution).
  
 
Let U<sub>1</sub> and U<sub>2</sub> be independent uniform (0,1) random variables. Then
 
Let U<sub>1</sub> and U<sub>2</sub> be independent uniform (0,1) random variables. Then
<math>X_{1} = (-2lnU)^0.5_{1}*cos(2\pi U_{2})</math>
+
<math>X_{1} = ((-2lnU_{1})^.5)*cos(2\pi U_{2})</math>
  
<math>X_{2} = (-2lnU)^0.5_{1}*sin(2\pi U_{2})</math>
+
<math>X_{2} = (-2lnU_{1})^0.5*sin(2\pi U_{2})</math>
 
are '''independent''' N(0,1) random variables.
 
are '''independent''' N(0,1) random variables.
  
Line 1,965: Line 2,168:
 
       u<sub>2</sub> = g<sub>2</sub> ^-1(x1,x2)
 
       u<sub>2</sub> = g<sub>2</sub> ^-1(x1,x2)
  
Inverting the above transformations, we have
+
Inverting the above transformation, we have
 
     u1 = exp^{-(x<sub>1</sub> ^2+ x<sub>2</sub> ^2)/2}
 
     u1 = exp^{-(x<sub>1</sub> ^2+ x<sub>2</sub> ^2)/2}
 
     u2 = (1/2pi)*tan^-1 (x<sub>2</sub>/x<sub>1</sub>)
 
     u2 = (1/2pi)*tan^-1 (x<sub>2</sub>/x<sub>1</sub>)
Line 1,972: Line 2,175:
 
   f(x1,x2) = {exp^(-(x1^2+x2^2)/2)}/2pi
 
   f(x1,x2) = {exp^(-(x1^2+x2^2)/2)}/2pi
 
which factors into two standard normal pdfs.
 
which factors into two standard normal pdfs.
 +
 +
 +
(The quote is from http://mathworld.wolfram.com/Box-MullerTransformation.html)
 +
(The proof is from http://www.math.nyu.edu/faculty/goodman/teaching/MonteCarlo2005/notes/GaussianSampling.pdf)
  
 
=== General Normal distributions ===
 
=== General Normal distributions ===
Line 1,993: Line 2,200:
 
where <math> \mu </math> is the mean or expectation of the distribution and <math> \sigma </math> is standard deviation <br />
 
where <math> \mu </math> is the mean or expectation of the distribution and <math> \sigma </math> is standard deviation <br />
  
The special case of the normal distribution is standard normal distribution, which the variance is 1 and the mean is zero. If X is a general normal deviate, then Z = (X − μ)/σ will have a standard normal distribution.
+
The probability density must be scaled by 1/sigma so that the integral is still 1.(Acknowledge: https://en.wikipedia.org/wiki/Normal_distribution)
 +
The special case of the normal distribution is standard normal distribution, which the variance is 1 and the mean is zero. If X is a general normal deviate, then <math> Z=\dfrac{X - (\mu)}{\sigma} </math> will have a standard normal distribution.
  
 
If Z ~ N(0,1), and we want <math>X </math>~<math> N(\mu, \sigma^2)</math>, then <math>X = \mu + \sigma * Z</math> Since <math>E(x) = \mu +\sigma*0 = \mu </math> and <math>Var(x) = 0 +\sigma^2*1</math>
 
If Z ~ N(0,1), and we want <math>X </math>~<math> N(\mu, \sigma^2)</math>, then <math>X = \mu + \sigma * Z</math> Since <math>E(x) = \mu +\sigma*0 = \mu </math> and <math>Var(x) = 0 +\sigma^2*1</math>
Line 2,126: Line 2,334:
 
   
 
   
 
The Bernoulli distribution is a special case of binomial distribution, where the variate x only has two outcomes; so that the Bernoulli also can use the probability density function of the binomial distribution with the variate x taking values 0 and 1.
 
The Bernoulli distribution is a special case of binomial distribution, where the variate x only has two outcomes; so that the Bernoulli also can use the probability density function of the binomial distribution with the variate x taking values 0 and 1.
 +
 +
The most famous example for the Bernoulli Distribution would be the "Flip Coin" question, which has only two possible outcomes(Success or Failure) with the same probabilities of 0.5
  
 
Let x1,x2 denote the lifetime of 2 independent particles, x1~exp(<math>\lambda</math>), x2~exp(<math>\lambda</math>)
 
Let x1,x2 denote the lifetime of 2 independent particles, x1~exp(<math>\lambda</math>), x2~exp(<math>\lambda</math>)
Line 2,183: Line 2,393:
 
for k=1:5000
 
for k=1:5000
 
     i = 1;
 
     i = 1;
     while (i <= n)
+
     for i=1:n
 
         u=rand();
 
         u=rand();
 
         if (u <= p)
 
         if (u <= p)
Line 2,190: Line 2,400:
 
             y(i) = 0;
 
             y(i) = 0;
 
         end
 
         end
        i = i + 1;
 
 
     end
 
     end
  
Line 2,202: Line 2,411:
  
  
</pre>
+
 
 
</div>
 
</div>
 
Note: We can also regard the Bernoulli Distribution as either a conditional distribution or <math>f(x)= p^{x}(1-p)^{(1-x)}</math>, x=0,1.
 
Note: We can also regard the Bernoulli Distribution as either a conditional distribution or <math>f(x)= p^{x}(1-p)^{(1-x)}</math>, x=0,1.
Line 2,223: Line 2,432:
 
Procedure:
 
Procedure:
  
1.Generate U~Unif [0, 1)<br>
+
1) Generate U~Unif (0, 1)<br>
2.set <math>x=F^{-1}(u)</math><br>
+
2) Set <math>x=F^{-1}(u)</math><br>
3.X~f(x)<br>
+
3) X~f(x)<br>
  
 
'''Remark'''<br>
 
'''Remark'''<br>
1. The preceding can be written algorithmically as
+
1) The preceding can be written algorithmically for discrete random variables as <br>
Generate a random number U
+
Generate a random number U ~ U(0,1] <br>
If U<<sub>p0</sub> set X=<sub>x0</sub> and stop
+
If U < p<sub>0</sub> set X = x<sub>0</sub> and stop <br>
If U<<sub>p0</sub>+<sub>p1</sub> set X=x1 and stop
+
If U < p<sub>0</sub> + p<sub>1</sub> set X = x<sub>1</sub> and stop <br>
...
+
... <br>
2. If the <sub>xi</sub>, i>=0, are ordered so that <sub>x0</sub><<sub>x1</sub><<sub>x2</sub><... and if we let F denote the distribution function of X, then <math>F(<sub>xk</sub>=<sub>/sum/pi</sub>)</math> and so X will equal <sub>xj</sub> if F(<sub>x(j-1)</sub>)<=U<F(<sub>xj</sub>)
+
2) If the x<sub>i</sub>, i>=0, are ordered so that x<sub>0</sub> < x<sub>1</sub> < x<sub>2</sub> <... and if we let F denote the distribution function of X, then X will equal x<sub>j</sub> if F(x<sub>j-1</sub>) <= U < F(x<sub>j</sub>)
  
 
'''Example 1'''<br>
 
'''Example 1'''<br>
Line 2,246: Line 2,455:
 
'''Solution:'''<br>
 
'''Solution:'''<br>
  
x~exp(<math>\lambda</math>)<br>
+
x<sub>1</sub>~exp(<math>\lambda_1</math>)<br>
 
+
x<sub>2</sub>~exp(<math>\lambda_2</math>)<br>
<math>f_{x}(x)=\lambda e^{-\lambda x},x\geq0 </math> <br>
+
<math>f_{x(x)}=\lambda e^{-\lambda x},x\geq0 </math> <br>
 +
<math>F_X(x)=1-e^{-\lambda x}, x\geq 0</math><br>
  
 
<math>1-F_Y(y) = P(Y>y)</math> = P(min(X<sub>1</sub>,X<sub>2</sub>) > y) = <math>\, P((X_1)>y) P((X_2)>y) = e^{\, -(\lambda_1 + \lambda_2) y}</math><br>
 
<math>1-F_Y(y) = P(Y>y)</math> = P(min(X<sub>1</sub>,X<sub>2</sub>) > y) = <math>\, P((X_1)>y) P((X_2)>y) = e^{\, -(\lambda_1 + \lambda_2) y}</math><br>
Line 2,259: Line 2,469:
  
 
Step1: Generate U~ U(0, 1)<br>
 
Step1: Generate U~ U(0, 1)<br>
Step2: set <math>y=\, {-\frac {1}{{\lambda_1 +\lambda_2}}} ln(u)</math><br>
+
 
 +
Step2: set <math>y=\, {-\frac {1}{{\lambda_1 +\lambda_2}}} ln(1-u)</math><br>
 +
 
 +
    or set <math>y=\, {-\frac {1} {{\lambda_1 +\lambda_2}}} ln(u)</math><br>
 +
Since it is a uniform distribution, therefore after generate a lot of times 1-u and u are the same.
 +
 
 +
 
 +
* '''Matlab Code'''<br />
 +
<pre style="font-size:16px">
 +
>> lambda1 = 1;
 +
>> lambda2 = 2;
 +
>> u = rand;
 +
>> y = -log(u)/(lambda1 + lambda2)
 +
</pre>
  
 
If we generalize this example from two independent particles to n independent particles we will have:<br>
 
If we generalize this example from two independent particles to n independent particles we will have:<br>
Line 2,295: Line 2,518:
 
'''Solution:'''<br>
 
'''Solution:'''<br>
 
<br>
 
<br>
1. generate u ~ Unif[0, 1)<br>
+
1. Generate <math>U ~\sim~ Unif[0, 1)</math><br>
2. Set x = U<sup>1/n</sup><br>
+
2. Set <math>X = U^{1/n}</math><br>
 
<br>
 
<br>
For example, when n = 20,<br>
+
For example, when <math>n = 20</math>,<br>
u = 0.6 => x = u<sub>1/20</sub> = 0.974<br>
+
<math>U = 0.6</math> => <math>X = U^{1/20} = 0.974</math><br>
u = 0.5 => x = u<sub>1/20</sub> = 0.966<br>
+
<math>U = 0.5 =></math> <math>X = U^{1/20} = 0.966</math><br>
u = 0.2 => x = u<sub>1/20</sub> = 0.923<br>
+
<math>U = 0.2</math> => <math>X = U^{1/20} = 0.923</math><br>
 
<br>
 
<br>
Observe from above that the values of X for n = 20 are close to 1, this is because we can view X<sup>n</sup> as the maximum of n independent random variables X, X~Unif(0,1) and is much likely to be close to 1 as n increases. This is because when n is large the exponent tends towards 0. This observation is the motivation for method 2 below.<br>
+
Observe from above that the values of X for n = 20 are close to 1, this is because we can view <math>X^n</math> as the maximum of n independent random variables <math>X,</math> <math>X~\sim~Unif(0,1)</math> and is much likely to be close to 1 as n increases. This is because when n is large the exponent tends towards 0. This observation is the motivation for method 2 below.<br>
  
 
Recall that
 
Recall that
Line 2,354: Line 2,577:
 
The general algorithm to generate random variables from a composition CDF is:
 
The general algorithm to generate random variables from a composition CDF is:
  
1)  Generate U, V ~ <math>U(0,1)</math>
+
1)  Generate U,V ~ <math> Unif(0,1)</math>
  
2)  If u < p<sub>1</sub>, v=<math>F_{X_{1}}(x)</math><sup>-1</sup>
+
2)  If U < p<sub>1</sub>, V = <math>F_{X_{1}}(x)</math><sup>-1</sup>
  
3)  Else if u < p<sub>1</sub>+p<sub>2</sub>, v=<math>F_{X_{2}}(x)</math><sup>-1</sup>
+
3)  Else if U < p<sub>1</sub> + p<sub>2</sub>, V = <math>F_{X_{2}}(x)</math><sup>-1</sup>
  
4)  ....
+
4)  Repeat from Step 1 (if N randomly generated variables needed, repeat N times)
  
 
<b>Explanation</b><br>
 
<b>Explanation</b><br>
 
Each random variable that is a part of X contributes <math>p_{i} F_{X_{i}}(x)</math> to <math>F_{X}(x)</math> every time.
 
Each random variable that is a part of X contributes <math>p_{i} F_{X_{i}}(x)</math> to <math>F_{X}(x)</math> every time.
 
From a sampling point of view, that is equivalent to contributing <math>F_{X_{i}}(x)</math> <math>p_{i}</math> of the time. The logic of this is similar to that of the Accept-Reject Method, but instead of rejecting a value depending on the value u takes, we instead decide which distribution to sample it from.
 
From a sampling point of view, that is equivalent to contributing <math>F_{X_{i}}(x)</math> <math>p_{i}</math> of the time. The logic of this is similar to that of the Accept-Reject Method, but instead of rejecting a value depending on the value u takes, we instead decide which distribution to sample it from.
 +
 +
 +
<b> Simplified Version </b><br>
 +
1) Generate <math>u \sim Unif(0,1)</math> <br>
 +
2) Set <math> X=0, s=P_0</math><br>
 +
3) While <math> u > s, </math><br>
 +
set <math> X = X+1</math> and <math> s=s+P_x </math> <br>
 +
4) Return <math> X </math>
  
 
=== Examples of Decomposition Method ===
 
=== Examples of Decomposition Method ===
Line 2,407: Line 2,638:
 
=== Example of Decomposition Method ===
 
=== Example of Decomposition Method ===
  
F<sub>x</sub>(x) = 1/3*x+1/3*x<sup>2</sup>+1/3*x<sup>3</sup>, 0<= x<=1
+
<math>F_x(x) = \frac {1}{3} x+\frac {1}{3} x^2+\frac {1}{3} x^3, 0\leq x\leq 1</math>
  
let U =F<sub>x</sub>(x) = 1/3*x+1/3*x<sup>2</sup>+1/3*x<sup>3</sup>, solve for x.
+
Let <math>U =F_x(x) = \frac {1}{3} x+\frac {1}{3} x^2+\frac {1}{3} x^3</math>, solve for x.
  
P<sub>1</sub>=1/3, F<sub>x1</sub>(x)= x, P<sub>2</sub>=1/3,F<sub>x2</sub>(x)= x<sup>2</sup>,  
+
<math>P_1=\frac{1}{3}, F_{x1} (x)= x, P_2=\frac{1}{3},F_{x2} (x)= x^2,  
P<sub>3</sub>=1/3,F<sub>x3</sub>(x)= x<sup>3</sup>
+
P_3=\frac{1}{3},F_{x3} (x)= x^3</math>
  
 
'''Algorithm:'''
 
'''Algorithm:'''
  
Generate U ~ Unif [0,1)
+
Generate <math>\,U \sim Unif [0,1)</math>
  
Generate V~ Unif [0,1)
+
Generate <math>\,V \sim  Unif [0,1)</math>
  
if 0<u<1/3, x = v
+
if <math>0\leq u \leq \frac{1}{3}, x = v</math>
  
else if u<2/3, x = v<sup>1/2</sup>
+
else if <math>u \leq \frac{2}{3}, x = v^{\frac{1}{2}}</math>
  
else x = v<sup>1/3</sup><br>
+
else <math>x=v^{\frac{1}{3}}</math> <br>
  
  
 
'''Matlab Code:'''  
 
'''Matlab Code:'''  
 
<pre style="font-size:16px">
 
<pre style="font-size:16px">
u=rand
+
u=rand # U is
 
v=rand
 
v=rand
 
if u<1/3
 
if u<1/3
Line 2,489: Line 2,720:
  
 
For More Details, please refer to http://www.stanford.edu/class/ee364b/notes/decomposition_notes.pdf
 
For More Details, please refer to http://www.stanford.edu/class/ee364b/notes/decomposition_notes.pdf
 
  
 
===Fundamental Theorem of Simulation===
 
===Fundamental Theorem of Simulation===
Line 2,497: Line 2,727:
 
(Basis of the Accept-Reject algorithm)
 
(Basis of the Accept-Reject algorithm)
  
The advantage of this method is that we can sample a unknown distribution from a easy distribution. The disadvantage of this method is that it may need to reject many points, which is inefficient.
+
The advantage of this method is that we can sample a unknown distribution from a easy distribution. The disadvantage of this method is that it may need to reject many points, which is inefficient.<br />
 +
Inverse each part of partial CDF, the partial CDF is divided by the original CDF, partial range is uniform distribution.<br />
 +
More specific definition of the theorem can be found here.<ref>http://www.bus.emory.edu/breno/teaching/MCMC_GibbsHandouts.pdf</ref>
  
inverse each part of partial CDF, the partial CDF is divided by the original CDF, partial range is uniform distribution.
+
Matlab code:
 +
 
 +
<pre style="font-size:16px">
 +
close all
 +
clear all
 +
ii=1;
 +
while ii<1000
 +
u=rand
 +
y=R*(2*U-1)
 +
if (1-U^2)>=(2*u-1)^2
 +
x(ii)=y;
 +
ii=ii+1
 +
end
 +
</pre>
  
 
===Question 2===
 
===Question 2===
Line 2,518: Line 2,763:
  
 
The beta distribution maximized at 0.5 with value <math>(1/4)^n</math>.
 
The beta distribution maximized at 0.5 with value <math>(1/4)^n</math>.
So, <math>c=b*(1/4)^n</math>
+
So, <math>c=b*(1/4)^n</math><br />
Algorithm:
+
Algorithm: <br />
1.Draw <math>U_1</math> from <math>U(0, 1)</math>.<math> U_2</math> from <math>U(0, 1)<math>
+
1.Draw <math>U_1</math> from <math>U(0, 1)</math>. <math> U_2</math> from <math>U(0, 1)</math> <br />
2.If <math>U_2<=b*(U_1)^n*(1-(U_1))^n/b*(1/4)^n=(4*(U_1)*(1-(U_1)))^n</math>
+
2.If <math>U_2<=b*(U_1)^n*(1-(U_1))^n/b*(1/4)^n=(4*(U_1)*(1-(U_1)))^n</math><br />
 
   then X=U_1
 
   then X=U_1
 
   Else return to step 1.
 
   Else return to step 1.
Line 2,542: Line 2,787:
 
===The Bernoulli distribution===
 
===The Bernoulli distribution===
  
The Bernoulli distribution is a special case of the binomial distribution, where n = 1. X ~ Bin(1, p) has the same meaning as X ~ Ber(p), where p is the probability if the event success, otherwise the probability is 1-p (we usually define a variate q, q= 1-p). The mean of Bernoulli is p, variance is p(1-p). Bin(n, p), is the distribution of the sum of n independent Bernoulli trials, Bernoulli(p), each with the same probability p, where 0<p<1. <br>
+
The Bernoulli distribution is a special case of the binomial distribution, where n = 1. X ~ Bin(1, p) has the same meaning as X ~ Ber(p), where p is the probability of success and 1-p is the probability of failure (we usually define a variate q, q= 1-p). The mean of Bernoulli is p and the variance is p(1-p). Bin(n, p), is the distribution of the sum of n independent Bernoulli trials, Bernoulli(p), each with the same probability p, where 0<p<1. <br>
 
For example, let X be the event that a coin toss results in a "head" with probability ''p'', then ''X~Bernoulli(p)''. <br>
 
For example, let X be the event that a coin toss results in a "head" with probability ''p'', then ''X~Bernoulli(p)''. <br>
P(X=1)=p,P(X=0)=1-p, P(x=0)+P(x=1)=p+q=1
+
P(X=1)= p
 +
P(X=0)= q = 1-p
 +
Therefore, P(X=0) + P(X=1) = p + q = 1
  
 
'''Algorithm: '''
 
'''Algorithm: '''
  
1. Generate u~Unif(0,1) <br>
+
1) Generate <math>u\sim~Unif(0,1)</math> <br>
2. If u p, then x = 1 <br>
+
2) If <math>u \leq p</math>, then <math>x = 1 </math><br>
else x = 0 <br>
+
else <math>x = 0</math> <br>
 
The answer is: <br>
 
The answer is: <br>
when U≤p, x=1 <br>
+
when <math> U \leq p, x=1</math> <br>
when U>p, x=0<br>
+
when <math>U \geq p, x=0</math><br>
3.Repeat as necessary
+
3) Repeat as necessary
 +
 
 +
* '''Matlab Code'''<br />
 +
<pre style="font-size:16px">
 +
>> p = 0.8    % an arbitrary probability for example
 +
>> for i = 1: 100
 +
>>  u = rand;
 +
>>  if u < p
 +
>>      x(ii) = 1;
 +
>>  else
 +
>>      x(ii) = 0;
 +
>>  end
 +
>> end
 +
>> hist(x)
 +
</pre>
  
 
===The Binomial Distribution===
 
===The Binomial Distribution===
  
If X~Bin(n,p), then its pmf is of form:
+
In general, if the random variable X follows the binomial distribution with parameters n and p, we write X ~ Bin(n, p).
 +
(Acknowledge: https://en.wikipedia.org/wiki/Binomial_distribution)
 +
If X ~ B(n, p), then its pmf is of form:
  
 
f(x)=(nCx) p<sup>x</sup>(1-p)<sup>(n-x)</sup>, x=0,1,...n<br />
 
f(x)=(nCx) p<sup>x</sup>(1-p)<sup>(n-x)</sup>, x=0,1,...n<br />
Line 2,604: Line 2,867:
 
For example,<br />
 
For example,<br />
 
If the success event showed at the first time, which x=1, then f(x)=p.<br />
 
If the success event showed at the first time, which x=1, then f(x)=p.<br />
If the success event showed at the second time and failed at the first time, which x=2, then f(x)=p(1-p).<br />
+
If the success event showed at the second time and failed at the first time, which x = 2, then f(x)= p(1-p).<br />
If the success event showed at the third time and failed at the first and second time, which x=3, then f(x)= p(1-p)<sup>(k-1)</sup>. etc.<br />
+
If the success event showed at the third time and failed at the first and second time, which x = 3, then f(x)= p(1-p)<sup>2 </sup>. etc.<br />
If the success event showed at the x time and all failed before time x, which x=x, then f(x)= p(1-p)<sup>(k-1)</sup><br />
+
If the success event showed at the k time and all failed before time k, which implies x = k, then f(k)= p(1-p)<sup>(k-1)</sup><br />
 
which is,<br />
 
which is,<br />
 
x    Pr<br />
 
x    Pr<br />
Line 2,616: Line 2,879:
 
.    .<br />
 
.    .<br />
 
n    P(1-P)<sup>(n-1)</sup><br />
 
n    P(1-P)<sup>(n-1)</sup><br />
 +
Also,  the sequence of the outputs of the probability is a geometric sequence.
 +
 
For example, suppose a die is thrown repeatedly until the first time a "6" appears. This is a question of geometric distribution of the number of times on the set { 1, 2, 3, ... } with p = 1/6.
 
For example, suppose a die is thrown repeatedly until the first time a "6" appears. This is a question of geometric distribution of the number of times on the set { 1, 2, 3, ... } with p = 1/6.
  
Line 2,655: Line 2,920:
  
  
If Y~Exp(<math>\lambda</math>) then X=floor(Y)+1 is geometric.<br />
+
If Y~Exp(<math>\lambda</math>) then <math>X=\left \lfloor Y \right \rfloor+1</math> is geometric.<br />
 
Choose e^(-<math>\lambda</math>)=1-p. Then X ~ geo (p) <br />
 
Choose e^(-<math>\lambda</math>)=1-p. Then X ~ geo (p) <br />
  
 
P (X > x) = (1-p)<sup>x</sup>(because first x trials are not successful) <br/>
 
P (X > x) = (1-p)<sup>x</sup>(because first x trials are not successful) <br/>
 +
 +
NB: An advantage of using this method is that nothing is rejected. We accept all the points, and the method is more efficient. Also, this method is closer to the inverse transform method as nothing is being rejected. <br />
  
 
'''Proof''' <br/>
 
'''Proof''' <br/>
  
P(X>x) = P( floor(Y) + 1 > X) = P(floor (Y) > x- 1) = P(Y>= x) = e<sup>(-<math>\lambda</math> * x)</sup> <br>
+
<math>P(X>x) = P( \left \lfloor Y \right \rfloor + 1 > X) = P(\left \lfloor Y \right \rfloor > x- 1) = P(Y>= x) = e^{-\lambda × x} </math> <br>
  
 
SInce p = 1- e<sup>-<math>\lambda</math></sup> or <math>\lambda</math>= <math>-log(1-p)</math>(compare the pdf of exponential distribution and Geometric distribution,we can look at e<sup>-<math>\lambda</math></sup> the probability of the fail trial), then <br>
 
SInce p = 1- e<sup>-<math>\lambda</math></sup> or <math>\lambda</math>= <math>-log(1-p)</math>(compare the pdf of exponential distribution and Geometric distribution,we can look at e<sup>-<math>\lambda</math></sup> the probability of the fail trial), then <br>
Line 2,705: Line 2,972:
 
<math>P(Y>=X)</math><br />
 
<math>P(Y>=X)</math><br />
 
Y ~ Exp (<math>\lambda</math>)<br />
 
Y ~ Exp (<math>\lambda</math>)<br />
pdf of Y :  <math>-\lambda e^{-\lambda}</math><br />
+
pdf of Y :  <math>\lambda e^{-\lambda}</math><br />
cdf of Y :  <math>1-\lambda e^{-\lambda}</math><br />
+
cdf of Y :  <math>1- e^{-\lambda}</math><br />
cdf <math>P(Y<x)=1-\lambda e^{-\lambda}</math><br />
+
cdf <math>P(Y<x)=1-e^{-\lambda x}</math><br />
<math>P(Y>=x)=1-(1-\lambda e^{-\lambda})=e^{-\lambda x}</math><br />
+
<math>P(Y>=x)=1-(1- e^{-\lambda x})=e^{-\lambda x}</math><br />
 
<math> e^{-\lambda}=1-p      ->    -log(1-p)=\lambda</math><br />
 
<math> e^{-\lambda}=1-p      ->    -log(1-p)=\lambda</math><br />
 
<math>P(Y>=x)=e^{-\lambda x}=e^{log(1-p)x}=(1-p)^x</math><br />
 
<math>P(Y>=x)=e^{-\lambda x}=e^{log(1-p)x}=(1-p)^x</math><br />
Line 2,754: Line 3,021:
 
We have X ~Geo(1/6), <math>f(x)=(1/6)*(5/6)^{x-1}</math>, x=1,2,3....  
 
We have X ~Geo(1/6), <math>f(x)=(1/6)*(5/6)^{x-1}</math>, x=1,2,3....  
  
Now, let <math>Y=e^{\lambda}</math> => x=floor(Y) +1  
+
Now, let <math>\left \lfloor Y \right \rfloor=e^{\lambda}</math> => x=floor(Y) +1  
  
 
Let <math>e^{-\lambda}=5/6</math>  
 
Let <math>e^{-\lambda}=5/6</math>  
Line 2,766: Line 3,033:
 
1) Let Y be <math>e^{\lambda}</math>, exponentially distributed  
 
1) Let Y be <math>e^{\lambda}</math>, exponentially distributed  
  
2) Set X= floor(Y)+1, to generate X  
+
2) Set <math>X= \left \lfloor Y \right \rfloor +1 </math>, to generate X  
  
 
<math> E[x]=6, Var[X]=5/6 /(1/6^2) = 30 </math>
 
<math> E[x]=6, Var[X]=5/6 /(1/6^2) = 30 </math>
Line 2,872: Line 3,139:
  
 
=== Beta Distribution ===
 
=== Beta Distribution ===
The beta distribution is a continuous probability distribution. There are two positive shape parameters in this distribution defined as alpha and beta, both parameters greater than 0, and X within the interval [0,1]. The parameter alpha is used as exponents of the random variable. The parameter beta is used to control the shape of the this distribution. We use the beta distribution to build the model of the behavior of random variables, which are limited to intervals of finite length. For example, we can use the beta distribution to analyze the time allocation of sunshine data and variability of soil properties.
+
The beta distribution is a continuous probability distribution. <br>
 +
PDF:<math>\displaystyle \text{ } f(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1} </math><br>  where <math>0 \leq x \leq 1</math> and <math>\alpha</math>>0, <math>\beta</math>>0<br/>
 +
<div style = "align:left; background:#F5F5DC; font-size: 120%">
 +
Definition:
 +
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parametrized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution.<br/.>
 +
More can be find in the link: <ref>http://en.wikipedia.org/wiki/Beta_distribution</ref>
 +
</div>
 +
 
 +
There are two positive shape parameters in this distribution defined as alpha and beta: <br>
 +
-Both parameters are greater than 0, and X is within the interval [0,1]. <br>
 +
-Alpha is used as exponents of the random variable. <br>
 +
-Beta is used to control the shape of the this distribution. We use the beta distribution to build the model of the behavior of random variables, which are limited to intervals of finite length. <br>
 +
-For example, we can use the beta distribution to analyze the time allocation of sunshine data and variability of soil properties. <br>
  
 
If X~Beta(<math>\alpha, \beta</math>) then its p.d.f. is of the form
 
If X~Beta(<math>\alpha, \beta</math>) then its p.d.f. is of the form
Line 2,881: Line 3,160:
 
Note: <math>\Gamma(\alpha)=(\alpha-1)! </math> if <math>\alpha</math> is a positive integer.
 
Note: <math>\Gamma(\alpha)=(\alpha-1)! </math> if <math>\alpha</math> is a positive integer.
  
 +
Note: Gamma Function Properties
 +
 +
If <math>\alpha=\frac{1}{2} ,
 +
 +
\Gamma(\frac {1}{2})=\sqrt\pi </math>
  
 
The mean of the beta distribution is <math>\frac{\alpha}{\alpha + \beta}</math>. The variance is <math>\frac{\alpha\beta}{(\alpha+\beta)^2 (\alpha + \beta + 1)}</math>
 
The mean of the beta distribution is <math>\frac{\alpha}{\alpha + \beta}</math>. The variance is <math>\frac{\alpha\beta}{(\alpha+\beta)^2 (\alpha + \beta + 1)}</math>
Line 2,914: Line 3,198:
 
:<math>\displaystyle \text{f}(x) = \frac{\Gamma(\alpha+1)}{\Gamma(\alpha)\Gamma(1)}x^{\alpha-1}(1-x)^{1-1}=\alpha x^{\alpha-1}</math><br>
 
:<math>\displaystyle \text{f}(x) = \frac{\Gamma(\alpha+1)}{\Gamma(\alpha)\Gamma(1)}x^{\alpha-1}(1-x)^{1-1}=\alpha x^{\alpha-1}</math><br>
  
The CDF is <math>F(x) = x^{\alpha}</math> (using integration of <math>f(x)</math>)
+
By integrating <math>f(x)</math>, we find the CDF of X is <math>F(x) = x^{\alpha}</math>.
WIth CDF F(x) = x^α, if U have CDF, it is very easy to sample:
+
As <math>F(x)^{-1} = x^\frac {1}{\alpha}</math>, using the inverse transform method, <math> X = U^\frac {1}{\alpha} </math> with U ~ U[0,1].
y=x^α --> x=y^α --> inverseF(x)= x^(1/α)
 
U~U(0,1) --> x=u^(1/α)
 
Applying the inverse transform method with <math>y = x^\alpha \Rightarrow x = y^\frac {1}{\alpha}</math>
 
 
 
<math>F(x)^{-1} = y^\frac {1}{\alpha}</math>
 
 
 
between case 1 and case 2, when alpha and beta be different value, the beta distribution can simplify to other distribution.
 
  
 
'''Algorithm'''
 
'''Algorithm'''
Line 2,937: Line 3,214:
 
</pre>
 
</pre>
  
'''Case 3:'''<br\> To sample from beta in general. we use the property that <br\>
+
'''Case 3:'''<br\> To sample from beta in general, we use the property that <br\>
  
 
:if <math>Y_1</math> follows gamma <math>(\alpha,1)</math><br\>
 
:if <math>Y_1</math> follows gamma <math>(\alpha,1)</math><br\>
Line 2,949: Line 3,226:
  
 
'''Algorithm'''<br\>
 
'''Algorithm'''<br\>
*1. Sample from Y1 ~ Gamma (<math>\alpha</math>,1) #<math>\alpha</math> is the shape, and 1 is the scale. <br\>
+
*1. Sample from Y1 ~ Gamma (<math>\alpha</math>,1) <math>\alpha</math> is the shape, and 1 is the scale. <br\>
 
*2. Sample from Y2 ~ Gamma (<math>\beta</math>,1)  <br\>
 
*2. Sample from Y2 ~ Gamma (<math>\beta</math>,1)  <br\>
 
*3. Set  
 
*3. Set  
Line 3,030: Line 3,307:
 
=== Random Vector Generation ===
 
=== Random Vector Generation ===
 
We want to sample from <math>X = (X_1, X_2, </math>…,<math> X_d)</math>, a d-dimensional vector from a known pdf <math>f(x)</math> and cdf <math>F(x)</math>.
 
We want to sample from <math>X = (X_1, X_2, </math>…,<math> X_d)</math>, a d-dimensional vector from a known pdf <math>f(x)</math> and cdf <math>F(x)</math>.
We need to take into account the following two cases:
+
We need to take into account the following two cases:  
  
 
====Case 1====
 
====Case 1====
Line 3,040: Line 3,317:
  
 
====Case 2====
 
====Case 2====
If <math>X_1, X_2,</math>, ..., <math>X_d</math> are not independent<br/>
+
If <math>X_1, X_2, \cdots , X_d</math> are not independent<br/>
<math>f(x) = f(x_1</math>, ..., <math>x_d) = f(x_1) f(x_2|x_1)</math>...<math>f(x_d|x_(d-1)</math>,,<math>x_1)</math><br/>
+
<math>f(x) = f(x_1, \cdots , x_d) = f(x_1) f(x_2|x_1) \cdots f(x_d|x_{d-1},\cdots ,x_1)</math><br/>
we need to know the conditional distributions of <math>f(x_2|x_1), f(x_3|x_2, x_1),</math> …, <math>f(x_d|x_(d-1)</math>,,<math>x_1)</math><br/>
+
we need to know the conditional distributions of <math>f(x_2|x_1), f(x_3|x_2, x_1),\cdots, f(x_d|x_{d-1}, \cdots, x_1)</math><br/>
 
This is generally a hard problem. Conditional probabilities are not easy to compute, then sampling from these would be based on your statistics knowledge.
 
This is generally a hard problem. Conditional probabilities are not easy to compute, then sampling from these would be based on your statistics knowledge.
 
In each case, we have to consider the previous cases.
 
In each case, we have to consider the previous cases.
Line 3,066: Line 3,343:
 
 
 
Algorithm: <br/>
 
Algorithm: <br/>
1)  for i = 1 to d <br/>
+
1)  For i = 1 to d <br/>
 
2)    U<sub>i</sub> ~ U(0,1) <br/>
 
2)    U<sub>i</sub> ~ U(0,1) <br/>
 
3)    x<sub>i</sub> = a<sub>i</sub> + U(b<sub>i</sub>-a<sub>i</sub>) <br/>
 
3)    x<sub>i</sub> = a<sub>i</sub> + U(b<sub>i</sub>-a<sub>i</sub>) <br/>
4)  end <br/>
+
4)  End <br/>
  
 
*Note: x<sub>i</sub> = a<sub>i</sub> + U(b<sub>i</sub>-a<sub>i</sub>) denotes X<sub>i</sub> ~U(a<sub>i</sub>,b<sub>i</sub>) <br/>
 
*Note: x<sub>i</sub> = a<sub>i</sub> + U(b<sub>i</sub>-a<sub>i</sub>) denotes X<sub>i</sub> ~U(a<sub>i</sub>,b<sub>i</sub>) <br/>
Line 3,075: Line 3,352:
 
An example of the 2-D case is given below:
 
An example of the 2-D case is given below:
  
<pre style='font-size:16px'>
+
<pre style='font-size:14px'>
 
 
 
>>a=[1 2];  
 
>>a=[1 2];  
 
>>b=[4 6];  
 
>>b=[4 6];  
Line 3,092: Line 3,368:
 
[[File:2d_ex.jpg|300px]]
 
[[File:2d_ex.jpg|300px]]
  
==== Code: ====
+
==== Matlab Code: ====
  
<pre style='font-size:16px'>
+
<pre style='font-size:14px'>
 
function x = urectangle (d,n,a,b)
 
function x = urectangle (d,n,a,b)
 
for ii = 1:d;
 
for ii = 1:d;
Line 3,101: Line 3,377:
 
     %keyboard                      #makes the function stop at this step so you can evaluate the variables
 
     %keyboard                      #makes the function stop at this step so you can evaluate the variables
 
end
 
end
 
  
 
>>x=urectangle(2, 100, 2, 5);
 
>>x=urectangle(2, 100, 2, 5);
Line 3,141: Line 3,416:
  
  
This is the picture of the example  
+
The following is a picture relating to the example
 +
 
 
[[File:Untitled.jpg]]
 
[[File:Untitled.jpg]]
  
matlab code:
+
Matlab code:
<pre>
+
<pre style='font-size:16px'>
 
u = rand(d,n);
 
u = rand(d,n);
 
z = 1- 2 *u;
 
z = 1- 2 *u;
Line 3,169: Line 3,445:
 
==Class 10 - Thursday June 6th 2013 ==  
 
==Class 10 - Thursday June 6th 2013 ==  
 
MATLAB code for using Acceptance/Rejection Method to sample from a d-dimensional unit ball.
 
MATLAB code for using Acceptance/Rejection Method to sample from a d-dimensional unit ball.
 +
G: d-dimensional unit ball G
 +
W: d-dimensional Hypercube
  
1. U<sub>1</sub>~UNIF(0, 1)
+
<pre style='font-size:16px'>
  U<sub>2</sub>~UNIF(0, 1)
+
1)  U1~UNIF(0,1)
  U<sub>d</sub>~UNIF(0, 1)
+
    U2~UNIF(0,1)
 
+
    ...
 +
    Ud~UNIF(0,1)
 +
2)  X1 = 1-2U1
 +
    X2 = 1-2U2
 +
    ...
 +
    Xd = 1-2Ud
 +
    R = sum(Xi^2)
 +
3)  If R<=1
 +
    X = (X1,X2,...,Xd),
 +
    else go to step 1
 +
</pre>
  
 
==== Code: ====
 
==== Code: ====
Line 3,210: Line 3,498:
 
z(:,ii) means all the numbers in the ii column starting from 1st column until the nth
 
z(:,ii) means all the numbers in the ii column starting from 1st column until the nth
 
column, which is the last one.
 
column, which is the last one.
 +
 +
higher dimension, less efficient and we need more data points
  
 
Save it with the name of the pattern.
 
Save it with the name of the pattern.
Line 3,278: Line 3,568:
  
 
</pre>
 
</pre>
3d unitaball
+
3d unit ball
 +
 
 
[[File:3-dimensional unitball.jpg|400px]]
 
[[File:3-dimensional unitball.jpg|400px]]
  
Note that the c increases exponentially as d increases, this cause more points being rejected.
+
Note that c increases exponentially as d increases, which will result in a lower acceptance rate and more points being rejected. So this method is not efficient for large values of d.
This method is not efficient for large values of d.  
 
  
In practice, when we need to vectorlise high quality image or genes then d would be very large.  So AR method is not an efficient way to solve the problem.
+
In practice, when we need to vectorlize a high quality image or genes then d would have to be very large.  So AR method is not an efficient way to solve the problem.
  
 
=== Efficiency ===
 
=== Efficiency ===
Line 3,290: Line 3,580:
 
In the above example, the efficiency of the vector A/R is equal to the ratio
 
In the above example, the efficiency of the vector A/R is equal to the ratio
  
<math>\frac{1}{C}=\frac{\text{volume of hyperball}}{\text{volume of hybercube}}= max {g(x)/f(x)} </math>
+
<math>\frac{1}{C}=\frac{\text{volume of hyperball}}{\text{volume of hybercube}}= \max \frac{g(x)}{f(x)} </math>
  
 
In general, the efficiency can be thought of as the total number of points accepted divided by the total number of points generated.
 
In general, the efficiency can be thought of as the total number of points accepted divided by the total number of points generated.
Line 3,298: Line 3,588:
 
For example, for approximating value of <math>\pi</math>, when <math>d \text{(dimension)} =2</math>, the efficiency is around 0.7869; when <math>d=3</math>, the efficiency is around 0.5244; when <math>d=10</math>, the efficiency is around 0.0026: it is getting close to 0.
 
For example, for approximating value of <math>\pi</math>, when <math>d \text{(dimension)} =2</math>, the efficiency is around 0.7869; when <math>d=3</math>, the efficiency is around 0.5244; when <math>d=10</math>, the efficiency is around 0.0026: it is getting close to 0.
  
Thus, when we want to generate high dimension vectors, Acceptance-Rejection Method is not efficient to be used.
+
A 'C' value of 1 implies an acceptance rate of 100% (most efficient scenario) but as we sample from higher dimensions, 'C' usually gets larger. Thus, when we want to generate high dimension vectors, Acceptance-Rejection Method is not efficient to be used.
  
 
<span style="color:red;padding:0 auto;"><br>The end of midterm coverage</span>
 
<span style="color:red;padding:0 auto;"><br>The end of midterm coverage</span>
 +
<div style="border:1px solid #cccccc;border-radius:10px;box-shadow: 0 5px 15px 1px rgba(0, 0, 0, 0.6), 0 0 200px 1px rgba(255, 255, 255, 0.5);padding:20px;margin:20px;background:#FFFFAD;">
 +
<h2 style="text-align:center;">Summary of vector acceptance-rejection sampling</h2>
 +
<p><b>Problem:</b> <math> f(x_1, x_2, ...x_n)</math> is difficult to sample from</p>
 +
<p><b>Plan:</b></p>
 +
Let W represent the sample space covered by <math> f(x_1, x_2, ...x_n)</math>
 +
<ol>
 +
<li>1.Draw <math>\vec{y}=y_1,y_2...y_n\sim~g()</math> where g has sample space G which is greater than W. g is a distribution that is easy to sample from (i.e. uniform)</li>
 +
<li>2.if <math>\vec{y} \subseteq W </math> then <math>\vec{x}=\vec{y} </math><br /> else go 1) </li>
 +
</ol>
 +
<p>x will have the desired distribution.</p>
 +
 +
</div>
  
 
==== Stochastic Process ====
 
==== Stochastic Process ====
Line 3,306: Line 3,608:
 
<math>\big\{X_t:t\in T\big\}</math>, where the set X is called the state space that each variable is in it and T is called the index set.  
 
<math>\big\{X_t:t\in T\big\}</math>, where the set X is called the state space that each variable is in it and T is called the index set.  
  
definition:In probability theory, a stochastic process /stoʊˈkæstɪk/, or sometimes random process (widely used) is a collection of random variables; this is often used to represent the evolution of some random value, or system, over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.(from Wikipedia)
+
'''Definition:''' In probability theory, a stochastic process /stoʊˈkæstɪk/, or sometimes random process (widely used) is a collection of random variables; this is often used to represent the evolution of some random value, or system, over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve. (from Wikipedia)
  
A stochastic process is non-deterministic. This means that there is some indeterminacy in the final state, even if the initial condition is known.
+
A stochastic process is non-deterministic. This means that even if we know the initial condition(state), and we know some possibilities of the states to follow, the exact value of the final state remains to be uncertain.  
  
 
We can illustrate this with an example of speech: if "I" is the first word in a sentence, the set of words that could follow would be limited (eg. like, want, am), and the same happens for the third word and so on. The words then have some probabilities among them such that each of them is a random variable, and the sentence would be a collection of random variables. <br>
 
We can illustrate this with an example of speech: if "I" is the first word in a sentence, the set of words that could follow would be limited (eg. like, want, am), and the same happens for the third word and so on. The words then have some probabilities among them such that each of them is a random variable, and the sentence would be a collection of random variables. <br>
Line 3,320: Line 3,622:
 
2. Markov Process- This is a stochastic process that satisfies the Markov property which can be understood as the memory-less property. The property states that the jump to a future state only depends on the current state of the process, and not of the process's history. This model is used to model random walks exhibited by particles, the health state of a life insurance policyholder, decision making by a memory-less mouse in a maze, etc. <br>
 
2. Markov Process- This is a stochastic process that satisfies the Markov property which can be understood as the memory-less property. The property states that the jump to a future state only depends on the current state of the process, and not of the process's history. This model is used to model random walks exhibited by particles, the health state of a life insurance policyholder, decision making by a memory-less mouse in a maze, etc. <br>
 
   
 
   
 
Stochastic Process means even we get some conditions at the beginning, we just can guess some variables followed the first, but at the end the variable would be unpredictable.
 
  
 
=====Example=====
 
=====Example=====
Line 3,328: Line 3,628:
 
stochastic process always has state space and the index set to limit the range.
 
stochastic process always has state space and the index set to limit the range.
  
The state space is the set of cars , while <math>x_t</math> are sport cars.
+
The state space is the set of cars, while <math>x_t</math> are sport cars.
 +
 
 +
Births in a hospital occur randomly at an average rate
 +
 
 +
The number of cases of a disease in different towns
  
 
==== Poisson Process ====
 
==== Poisson Process ====
The Poisson process, which is discrete, arises when we count the number of occurrences of events over time.
+
[[File:Possionprocessidiagram.png‎]]
 +
 
 +
The Poisson process is a discrete counting process which counts the number of<br\>
 +
of events and the time that these occur in a given time interval.<br\>
 +
 
 +
e.g traffic accidents , arrival of emails. Emails arrive at random time <math>T_1, T_2 ... T_n</math> for example (2, 7, 3) is the number of emails received on day 1, day 2, day 3. This is a stochastic process and Poisson process with condition.
  
e.g traffic accidents , arrival of emails. Emails arrive at random time <math>T_1, T_2</math> ...
+
The probability of observing x events in a given interval is given by
 +
<math> P(X = x) = e^{-\lambda}* \lambda^x/ x! </math>
 +
where x = 0; 1; 2; 3; 4; ....
  
-Let <math>N_t</math> denote the number of arrivals in the time interval <math>(0,t]</math><br\>
+
-Let <math>N_t</math> denote the number of arrivals within the time interval <math>(0,t]</math><br\>
 
-Let <math>N(a,b]</math> denote the number of arrivals in the time interval (a,b]<br\>
 
-Let <math>N(a,b]</math> denote the number of arrivals in the time interval (a,b]<br\>
 
-By definition, <math>N(a,b]=N_b-N_a</math><br\>
 
-By definition, <math>N(a,b]=N_b-N_a</math><br\>
Line 3,349: Line 3,660:
 
E[N<sub>t</sub>] = <math>\lambda t</math> and Var[N<sub>t</sub>] = <math>\lambda t</math>
 
E[N<sub>t</sub>] = <math>\lambda t</math> and Var[N<sub>t</sub>] = <math>\lambda t</math>
  
==== ====
+
the rate parameter may change over time; such a process is called a non-homogeneous Poisson process
 +
 
 +
==== Examples ====
 
<br />
 
<br />
'''How to generate a multivariate normal with build in function "randn": (example)'''<br />
+
'''How to generate a multivariate normal with the built-in function "randn": (example)'''<br />
 
(please check the general idea at the end of lecture 6 course note.)
 
(please check the general idea at the end of lecture 6 course note.)
  
Line 3,361: Line 3,674:
 
                       %matrix to 1*n matrix;
 
                       %matrix to 1*n matrix;
 
</pre>
 
</pre>
 +
For example, if we use mu = [2 5], we would get <br/>
 +
<math> = \left[ \begin{array}{ccc}
 +
3.8214 & 0.3447 \\
 +
6.3097 & 5.6157 \end{array} \right]</math>
  
and if we want to use box-muller to generate a multivariate normal, we could use the code in lecture 6:
+
 
 +
If we want to use box-muller to generate a multivariate normal, we could use the code in lecture 6:
 
<pre style='font-size:16px'>
 
<pre style='font-size:16px'>
 
d = length(mu);
 
d = length(mu);
Line 3,376: Line 3,694:
 
X = Z*R + ones(n,1)*mu';
 
X = Z*R + ones(n,1)*mu';
 
</pre>
 
</pre>
 
  
 
==== '''Central Limit Theorem''' ====
 
==== '''Central Limit Theorem''' ====
  
We can use simulation to test results in probability and statistics. For example, with the central limit theorem, if we sample from a sufficiently large number of independently distributed random variables, the mean will be approximately normally distributed. We illustrate with an example using 1000 observations each of 20 independent exponential random variables.
+
Theorem: "Given a distribution with mean μ and variance σ², the sampling distribution of the mean approaches a  normal distribution with a mean (μ) and a variance σ²/N as N, the sample size, increases". Furthermore, the original distribution can be of any arbitrary shape, the sampling distribution of the mean will approach a normal distribution with a large enough N.<ref>
 +
http://davidmlane.com/hyperstat/A14043.html
 +
</ref>
  
'''Definition:'''
+
Applying the central limit theorem to simulations, we may revise the definition to be the following: Given certain conditions, the mean of a sufficiently large number of independent random variables, each with a well defined mean and variance, will be approximately normal distributed.(i.e. if we simulate sufficiently many independent observations based on well defined mean and variance, the mean of these observations will follow an approximately normal distribution.)
  
Given certain conditions, the mean of a sufficiently large number of independent random variables, each with a well defined mean and variance, will be approximately normal distributed.
+
We illustrate with an example using 1000 observations each of 20 independent exponential random variables.
  
 
<pre style='font-size:16px'>
 
<pre style='font-size:16px'>
Line 3,393: Line 3,712:
 
>>hist(X(1:20,:)) -> approaches to normal
 
>>hist(X(1:20,:)) -> approaches to normal
 
</pre>
 
</pre>
 +
 +
(The definition of CLT is from http://en.wikipedia.org/wiki/Central_limit_theorem)
 +
 +
<math> \lim_{n \to \infty} P*[{\frac{X_1 + ... + X_n -n*\mu}{\sigma*\surd n}} < x] = \Phi (x)</math>
  
 
==Class 11 - Tuesday,June 11, 2013==
 
==Class 11 - Tuesday,June 11, 2013==
Line 3,399: Line 3,722:
  
 
===Poisson Process===
 
===Poisson Process===
 +
A Poisson Process is a stochastic approach to count number of events in a certain time period. <s>Strike-through text</s>
 
A discrete stochastic variable ''X'' is said to have a Poisson distribution with parameter ''λ'' > 0 if
 
A discrete stochastic variable ''X'' is said to have a Poisson distribution with parameter ''λ'' > 0 if
:<math>\!f(n)= \frac{\lambda^n e^{-\lambda}}{n!}  \qquad n= 0,1,\ldots,</math>.
+
:<math>\!f(n)= \frac{\lambda^n e^{-\lambda}}{n!}  \qquad n= 0,1,2,3,4,5,\ldots,</math>.
 +
 
 +
<math>\{X_t:t\in T\}</math>  where <math>\ X_t </math> is state space and T is index set.
  
  
 
'''Properties of Homogeneous Poisson Process'''<br>
 
'''Properties of Homogeneous Poisson Process'''<br>
 
(a) '''Independence:''' The numbers of arrivals in non-overlapping intervals are independent  <br>
 
(a) '''Independence:''' The numbers of arrivals in non-overlapping intervals are independent  <br>
(b) '''Homogeneity or Uniformity:''' The number of arrival in each interval I(a,b] is Poisson distribution with rate <math>\lambda (b-a)</math><br/>
+
(b) '''Homogeneity or Uniformity:''' The number of arrivals in each interval I(a,b] is Poisson distribution with rate <math>\lambda (b-a)</math><br/>
 
(c) '''Individuality:'''  for a sufficiently short time period of length h, the probability of 2 or more events occurring in the interval is close to 0, or formally <math>\mathcal{O}(h)</math><br>
 
(c) '''Individuality:'''  for a sufficiently short time period of length h, the probability of 2 or more events occurring in the interval is close to 0, or formally <math>\mathcal{O}(h)</math><br>
  
 
+
NOTE: it is very important to note that the time between the occurrence of consecutive events (in a Poisson Process) is exponentially distributed with the same parameter as that in the Poisson distribution. This characteristic is used when trying to simulate a Poisson Process.
'''Notation'''<br>
 
N<sub>t</sub> denotes the number of arrivals up to t, i.e.(0,t] <br>
 
N<sub>[b-a)</sub> = N<sub>b</sub> - N<sub>a</sub> denotes the number of arrivals in I(a, b]. <br>
 
 
 
  
 
For a small interval (t,t+h], where h is small<br>
 
For a small interval (t,t+h], where h is small<br>
1. The number of arrivals in this interval is independent of the number of arrivals up to t(N<sub>t</sub>)<br>
+
1. The number of arrivals up to time t(N<sub>t</sub>) is independent of the number of arrival in the interval<br>
2. <math> P (N(t,t+h)=1|N_{t} ) = P (N(t,t+h)=1) =\frac{e^{-\lambda h} (\lambda h)^1}{1!} =e^{-\lambda h} {\lambda h} \approx \lambda h </math> since <math>e^{\lambda h} \approx 1</math> when h is small.<br>
+
2. <math> P (N(t,t+h)=1|N_{t} ) = P (N(t,t+h)=1) =\frac{e^{-\lambda h} (\lambda h)^1}{1!} =e^{-\lambda h} {\lambda h} \approx \lambda h </math> since <math>e^{-\lambda h} \approx 1</math> when h is small.<br>
  
 
<math>\lambda h</math> can be thought of as the probability of observing an arrival in the interval t to t+h.<br>
 
<math>\lambda h</math> can be thought of as the probability of observing an arrival in the interval t to t+h.<br>
Line 3,425: Line 3,747:
 
'''Generate a Poisson Process'''<br />
 
'''Generate a Poisson Process'''<br />
  
<math>U_n \sim U(0,1)</math><br>
+
1. set <math>T_{0}=0</math> and n=1<br/>
<math>T_n-T_{n-1}=-\frac {1}{\lambda} log(U_n)</math><br>
 
  
1. set T<sub>0</sub>=0 and n=1<br />
+
2. <math>U_{n} \sim~ U(0,1)</math><br />
  
2. U<sub>n</sub>~U(0,1)<br />
+
3. <math>T_{n} = T_{n-1}-\frac {1}{\lambda} log (U_{n})  </math> (declare an arrival)<br />
  
3. T<sub>n</sub> = T<sub>n-1</sub> - 1/lambda (declare an arrival)<br />
+
4. if <math>T_{n} \gneq T</math> stop<br />
 
 
4. if T<sub>n</sub>>T stop<br />
 
 
&nbsp;&nbsp;&nbsp;&nbsp;else<br />
 
&nbsp;&nbsp;&nbsp;&nbsp;else<br />
 
&nbsp;&nbsp;&nbsp;&nbsp;n=n+1 go to step 2<br />
 
&nbsp;&nbsp;&nbsp;&nbsp;n=n+1 go to step 2<br />
  
 
Since <math>P(N(t,t+h)=1) = e^{-{\lambda} h}\lambda h</math>, we can regard <math>\lambda </math>h as a exponential distribution, and according to what we learnt, <math>T_n-T_{n-1} = h = -\frac {1}{\lambda} log(U_n)</math>.<br>
 
Since <math>P(N(t,t+h)=1) = e^{-{\lambda} h}\lambda h</math>, we can regard <math>\lambda </math>h as a exponential distribution, and according to what we learnt, <math>T_n-T_{n-1} = h = -\frac {1}{\lambda} log(U_n)</math>.<br>
 
+
*Note : Recall that exponential random variable is the waiting time  until one event of interested occurs.
  
 
'''Review of Poisson - Example'''
 
'''Review of Poisson - Example'''
  
Let X be the r.v of the number of accidents in an hour. It is distributed Poisson(1.8).
+
Let X be the r.v of the number of accidents in an hour, following the Poisson distribution with mean 1.8.
  
 
<math>P(X=0)=e^{-1.8} </math>
 
<math>P(X=0)=e^{-1.8} </math>
Line 3,452: Line 3,771:
  
 
<span style="background:#F5F5DC">
 
<span style="background:#F5F5DC">
P(N<sub>3</sub>>3|N<sub>2</sub>)=P(N<sub>1</sub>>2)
+
<math>P(N_3> 3 | N_2)=P(N_1 > 2)</math>
 
</span>
 
</span>
  
when we use the inverse-transfer method, we can assume the poisson process to be exp distribution, and get the h function from the  inverse method, and sometimes we assume h is very small.
+
When we use the inverse-transform method, we can assume the poisson process to be an exponential distribution, and get the h function from the  inverse method. Sometimes we assume that h is very small.
 +
 
 +
'''Multi-dimensional Poisson Process'''<br>
 +
 
 +
The poisson distribution arises as the distribution of counts of occurrences of events in (multidimensional) intervals in multidimensional poisson process in a directly equivalent way to the result for unidimensional processes. This,is ''D'' is any region the multidimensional space for which |D|, the area or volume of the region, is finite, and if {{nowrap|''N''(''D'')}} is count of the number of events in ''D'', then
 +
 
 +
<math> P(N(D)=k)=\frac{(\lambda|D|)^k e^{-\lambda|D|}}{k!} .</math>
  
 
=== Generating a Homogeneous Poisson Process ===
 
=== Generating a Homogeneous Poisson Process ===
Line 3,477: Line 3,802:
 
h is the a range and we assume the probability of every point in this rang is the same by uniform ditribution.(cause h is small)
 
h is the a range and we assume the probability of every point in this rang is the same by uniform ditribution.(cause h is small)
 
and we test the rang is Tn smaller than T. And <math> -\frac {1}{\lambda} </math>  log (U<sub>n</sub>)represents that chance that one arrival arrives.
 
and we test the rang is Tn smaller than T. And <math> -\frac {1}{\lambda} </math>  log (U<sub>n</sub>)represents that chance that one arrival arrives.
 +
 +
<b>Higher Dimensions:</b><br>
 +
To sample from higher dimensional Poisson process:<br>
 +
1. Generate a random number N that is Poisson distributed with parameter <math>{\lambda}</math>*A<sub>d</sub>, where A<sub>d</sub> is the area under the bounded region. (ie A<sub>2</sub> is area of the region, A<sub>3</sub> is the volume of the 3-d space.<br>
 +
2. Given N=n, generate n random (uniform) points in the region.<br>
  
 
'''[[At the end, it generates n (cumulative) arrival times, up to time TT.]]'''
 
'''[[At the end, it generates n (cumulative) arrival times, up to time TT.]]'''
Line 3,497: Line 3,827:
  
 
</pre>
 
</pre>
 
+
<br>
  
 
The following plot is using TT = 50.<br>
 
The following plot is using TT = 50.<br>
 
The number of points generated every time on average should be <math>\lambda</math> * TT. <br>
 
The number of points generated every time on average should be <math>\lambda</math> * TT. <br>
 
The maximum value of the points should be TT. <br>
 
The maximum value of the points should be TT. <br>
[[File:Poisson.jpg]]
+
[[File:Poisson.jpg]]<br>
when TT be big, the plot of the graph will be linear, when we set the TT be 5 or small number, the plot graph looks like discrete dietribution.
+
when TT be big, the plot of the graph will be linear, when we set the TT be 5 or small number, the plot graph looks like discrete distribution.
  
 
===Markov chain===
 
===Markov chain===
A Markov Chain is a stochastic process where: <br/>
+
"A Markov Chain is a stochastic process where: <br/>
  
 
1) Each stage has a fixed number of states, <br/>
 
1) Each stage has a fixed number of states, <br/>
2) the (conditional) probabilities at each stage only depend on the previous state. <br/>
+
2) the (conditional) probabilities at each stage only depend on the previous state." <br/>
  
 
Source: "http://math.scu.edu/~cirving/m6_chapter8_notes.pdf" <br/>
 
Source: "http://math.scu.edu/~cirving/m6_chapter8_notes.pdf" <br/>
  
Markov Chain is the simplification of assumption, for instance, the result of GPA in university depend on the GPA's in high school, middle school, elementary school, etc., but during a job interview after university graduation, the interviewer would most likely to ask about the GPA in university of the interviewee but not the GPA from early years because they assume what happened before are summarized and adequately represented by the information of the GPA earned during university. Therefore, it's not necessary to look back to elementary school. A Markov Chain works the same way, we assume that everything that has occurred earlier in the process is only important to finding out where we are now, the future only depends on the present of where we are now, not the past of how we got here. So the n<sub>th</sub>random variable would only depend on the n-1<sub>th</sub>term but not all the previous ones. A Markov process exhibits the memoryless property.  
+
A Markov Chain is said to be irreducible if for each pair of states i and j there is a positive probability, starting in state i, that the process will ever enter state j.(source:"https://en.wikipedia.org/wiki/Markov_chain")
A good real world application using Markov Chain is the google link analysis algorithm "PageRank".
+
 
 +
Markov Chain is the simplification of assumption, for instance, the result of GPA in university depend on the GPA's in high school, middle school, elementary school, etc., but during a job interview after university graduation, the interviewer would most likely to ask about the GPA in university of the interviewee but not the GPA from early years because they assume what happened before are summarized and adequately represented by the information of the GPA earned during university. Therefore, it's not necessary to look back to elementary school. A Markov Chain works the same way, we assume that everything that has occurred earlier in the process is only important for finding out where we are now, the future only depends on the present of where we are now, not the past of how we got here. So the n<sub>th</sub>random variable would only depend on the n-1<sub>th</sub>term but not all the previous ones. A Markov process exhibits the memoryless property.<br/>
 +
 
 +
Examples of Markov Chain applications in various fields:<br />
 +
*Physics: The movement of a particle (memory-less random walk)<br />
 +
*Finance: The volatility of prices of financial securities or commodities<br />
 +
*Actuarial Science: The pricing and valuation of multiple-state and multiple-lives insurance and annuity contracts<br />
 +
*Technology: The Google link analysis algorithm "PageRank"<br />
 +
 
  
 +
'''Definition''' An irreducible Markov Chain is said to be aperiodic if for some n <math>\ge 0 </math> and some state j.<br />
 +
<math> P*(X_n=j | X_0 =j) > 0 </math>    and    <math>  P*(X_{n+1} | X_0=j) > 0 </math> <br />
 +
 +
It can be shown that if the Markov Chain is irreducible and aperiodic then, <br />
 +
<math> \pi_j = \lim_{n -> \infty} P*(X_n = j) for j=1...N </math> <br />
 +
Source: From Simulation textbook <br />
  
 
Product Rule (Stochastic Process):<br />
 
Product Rule (Stochastic Process):<br />
Line 3,521: Line 3,865:
  
 
In Markov Chain<br />
 
In Markov Chain<br />
<math> f(x_1,x_2,...,x_n)=f(x_1)f(x_\mid x_1)f(x_3\mid x_2)...f(x_n\mid x_{n-1}) </math>
+
<math> f(x_1,x_2,...,x_n)=f(x_1)f(x_2\mid x_1)f(x_3\mid x_2)...f(x_n\mid x_{n-1}) </math>
  
 
Concept: The current status of a subject must be relative to the past.However, it will only depend on the previous result only. In other words, if an event occurring tomorrow follows a Markov process it will depend on today and yesterday (the past) is negligible. The past (not including the current state of course) is negligible  since its information is believed to have been captured and reflected in the current state.   
 
Concept: The current status of a subject must be relative to the past.However, it will only depend on the previous result only. In other words, if an event occurring tomorrow follows a Markov process it will depend on today and yesterday (the past) is negligible. The past (not including the current state of course) is negligible  since its information is believed to have been captured and reflected in the current state.   
  
A Markov Chain is a stochastic Process for which the distribution of <math>x_t</math> depends only on <math>x_{t-1}</math>.
+
A Markov Chain is a Stochastic Process for which the distribution of <math>x_t</math> depends only on <math>x_{t-1}</math>.
  
Given <math> x_t</math>, <math>x_{t-1}</math> and <math>x_{t+1} </math> are independent. The process of getting <math> x_n </math> is drawn as follows. It implies the concept of markov chain. The distribution of <math>x_n</math> only depends on the value of <math>x_{n-1}</math>.
+
Given <math> x_t</math>, <math>x_{t-1}</math> and <math>x_{t+1} </math> are independent. The process of getting <math> x_n </math> is drawn as follows. The distribution of <math>x_n</math> only depends on the value of <math>x_{n-1}</math>.
  
 
<math> x_1 \rightarrow x_2\rightarrow...\rightarrow x_n</math>  
 
<math> x_1 \rightarrow x_2\rightarrow...\rightarrow x_n</math>  
Line 3,538: Line 3,882:
  
 
A continuous time markov process would be one where the time spent in each state is not discrete, but can take  
 
A continuous time markov process would be one where the time spent in each state is not discrete, but can take  
on positive real values, that is the index set is the positive real numbers. If the process is homogeneous, then the time spent will have an exponential distribution. In this case we will have a transition  
+
on positive real values. In other words, the index set is the positive real numbers. If the process is homogeneous, then the time spent will have an exponential distribution. In this case we will have a transition rate matrix that captures the rate at which we move between two states. An example will be the homogeneous Poisson process.<br />
rate matrix, which captures the rate at which we move between two states. An example will be the homogeneous Poisson process.<br />
 
  
  
Line 3,565: Line 3,908:
 
0.2 & 0.8
 
0.2 & 0.8
 
\end{matrix} \right] </math>
 
\end{matrix} \right] </math>
 +
 +
Note: Column 1 corresponds to the state at time t and Column 2 corresponds to the state at time t+1.
  
 
The above matrix can be drawn into a state transition diagram
 
The above matrix can be drawn into a state transition diagram
Line 3,574: Line 3,919:
 
2. <math>\sum_{j}^{}{P_{ij}=1}</math>  which means the rows of P should sum to 1.<br />
 
2. <math>\sum_{j}^{}{P_{ij}=1}</math>  which means the rows of P should sum to 1.<br />
  
 +
Remark: <math>\sum_{i}^{}{P_{ij}\neq1}</math> in general. If equality holds, the matrix is called a doubly stochastic matrix.
  
 
In general, one would consider a (finite) set of elements <math> \Omega </math> such that: <br>
 
In general, one would consider a (finite) set of elements <math> \Omega </math> such that: <br>
Line 3,591: Line 3,937:
 
Then one might consider the periodicity of the chain and derive a notion of cyclic behavior. <br>
 
Then one might consider the periodicity of the chain and derive a notion of cyclic behavior. <br>
  
=== Example of Transition Matrix ===
+
=== Examples of Transition Matrix ===
  
[[File:Mark13.png]]
+
[[File:Mark13.png]]<br>
 +
The picture is from http://www.google.ca/imgres?imgurl=http://academic.uprm.edu/wrolke/esma6789/graphs/mark13.png&imgrefurl=http://academic.uprm.edu/wrolke/esma6789/mark1.htm&h=274&w=406&sz=5&tbnid=6A8GGaxoPux9kM:&tbnh=83&tbnw=123&prev=/search%3Fq%3Dtransition%2Bmatrix%26tbm%3Disch%26tbo%3Du&zoom=1&q=transition+matrix&usg=__hZR-1Cp6PbZ5PfnSjs2zU6LnCiI=&docid=PaQvi1F97P2urM&sa=X&ei=foTxUY3DB-rMyQGvq4D4Cg&sqi=2&ved=0CDYQ9QEwAQ&dur=5515)
  
There are three states: 0,1,2, and 3.
+
1.There are four states: 0,1,2, and 3.
  
Each row adds up to 1, and all the entries are between 0 and 1(which means the transition probability).
+
Each row adds up to 1, and all the entries are between 0 and 1(since the transition probability is always between 0 and 1).<br />
 +
This matrix means that <br />
 +
- at state 0, can only go to state 1, since probability is 1.<br />
 +
- at state 1, can go to state 0 with a probability of 1/3 and state 2 with a probability of 2/3<br />
 +
- at state 2, can go to state 1 with a probability of 2/3 and state 3 with a probability of 1/3<br />
 +
- at state 3, can only go to state 2, since probability is 1.<br />
 +
 
 +
2. Consider a Markov chain with state space {0, 1} and transition matrix
 +
<math> P= \left [ \begin{matrix}
 +
1/3 & 2/3 \\
 +
3/4 & 1/4
 +
\end{matrix} \right] </math>.
 +
Assuming that the chain starts in state 0 at time n = 0, what is the probability that it is  in state 1 at time n = 2?
 +
 
 +
 
 +
<math>\begin{align}
 +
P(X_{2} &=1 &\mid X_{0} &=0) & =P(X_{1} &=0,X_{2} &=1 &\mid X_{0} &=0)+P(X_{1} &=1,X_{2} &=1 &\mid X_{0} &=0)\end{align}  </math>
 +
<math> \begin{align} P(X_{1} &=0 &\mid X_{0} &=0) * P(X_{2} &=1 &\mid X_{1} &=0)+P(X_{1} &=1 &\mid X_{0} &=0) * P(X_{2} &=1 &\mid X_{1}&=1) &=1/3*2/3+ 2/3*1/4 &=7/18 \\
 +
\end{align}</math><br />
  
 
== Class 12 - Thursday,June 13, 2013 ==
 
== Class 12 - Thursday,June 13, 2013 ==
Line 3,605: Line 3,970:
  
 
=== Multiplicative Congruential Algorithm ===
 
=== Multiplicative Congruential Algorithm ===
x<sub>k+1</sub>= (ax<sub>k</sub>+c) mod m
+
<div style="border:1px solid red">
 +
A Linear Congruential Generator (LCG) yields a sequence of randomized numbers calculated with a linear equation. The method represents one of the oldest and best-known pseudorandom number generator algorithms.[1] The theory behind them is easy to understand, and they are easily implemented and fast, especially on computer hardware which can provide modulo arithmetic by storage-bit truncation.<br>
 +
from wikipedia
 +
</div>
 +
 
 +
<math>\begin{align}x_k+1= (ax_k+c) \mod  m\end{align}</math><br />
 +
 
 +
Where a, c, m and x<sub>1</sub> (the seed) are values we must chose before running the algorithm. While there is no set value for each, it is best for m to be large and prime. For example, Matlab uses a = 75,b = 0,m = 231 − 1.
  
Where a, c, m and x<sub>1</sub> (the seed) are values we must chose before running the algorithm. While there is no set value for each, it is best for m to be large and prime.
+
'''Examples:'''<br>
 +
1. <math>\begin{align}X_{0} = 10 ,a = 2 , c = 1 , m = 13 \end{align}</math><br>
 +
   
 +
<math>\begin{align}X_{1} = 2 * 10 + 1\mod 13 = 8\end{align}</math><br>
  
Examples:
+
<math>\begin{align}X_{2} = 2 * 8  + 1\mod 13 = 4\end{align}</math> ... and so on<br>
      X<sub>0</sub> = 10 ,a = 2 , c = 1 , m = 13
+
 
 +
 
 +
2. <math>\begin{align}X_{0} = 44 ,a = 13 , c = 17 , m = 211\end{align}</math><br>
 
        
 
        
          X<sub>1</sub> = 2 * 10 + mod 13 = 8
+
<math>\begin{align}X_{1} = 13 * 44 + 17\mod 211 = 167\end{align}</math><br>  
          X<sub>2</sub> = 2 * 8  + 1  mod 13 = 4
 
          ... and so on
 
  
      X<sub>0</sub> = 44 ,a = 13 , c = 17 , m = 211
+
<math>\begin{align}X_{2} = 13 * 167  + 17\mod 211 = 78\end{align}</math><br>  
     
+
 
          X<sub>1</sub> = 13 * 44 + 17 mod 211 = 167
+
<math>\begin{align}X_{3} = 13 * 78  + 17\mod 211 = 187\end{align}</math> ... and so on<br>
          X<sub>2</sub> = 13 * 167  + 17  mod 211 = 78
 
          X<sub>3</sub> = 13 * 78  + 17 mod 211 = 187
 
          ... and so on
 
  
 
=== Inverse Transformation Method ===
 
=== Inverse Transformation Method ===
Line 3,629: Line 4,001:
  
 
'''Note:''' <br/>
 
'''Note:''' <br/>
In Uniform Distribution P(X<=a)=a<br/ >
+
In Uniform Distribution <math>P(X<=a)=a </math><br/ >
proof:P(X<=x) = P(F<sup>-1</sup>(u)<=x)=P(u<=F(x)) = F(x)<br/ >
+
proof:<math>P(X<=x) = P(F^{-1}(u)<=x)=P(u<=F(x)) = F(x)</math><br/ >
  
 
'''For discrete cases:''' <br/ >
 
'''For discrete cases:''' <br/ >
Line 3,660: Line 4,032:
  
 
- Sample uniformly from a space W that contains the sample space G of interest<br/>
 
- Sample uniformly from a space W that contains the sample space G of interest<br/>
-Accept if the point is inside G<br/>
+
- Accept if the point is inside G <br/>
Sample uniformly from W<br/>
+
 
 +
Steps:
 +
1. Sample uniformly from W <br/>
 
g(x)=<math>\frac{1}{A_W}</math>, where A<sub>W</sub> is the area of W.<br/>
 
g(x)=<math>\frac{1}{A_W}</math>, where A<sub>W</sub> is the area of W.<br/>
 
f(x)=<math>\frac{1}{A_G}</math>, where A<sub>G</sub> is the area of G.<br/>
 
f(x)=<math>\frac{1}{A_G}</math>, where A<sub>G</sub> is the area of G.<br/>
 +
2. If the point is inside G, accept the point. Else, reject and repeat step 1.
  
  
Line 3,674: Line 4,049:
 
===Exponential===
 
===Exponential===
  
Models the waiting until the first success.<br>
+
Models the waiting time until the first success.<br>
 +
<math>X\sim~Exp(\lambda)</math> <br />
  
X~Exp<math>(\lambda) </math><br>
+
<math>f(x) = \lambda e^{-\lambda x} \, , x>0 </math><br/>
<math> f (x) = \lambda e^{-\lambda x}</math> , <math>x>0 </math><br/>
 
  
1. U~Unif(0,1)
+
<math>1.\, U\sim~U(0,1)</math>
 
+
<br />
2. The inverse of exponential function is x = <math>\frac{-1}{\lambda} log(U)</math>
+
<math>2.\, x = \frac{-1}{\lambda} log(U)</math>
  
 
===Normal===
 
===Normal===
Line 3,699: Line 4,074:
 
In the multivariate case,<br/ >
 
In the multivariate case,<br/ >
 
<math>\underline{Z}\sim N(\underline{0},I)\rightarrow  \underline{X} \sim N(\underline{\mu},\Sigma)</math> <br/ >
 
<math>\underline{Z}\sim N(\underline{0},I)\rightarrow  \underline{X} \sim N(\underline{\mu},\Sigma)</math> <br/ >
<math>\underline{X} = \underline{\mu} +\Sigma ^{1/2} \underline{Z}</math>
+
<math>\underline{X} = \underline{\mu} +\Sigma ^{1/2} \underline{Z}</math><br/>
 +
Note: <math>\Sigma^{1/2}</math> can be obtained from Cholesky decomposition (chol(A) in MATLAB), which is guaranteed to exist, as  <math>\Sigma</math> is positive semi-definite.
  
 
=== Gamma ===
 
=== Gamma ===
Line 3,707: Line 4,083:
  
 
Also, Gamma(t,λ) can be expressed into a summation of t exp(λ).<br>
 
Also, Gamma(t,λ) can be expressed into a summation of t exp(λ).<br>
<math>x=\frac {-1}{\lambda}log(u_1)-\frac {1}{\lambda}log(u_2)-.......-\frac {1}{\lambda}log(u_t)</math>
+
<math>x=\frac {-1}{\lambda}\log(u_1)-\frac {1}{\lambda}\log(u_2)-.......-\frac {1}{\lambda}\log(u_t)</math>
  
<math>=\frac {-1}{\lambda}[log(u_1)+log(u_2)+.....+log(u_t)]</math>
+
<math>=\frac {-1}{\lambda}[\log(u_1)+\log(u_2)+.....+\log(u_t)]</math>
  
<math>=\frac {-1}{\lambda}log</math>(∏<math>_{j=1}^{t} U_j)</math>
+
<math>=\frac {-1}{\lambda}\log(\prod_{j=1}^{t} U_j)</math>
 +
 
 +
This is a special property of gamma distribution.
  
 
=== Bernoulli ===
 
=== Bernoulli ===
Line 3,717: Line 4,095:
 
A Bernoulli random variable can only take two possible values: 0 and 1. 1 represents "success" and 0 represents "failure." If p is the probability of success, we have pdf
 
A Bernoulli random variable can only take two possible values: 0 and 1. 1 represents "success" and 0 represents "failure." If p is the probability of success, we have pdf
  
<math> f(x)= p^x (1-p)^{1-x}, x=0,1 </math><br>
+
<math> f(x)= p^x (1-p)^{1-x},\,  x=0,1 </math><br>
  
 
To generate a Bernoulli random variable we use the following procedure:
 
To generate a Bernoulli random variable we use the following procedure:
  
sample u~U(0,1)<br>
+
<math> 1. U\sim~U(0,1)</math><br>
if u <= p, then x=1<br>
+
<math> 2. if\, u <= p, then\, x=1\,</math><br />  
else x=0<br>
+
<math> else\, x=0</math><br/>
 
where 1 stands for success and 0 stands for failure.<br>
 
where 1 stands for success and 0 stands for failure.<br>
  
Line 3,730: Line 4,108:
 
The sum of n independent Bernoulli trials
 
The sum of n independent Bernoulli trials
 
<br\>
 
<br\>
X~ Bin(n,p)<br/>
+
<math> X\sim~ Bin(n,p)</math><br/>
1. U1, U2, ... Un ~ U(0,1)<br/>
+
1.<math> U1, U2, ... Un \sim~U(0,1)</math><br/>
 
2. <math> X= \sum^{n}_{1} I(U_i \leq p) </math> ,where <math>I(U_i \leq p)</math> is an indicator for a successful trial.<br/>
 
2. <math> X= \sum^{n}_{1} I(U_i \leq p) </math> ,where <math>I(U_i \leq p)</math> is an indicator for a successful trial.<br/>
 
Return to 1<br/>
 
Return to 1<br/>
  
I is an indicator variable if for U <= P, then I(U<=P)=1; else I(U>P)=0.
+
I is an indicator variable if for <math>U \leq P,\, then\, I(U\leq P)=1;\, else I(U>P)=0.</math>
  
 
Repeat this N times if you need N samples.
 
Repeat this N times if you need N samples.
Line 3,749: Line 4,127:
 
simulate this binomial distribution.
 
simulate this binomial distribution.
  
1) Generate <math> U_1....U_{10} </math> ~ <math> U(0,1) </math>  <br>
+
1) Generate <math>U_1....U_{10} \sim~ U(0,1) </math>  <br>
 
2) <math> X= \sum^{10}_{1} I(U_i \leq \frac{1}{6}) </math> <br>
 
2) <math> X= \sum^{10}_{1} I(U_i \leq \frac{1}{6}) </math> <br>
3)Return to one.
+
3)Return to 1)
  
 
=== Beta Distribution ===
 
=== Beta Distribution ===
Line 3,797: Line 4,175:
 
<math>y-1=-(1-p)^x</math><br>
 
<math>y-1=-(1-p)^x</math><br>
 
<math>1-y=(1-p)^x</math><br>
 
<math>1-y=(1-p)^x</math><br>
<math>1-y=(e^(-\lambda))^x=e^(-\lambda*x)</math> since <math>1-p=e^-\lambda</math><br>
+
<math>1-y=(e^{-\lambda})^x=e^{-\lambda*x}</math> since <math>1-p=e^{-\lambda}</math><br>
 
<math>log(1-y)=-\lambda*x</math><br>
 
<math>log(1-y)=-\lambda*x</math><br>
 
<math>x=-1/(\lambda)*log(1-y)</math><br>
 
<math>x=-1/(\lambda)*log(1-y)</math><br>
<math>F^(-1)(x)=-1/(\lambda)*log(1-x)</math><br>
+
<math>F^-1(x)=-1/(\lambda)*log(1-x)</math><br>
 
<br>
 
<br>
 
'''Algorithm:'''<br />
 
'''Algorithm:'''<br />
Line 3,812: Line 4,190:
 
If X~Unif (0,1), Y= floor(5U)-2 = [5U]-2 -> Y~ DU[-2,2]  
 
If X~Unif (0,1), Y= floor(5U)-2 = [5U]-2 -> Y~ DU[-2,2]  
 
<br>
 
<br>
 +
 +
There is also another intuitive method:<br>
 +
1. Draw U ~ U(0,1)<br>
 +
2. i = 1, Pi = 1 - (1 - P)^i. <br>
 +
3. If u <= Pi = 1 - (1 - P)^i, set X = i.
 +
Else, i = i + 1. <br>
  
 
===Poisson===
 
===Poisson===
Line 3,850: Line 4,234:
  
 
== Class 13 - Tuesday June 18th 2013 ==
 
== Class 13 - Tuesday June 18th 2013 ==
n-step transition matrix is matrix <math> P_n </math> whose elements are the probability of moving to state j from state i in n steps. <br/>
+
'''Markov Chain'''
 +
<br>N-Step Transition Matrix: a matrix <math> P_n </math> whose elements are the probability of moving from state i to state j in n steps. <br/>
 
<math>P_n (i,j)=Pr⁡(X_{m+n}=j|X_m=i)</math> <br/>
 
<math>P_n (i,j)=Pr⁡(X_{m+n}=j|X_m=i)</math> <br/>
 +
 +
Explanation: (with an example) Suppose there 10 states { 1, 2, ..., 10}, and suppose you are on state 2, then P<sub>8</sub>(2, 5) represent the probability of moving from state 2 to state 5 in 8 steps.
 +
 +
One-step transition probability:<br/>
 +
The probability of  X<sub>n+1</sub> being in state j given that X<sub>n</sub> is in state i is called the
 +
one-step transition probability  and is denoted by P<sub>i,j</sub><sup>n,n+1</sup>. That is <br/>
 +
P<sub>i,j</sub><sup>n,n+1</sup> = Pr(X<sub>n+1</sub> =j/X<sub>n</sub> =i)
  
 
Example from previous class: <br/>
 
Example from previous class: <br/>
Line 3,860: Line 4,252:
 
\end{matrix} \right] </math>
 
\end{matrix} \right] </math>
  
The two-steps transition probability matrix is: <br/>
+
The two step transition probability matrix is:
<math>P_2 (a,b)=Pr⁡ (X_{m+2}=b| X_m=a)=Pr⁡(X_{m+1}=a| X_m=a)Pr⁡(X_{m+2}=a|X_{m+1}=a)+ Pr⁡(X_{m+1}=b|X_m=a)Pr⁡(X_{m+2}=a|X_{m+1}=b)</math> <br/>
 
 
 
 
 
'''NOTE:''' <math>Pr⁡ (X_{m+2}=a| X_m=a)</math> is the probability of moving from a to a in 2 steps
 
  
<math> =0.7(0.7)+0.3(0.2)=0.55 </math><br/>
+
<math> P P= \left [ \begin{matrix}
 
 
<math> P \times P= \left [ \begin{matrix}
 
 
0.7 & 0.3 \\
 
0.7 & 0.3 \\
 
0.2 & 0.8
 
0.2 & 0.8
\end{matrix} \right] \times \left [ \begin{matrix}
+
\end{matrix} \right] \left [ \begin{matrix}
 
0.7 & 0.3 \\
 
0.7 & 0.3 \\
 
0.2 & 0.8
 
0.2 & 0.8
Line 3,882: Line 4,268:
 
\end{matrix} \right] </math><br\>
 
\end{matrix} \right] </math><br\>
  
<math>P_2 = P_1 \times P_1 </math><br\>
+
Interpretation:<br\>
 +
- If at time 0 we are in state 1, then the probability of us being in state 1 at time 2 is 0.55 and 0.45 for state 2.<br\>
 +
- If at time 0 we are in state 2, then the probability of us being in state 1 at time 2 is 0.3 and 0.7 for state 2.<br\>
  
<math>P_n = P_1^n </math><br\>
+
<math>P_2 = P_1 P_1 </math><br\>
  
=== n-step transition matrix ===
+
<math>P_3 = P_1 P_2 </math><br\>
It is the matrix P<sub>n</sub> whose elements is probability of moving to state j from state i in n steps
 
  
In general <math>P_n = P^n</math> with <math>P_n(i,j) \geq 0</math> and <math>\sum_{j} P_n(i,j) = 1</math>.<br />
+
<math>P_n = P_1 P_(n-1) </math><br\>
 +
 
 +
<math>P_n = P_1^n </math><br\>
 +
 
 +
 
 +
 
 +
The two-step transition probability of moving from state a to state a:
 +
<br/>
 +
<math>P_2 (a,a)=Pr⁡ (X_{m+2}=a| X_m=a)=Pr⁡(X_{m+1}=a| X_m=a)Pr⁡(X_{m+2}=a|X_{m+1}=a)+ Pr⁡(X_{m+1}=b|X_m=a)Pr⁡(X_{m+2}=a|X_{m+1}=b)</math> <br/>
 +
 
 +
<math> =0.7(0.7)+0.3(0.2)=0.55 </math><br/>
 +
 
 +
Another Example: <br/>
 +
 
 +
<math> P= \left [ \begin{matrix}
 +
1 & 0 \\
 +
0.7 & 0.3
 +
\end{matrix} \right] </math>
 +
 
 +
The two step transition probability matrix is:
 +
 
 +
<math> P P= \left [ \begin{matrix}
 +
1 & 0 \\
 +
0.7 & 0.3
 +
\end{matrix} \right] \left [ \begin{matrix}
 +
1 & 0 \\
 +
0.7 & 0.3
 +
\end{matrix} \right] </math>=<math>\left [ \begin{matrix}
 +
1(1)+ 0(0.7) & 1(0) + 0(0.3)              \\
 +
1(0.7)+0.7(0.3) & 0(0.7)+0.3(0.3)
 +
\end{matrix} \right] </math>=<math>\left [ \begin{matrix}
 +
1 &  0                  \\
 +
0.91  & 0.09
 +
\end{matrix} \right] </math><br\>
 +
 
 +
This is the two-step transition matrix.
 +
 
 +
=== n-step transition matrix ===
 +
The elements of matrix P<sub>n</sub> (i.e. the ij<sub>th</sub> entry P<sub>ij</sub>) is the probability of moving to state j from state i in n steps
 +
 
 +
In general <math>P_n = P^n</math> with <math>P_n(i,j) \geq 0</math> and <math>\sum_{j} P_n(i,j) = 1</math><br />
 
Note: <math>P_2 = P_1\times P_1; P_n = P^n</math><br />
 
Note: <math>P_2 = P_1\times P_1; P_n = P^n</math><br />
The equation above is a special case of the chapman-Kolmogorov equations.<br />
+
The equation above is a special case of the Chapman-Kolmogorov equations.<br />
The reason as to why the equation above is true is because of the markov property or<br />
+
It is true because of the Markov property or the memoryless property of Markov chains, where the probabilities of going forward to the next state <br />
memorylessness property of markov chains, where the probabilities of going forward to next state <br />
+
only depends on your current state, not your previous states. By intuition, we can multiply the 1-step transition <br />
only depends on your current state. Which would be, by intuition, why we can multiply the 1-step transition <br />
+
matrix n-times to get a n-step transition matrix.<br />
matrix n-times to get n-step transition matrix.<br />
 
 
   
 
   
Example: we can see how <math>P_n = P^n</math> from the following:
+
Example: We can see how <math>P_n = P^n</math> from the following:
 
<br/>
 
<br/>
<math>mu_1=mu_0\cdot P</math> <br/>
+
<math>\vec{\mu_1}=\vec{\mu_0}\cdot P</math> <br/>
<math>mu_2=mu_1\cdot P</math> <br/>
+
<math>\vec{\mu_2}=\vec{\mu_1}\cdot P</math> <br/>
<math>mu_3=mu_2\cdot P</math> <br/>
+
<math>\vec{\mu_3}=\vec{\mu_2}\cdot P</math> <br/>
Thus, we get that
+
Therefore,  
 
<br/>
 
<br/>
<math>mu_3=mu_0\cdot P^3
+
<math>\vec{\mu_3}=\vec{\mu_0}\cdot P^3
 
</math> <br/>
 
</math> <br/>
  
 
<math>P_n(i,j)</math> is called n-steps transition probability. <br>
 
<math>P_n(i,j)</math> is called n-steps transition probability. <br>
 +
<math>\vec{\mu_0} </math> is called the '''initial distribution'''. <br>
 +
<math>\vec{\mu_n} = \vec{\mu_0}* P^n </math> <br />
  
The transition probability matrix is <math> P= \left [ \begin{matrix}
+
Example with Markov Chain:
0.7 & 0.3 \\
+
Consider a two-state Markov chain {<math>X_t; t = 0, 1, 2,...</math>} with states {1,2} and transition probability matrix
0.2 & 0.8
 
\end{matrix} \right] </math> <br> <br>
 
  
There are two possibilities: Go to b and come back to a, or stay there the whole time.
+
<math> P= \left [ \begin{matrix}
 +
1/2 & 1/2 \\
 +
1/3 & 2/3
 +
\end{matrix} \right] </math>
  
 +
Given <math> X_0 = 1 </math>. Compute the following:
  
Example with Markov Chain:
+
a)<math> P(X_1=1 | X_0=1) = P(1,1) = 1/2 </math>
Consider a two-state Markov chain {Xt; t = 0, 1, 2,...} with states {1,2} and transition probability matrix
 
  
P=[1/2 1/2; 1/3 2/3], given Xo=1. Compute the following:
+
b)<math> P(X_2=1, X_1=1 |X_0=1) = P(X_2=1|X_1=1)*P(X_1=1|X_0=1)= 1/2 * 1/2 = 1/4 </math>
  
a)P(X1=1 | Xo=1) = P11 = 1/2
+
c)<math> P(X_2=1|X_0=1)= P_2(1,1) = 5/12 </math>
  
b)P(X2=1, X1=1 |Xo=1) = P(X2=1|X1=1)*P(X1=1|Xo=1)=1/2 * 1/2 =1/4
+
d)<math> P^2=P*P= \left [ \begin{matrix}
 
+
5/12 & 7/12 \\
c) P(X2=1|X0=1)= P11^2 =5/12
+
7/18 & 11/18
 +
\end{matrix} \right] </math>
  
 
=== Marginal Distribution of Markov Chain ===
 
=== Marginal Distribution of Markov Chain ===
We represent the probability of all states at time t with a vector <math>\mu_t</math><br/>
+
We represent the probability of all states at time t with a vector <math>\underline{\mu_t}</math><br/>
<math>\mu_t~=(\mu_t(1), \mu_t(2),...\mu_t(n))</math> where <math>\mu_t(1)</math> is the probability of being on state 1 at time t.<br/>
+
<math>\underline{\mu_t}~=(\mu_t(1), \mu_t(2),...\mu_t(n))</math> where <math>\underline{\mu_t(1)}</math> is the probability of being on state 1 at time t.<br/>
and in general, <math>\mu_t(i)</math> shows  the probability of being on state i at time t.<br/>
+
and in general, <math>\underline{\mu_t(i)}</math> shows  the probability of being on state i at time t.<br/>
For example, if there are two states a and b, then <math>\mu_5</math>=(0.1, 0.9) means that the chance of being in state a at time 5 is 0.1 and the chance of being on state b at time 5 is 0.9. <br/>
+
For example, if there are two states a and b, then <math>\underline{\mu_5}</math>=(0.1, 0.9) means that the chance of being in state a at time 5 is 0.1 and the chance of being on state b at time 5 is 0.9. <br/>
 
If we generate a chain for many times, the frequency of states at each time shows marginal distribution of the chain at that time. <br/>
 
If we generate a chain for many times, the frequency of states at each time shows marginal distribution of the chain at that time. <br/>
The vector <math>\mu_0</math> is called the initial distribution. <br/>
+
The vector <math>\underline{\mu_0}</math> is called the initial distribution. <br/>
  
<math> P_2~=P_1*P_1 (as verified above) </math>
+
<math> P^2~=P\cdot P </math> (as verified above)  
  
 
In general,
 
In general,
<math> P_n~=(P_1)^n </math> **Note that <math>P_1</math> is equal to the matrix P <br/>
+
<math> P^n~= \Pi_{i=1}^{n} P</math> (P multiplied n times)<br/>
<math>\mu_n~=\mu_0*P_n</math><br/>
+
<math>\mu_n~=\mu_0 P^n</math><br/>
 
where <math>\mu_0</math> is the initial distribution,
 
where <math>\mu_0</math> is the initial distribution,
 +
and <math>\mu_{m+n}~=\mu_m P^n</math><br/>
 +
N can be negative, if P is invertible.
  
 
<br/>
 
<br/>
Line 3,952: Line 4,384:
 
<math>\mu_4~=(3/4, 1/4)</math><br/>
 
<math>\mu_4~=(3/4, 1/4)</math><br/>
  
if we simulate a chain many times, frequency of states at time t show the marginal distribution at time t <br/>
+
if we simulate a chain many times, frequency of states at time t show the marginal distribution at time t <br />
  
  
Line 3,959: Line 4,391:
 
Marginal Distribution
 
Marginal Distribution
  
<math>\mu_1~ = \mu_0P</math>
+
<math>\mu_1~ = \mu_0P</math> <br>
<math>\mu_2~ = \mu_1P = \mu_0PP = \mu_0P^2</math>
+
<math>\mu_2~ = \mu_1P = \mu_0PP = \mu_0P^2</math> <br>
  
In general, <math>\mu_n~ = \mu_0P^n</math>
+
In general, <math>\mu_n~ = \mu_0P^n</math><br />
 +
Property: If <math>\mu_n~\neq\mu_t~</math>(for any t less than n), then we say P does not converge. <br />
  
===Stationary Distribution ===
 
  
  
  
<math>\pi</math> is stationary dist of the chain if <math>\pi</math>P = <math>\pi</math>
+
==== Stationary Distribution ====
 +
 
 +
 
 +
 
 +
 
 +
<math>\pi</math> is stationary distribution of the chain if <math>\pi</math>P = <math>\pi</math> In other words, a stationary distribution is when the markov process that have equal probability of moving to other states as its previous move.
 +
 
 +
where <math>\pi</math> is a probability vector <math>\pi</math>=(<math>\pi</math><sub>i</sub> | <math>i \in X</math>) such that all the entries are nonnegative and sum to 1. It is the eigenvector in this case.
 +
 
 +
In other words, if X''<sub>0</sub>'' is draw from <math>\pi</math>. Then marginally, X''<sub>n</sub>'' is also drawn from the same distribution <math>\pi</math> for every n≥0.
 +
 
 +
The above conditions are used to find the stationary distribution
 +
In matlab, we could use <math>P^n</math> to find the stationary distribution.(n is usually larger than 100)<br/>
  
where <math>\pi</math> is a probability vector <math>\pi</math>=(<math>\pi</math><sub>i</sub> | <math>i \in X</math>) such that all the entries are nonnegative and sum to 1.
 
  
 
'''Comments:'''<br/>
 
'''Comments:'''<br/>
Line 3,976: Line 4,419:
  
 
Comments: <br/>
 
Comments: <br/>
1. <math>\pi</math> may not exist and not always be unique. <br/>
+
1. <math>\pi</math> may not exist and even if it exists, it may not always be unique. <br/>
2. If <math>\pi</math> exists and is unique, then <math>\pi</math><sub>i</sub> is called the long-run proportion of the process in state i.<br/>
+
2. If <math>\pi</math> exists and is unique, then <math>\pi</math><sub>i</sub> is called the long-run proportion of the process in state i and the stationary distribution is also the limiting distribution of the process.<br/>
 +
 
 +
How long do you have to wait until you reach a steady sate?
 +
Ans: There is not clear way to find that out
 +
 
 +
How do you increase the time it takes to reach the steady state?
 +
Ans: Make the probabilities of transition much smaller, to reach from state 0 to state 1 and vice-versa p=0.005. And make the probabilities of staying in the same state extremely high. To stay in state 0 or state 1 p=0.995, then the matrix is in a "sticky state"
 +
 
  
 +
EXAMPLE : Random Walk on the cycle S={0,1,2}
 +
 +
<math>P^2 = \left[ \begin{array}{ccc}
 +
2pq & q^2 & p^2 \\
 +
p^2 & 2pq & q^2 \\
 +
q^2 & p^2 & 2pq \end{array} \right]</math>
 +
 +
Suppose<br/>
 +
<math>P(x_0=0)=\frac{1}{4}</math><br/>
 +
<math>P(x_0=1)=\frac{1}{2}</math><br/>
 +
<math>P(x_0=2)=\frac{1}{4}</math><br/>
 +
Thus<br/>
 +
<math>\pi_0 = \left[ \begin{array}{c} \frac{1}{4} \\ \frac{1}{2} \\ \frac{1}{4} \end{array} \right]</math><br/>
 +
so<br/>
 +
<math>\,\pi^2 = \pi_0 * P^2 </math>
 +
<math>= \left[ \begin{array}{c} \frac{1}{4} \\ \frac{1}{2} \\ \frac{1}{4} \end{array} \right] * \left[ \begin{array}{ccc}
 +
2pq & q^2 & p^2 \\
 +
p^2 & 2pq & q^2 \\
 +
q^2 & p^2 & 2pq \end{array} \right]</math>
 +
<math>= \left[ \begin{array}{c} \frac{1}{2}pq + \frac{1}{2}p^2+\frac{1}{4}q^2 \\ \frac{1}{4}q^2+pq+\frac{1}{4}p^2 \\ \frac{1}{4}p^2+\frac{1}{2}q^2+\frac{1}{2}pq\end{array} \right]</math>
  
 
==== MatLab Code ====
 
==== MatLab Code ====
Line 4,033: Line 4,503:
  
 
</pre>
 
</pre>
The definition of stationary distribution is that <math>\pi</math> is the stationary distribution of the chain if <math>\pi=\pi~P</math>, where <math>\pi</math> is a probability vector.
+
The definition of stationary distribution is that <math>\pi</math> is the stationary distribution of the chain if <math>\pi=\pi~P</math>, where <math>\pi</math> is a probability vector. For every n<math>>=</math>0.
  
 +
However, just because X<sub>''n''</sub> ~ <math>\pi</math> for every n<math>>=</math>0 does ''not'' mean every state is independently identically distributed.
 +
 +
'''Limiting distribution''' of the chain refers the transition matrix that reaches the stationary state. If the lim(n-> infinite)P^n -> c, where c is a constant, then, we say this Markov chain is coverage;  otherwise, it's not coverage.
  
 
Example: Find the stationary distribution of P= <math>\left[ {\begin{array}{ccc}
 
Example: Find the stationary distribution of P= <math>\left[ {\begin{array}{ccc}
Line 4,076: Line 4,549:
  
 
Solving the 4 equations for the 3 unknowns gets, <br>
 
Solving the 4 equations for the 3 unknowns gets, <br>
<math>\pi_{0}~=6/25</math>, <math>\pi_{1}~=2/5</math>, and <math>\pi_{2}~=9/25</math> <br>
+
<math>\pi_{0}=\frac {6}{25}</math>, <math>\pi_{1}~=\frac {2}{5}</math>, and <math>\pi_{2}~=\frac {9}{25}</math> <br>
 
Therefore <math>\pi=\left[ {\begin{array}{ccc}
 
Therefore <math>\pi=\left[ {\begin{array}{ccc}
6/25 & 2/5 & 9/25 \end{array} } \right]</math>
+
\frac {6}{25} & \frac {2}{5} & \frac {9}{25} \end{array} } \right]</math>
 +
 
 +
The above two examples are designed to solve for the stationary distribution of the matrix P however they also give us the limiting distribution of the matrices as we have mentioned earlier that the stationary distribution is equivalent to the limiting distribution.
 +
 
 +
'''Alternate Method of Computing the Stationary Distribution''' <br>
 +
 
 +
Recall that if <math>\lambda v = A v</math>, then <math>\lambda</math> is the eigenvalue of <math>A</math> corresponding to the eigenvector <math>v</math><br>
 +
 
 +
By definition of stationary distribution,  <math>\pi = \pi  P</math><br>
 +
Taking the transpose, <math>\pi^T  = (\pi  P)^T </math><br>
 +
then  <math>I \pi^T  = P^T \pi^T \Rightarrow (P^T-I) \pi^T = 0 </math><br>
 +
So <math>\pi^T </math> is an eigenvector of <math>P^T</math> with corresponding eigenvalue 1. <br>
 +
 
 +
the transpose method to calculate the pi matrix probability.
 +
 
 +
It is thus possible to compute the stationary distribution by taking the eigenvector of the transpose of the transition matrix corresponding to 1, and normalize it such that all elements are non-negative and sum to one so that the elements satisfy the definition of a stationary distribution. The transformed vector is still an eigenvector since a linear transformation of an eigenvector is still within the eigenspace. Taking the transpose of this transformed eigenvector gives the stationary distribution. <br>
 +
 
 +
 
 +
 
  
 
<span style="background:#F5F5DC">
 
<span style="background:#F5F5DC">
 
Generating Random Initial distribution<br>
 
Generating Random Initial distribution<br>
 
<math>\mu~=rand(1,n)</math><br>
 
<math>\mu~=rand(1,n)</math><br>
<math>\mu~=\mu/\Sigma(\mu)</math></span>
+
<math>\mu~=\frac{\mu}{\Sigma(\mu)}</math></span>
  
 
<span style="background:#F5F5DC">
 
<span style="background:#F5F5DC">
 
Doubly Stochastic Matrices<br></span>
 
Doubly Stochastic Matrices<br></span>
 
We say that the transition matrix <math>\, P=(p_{ij})</math> is doubly stochastic if both rows and columns sum to 1, i.e.,<br>
 
We say that the transition matrix <math>\, P=(p_{ij})</math> is doubly stochastic if both rows and columns sum to 1, i.e.,<br>
<math>\, \sum_{i} p_{ij} = \sum_{j} p_{ij} = 1 </math><br>
+
<math>\, \sum_{i} p_{ji} = \sum_{j} p_{ij} = 1 </math><br>
 
It is easy to show that the stationary distribution of an nxn doubly stochastic matrix P is:<br>
 
It is easy to show that the stationary distribution of an nxn doubly stochastic matrix P is:<br>
 
<math> (\frac{1}{n}, \ldots , \frac{1}{n}) </math>
 
<math> (\frac{1}{n}, \ldots , \frac{1}{n}) </math>
  
 +
=== Properties of Markov Chain ===
  
== Class 14 - Thursday June 20th 2013 ==
+
A Markov chain is a random process usually characterized as '''memoryless''': the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes.
  
Example: Find the stationary distribution of P= <math>\left[ {\begin{array}{ccc}
+
1. Reducibility <br>
1/3 & 1/3 & 1/3 \\
+
State <math>j</math> is said to be accessible from State <math>i</math> (written <math>i \rightarrow j</math>) if a system started in State <math>i</math> has a non-zero probability of transitioning into State <math>j</math> at some point. Formally, State <math>j</math> is accessible from State <math>i</math> if there exists an integer <math>n_{ij} \geq 0</math> such that <br>
1/4 & 3/4 & 0 \\
+
<math>P(X_{n_{ij}} =j \vert X_0 =i) > 0</math><br>
1/2 & 0 & 1/2 \end{array} } \right]</math>
 
  
<math>\pi=\pi*p</math>
+
This integer is allowed to be different for each pair of states, hence the subscripts in <math>n_{ij}</math>. By allowing n to be zero, every state is defined to be accessible from itself.<br />
  
Solve the system of linear equations to find
+
2. Periodicity <br>
 +
State <math>i</math> has period <math>k</math> if any return to State <math>i</math> must occur in multiples of <math>k</math> time steps. Formally, the period of a state is defined as <br>
 +
<math>k= \gcd\{n:P(X_n =j \vert X_0 =i)>0\} </math><br />
  
<math>\pi=(1/3,4/9,2/9)</math>
+
3. Recurrence <br>
 +
State <math>i</math> is said to be transient if, given that we start in State <math>i</math>, there is a non-zero probability that we will never return to <math>i</math>. Formally, let the random variable <math>T_i</math> be the first return time to State <math>i</math> (the "hitting time"): <br>
 +
<math>T_i = \min\{n \geq 1:X_n=i \vert X_0=i\}</math><br />
  
<math>\lambda*u=A*u</math>
+
(The properties are from
 +
http://www2.math.uu.se/~takis/L/McRw/mcrw.pdf)
  
<math>\pi</math><sup>T</sup>= p<sup>T</sup><math>\pi</math><sup>T</sup>
+
CHAPMAN-KOLMOGOROV EQUATION
==== MatLab Code ====
+
For all <math>n</math> and <math>m</math>, and any state <math>i</math> and <math>j</math>,
 +
<math>P^{n+m}(X_n+m = j \vert X_0 =i)= \sum_{k} P^n(X_1 = k \vert X_0 = i)*P^m(X_1 = j \vert X_0 =k)</math>
  
==== Limiting distribution ====
+
== Class 14 - Thursday June 20th 2013 ==
A Markov chain has limiting distribution <math>\pi</math> if
 
  
limit n-> infinity p<sup><math>\pi</math></sup>=P= <math>\left[ {\begin{array}{ccc}
+
Example: Find the stationary distribution of <math> P= \left[ {\begin{array}{ccc}
\pi & ... & \pi \end{array} } \right]</math>
+
\frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\[6pt]
 +
\frac{1}{4} & \frac{3}{4} & 0 \\[6pt]
 +
\frac{1}{2} & 0 & \frac{1}{2} \end{array} } \right]</math>
  
=== MatLab Code ===
+
<math>\displaystyle \pi=\pi  p</math>
<pre style='font-size:14px'>
 
>> P=[1/3, 1/3, 1/3; 1/4, 3/4, 0; 1/2, 0, 1/2]
 
  
P =
+
Solve the system of linear equations to find a stationary distribution
  
    0.3333    0.3333    0.3333
+
<math>\displaystyle \pi=(\frac{1}{3},\frac{4}{9}, \frac{2}{9})</math>
    0.2500    0.7500        0
 
    0.5000        0    0.5000
 
  
>> P^2
+
Note that <math>\displaystyle \pi=\pi  p</math> looks similar to eigenvectors/values <math>\displaystyle \lambda vec{u}=A vec{u}</math>
  
ans =
+
<math>\pi</math> can be considered as an eigenvector of P with eigenvalue = 1. But note that the vector <math>vec{u}</math> is a column vector and o we need to transform our <math>\pi</math> into a column vector.
  
    0.3611    0.3611    0.2778
+
<math>=> \pi</math><sup>T</sup>= P<sup>T</sup><math>\pi</math><sup>T</sup><br/>
    0.2708    0.6458    0.0833
+
Then <math>\pi</math><sup>T</sup> is an eigenvector of P<sup>T</sup> with eigenvalue = 1. <br />
    0.4167    0.1667    0.4167
+
MatLab tips:[V D]=eig(A), where D is a diagonal matrix of eigenvalues and V is a matrix of eigenvectors of matrix A<br />
 +
==== MatLab Code ====
 +
<pre style='font-size:14px'>
  
>> P^3
+
P = [1/3 1/3 1/3; 1/4 3/4 0; 1/2 0 1/2]
  
ans =
+
pii = [1/3 4/9 2/9]
  
    0.3495    0.3912    0.2593
+
[vec val] = eig(P')            %% P' is the transpose of matrix P
    0.2934    0.5747    0.1319
+
    0.3889    0.2639    0.3472
+
vec(:,1) = [-0.5571 -0.7428 -0.3714]      %% this is in column form
  
>> P^10
+
a = -vec(:,1)
  
ans =
+
>> a =  
 +
[0.5571 0.7428 0.3714]   
  
    0.3341    0.4419    0.2240
+
%% a is in column form
    0.3314    0.4507    0.2179
 
    0.3360    0.4358    0.2282
 
  
>> P^100
+
%% Since we want this vector a to sum to 1, we have to scale it
  
ans =
+
b = a/sum(a)
 +
 
 +
>> b =
 +
[0.3333 0.4444 0.2222] 
 +
 
 +
%% b is also in column form
 +
 
 +
%% Observe that b' = pii
  
    0.3333    0.4444    0.2222
 
    0.3333    0.4444    0.2222
 
    0.3333    0.4444    0.2222
 
 
</pre>
 
</pre>
 +
</br>
 +
==== Limiting distribution ====
 +
A Markov chain has limiting distribution <math>\pi</math> if
  
 +
<math>\lim_{n\to \infty} P^n= \left[ {\begin{array}{ccc}
 +
\pi_1 \\
 +
\vdots \\
 +
\pi_n \\
 +
\end{array} } \right]</math>
  
Example: Find the stationary distribution of P= <math>\left[ {\begin{array}{ccc}
+
That is <math>\pi_j=\lim[P^n]_{ij}</math> exists and is independent of i.<br/>
0 & 1 & 0 \\
+
 
0 & 0 & 1 \\
+
A Markov Chain is convergent if and only if its limiting distribution exists. <br/>
1 & 0 & 0 \end{array} } \right]</math>
+
 
 +
If the limiting distribution <math>\pi</math> exists, it must be equal to the stationary distribution.<br/>
  
<math>\pi=\pi~P</math><br>
+
This convergence means that,in the long run(n to infinity),the probability of finding the <br/>
 +
Markov chain in state j is approximately <math>\pi_j</math> no matter in which state <br/>
 +
the chain began at time 0. <br/>
  
0*<math>\pi</math><sub>0</sub>+0*<math>\pi</math><sub>1</sub>+1*<math>\pi</math><sub>2</sub>=<math>\pi</math><sub>0</sub><br>
+
'''Example:'''
1*<math>\pi</math><sub>0</sub>+0*<math>\pi</math><sub>1</sub>+0*<math>\pi</math><sub>2</sub>=<math>\pi</math><sub>1</sub><br>
+
For a transition matrix <math> P= \left [ \begin{matrix}
0*<math>\pi</math><sub>0</sub>+1*<math>\pi</math><sub>1</sub>+0*<math>\pi</math><sub>2</sub>=<math>\pi</math><sub>1</sub><br>
+
0 & 1 & 0 \\[6pt]
<math>\pi</math><sub>0</sub>+<math>\pi</math><sub>1</sub>+<math>\pi</math><sub>2</sub>=1<br>
+
0 & 0 & 1 \\[6pt]
 +
1 & 0 & 0 \\[6pt]
 +
\end{matrix} \right] </math>
 +
, find stationary distribution.<br/>
 +
We have:<br/>
 +
<math>0\times \pi_0+0\times \pi_1+1\times \pi_2=\pi_0</math><br/>
 +
<math>1\times \pi_0+0\times \pi_1+0\times \pi_2=\pi_1</math><br/>
 +
<math>0\times \pi_0+1\times \pi_1+0\times \pi_2=\pi_2</math><br/>
 +
<math>\,\pi_0+\pi_1+\pi_2=1</math><br/>
 +
this gives <math>\pi = \left [ \begin{matrix}
 +
\frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\[6pt]
 +
\end{matrix} \right] </math> <br/>
 +
However, there does not exist a limiting distribution. <math> \pi </math> is stationary but is not limiting.<br/>
 +
<br/>
 +
In general, there are chains with stationery distributions that don't converge, this means that they have stationary distribution but are not limiting.<br/>
  
 
=== MatLab Code ===
 
=== MatLab Code ===
 
<pre style='font-size:14px'>
 
<pre style='font-size:14px'>
 
MATLAB
 
MATLAB
>> P=[0 1 0;0 0 1; 1 0 0]
+
>> P=[0, 1, 0;0, 0, 1; 1, 0, 0]
  
 
P =
 
P =
Line 4,229: Line 4,753:
  
 
>> %P^10000 = P^10003
 
>> %P^10000 = P^10003
>> % This chain does not have limiting distribution, it has a stationary distribution.
+
>> % This chain does not have limiting distribution, it has a stationary distribution.
  
 
This chain does not converge, it has a cycle.
 
This chain does not converge, it has a cycle.
 
</pre>
 
</pre>
  
 +
The first condition of limiting distribution is satisfied; however, the second condition where <math>\pi</math><sub>j</sub> has to be independent of i (i.e. all rows of the matrix are the same) is not met.<br>
 +
 +
This example shows the distinction between having a stationary distribution and convergence(having a limiting distribution).Note: <math>\pi=(1/3,1/3,1/3)</math> is the stationary distribution as <math>\pi=\pi*p</math>. However, upon repeatedly multiplying P by itself (repeating the step <math>P^n</math> as n goes to infinite) one will note that the results become a cycle (of period 3) of the same sequence of matrices. The chain has a stationary distribution, but does not converge to it. Thus, there is no limiting distribution.<br>
 +
 +
'''Example:'''
 +
 +
<math> P= \left [ \begin{matrix}
 +
\frac{4}{5} & \frac{1}{5} & 0 & 0 \\[6pt]
 +
\frac{1}{5} & \frac{4}{5} & 0 & 0 \\[6pt]
 +
0 & 0 & \frac{4}{5} & \frac{1}{5} \\[6pt]
 +
0 & 0 & \frac{1}{10} & \frac{9}{10} \\[6pt]
 +
\end{matrix} \right] </math>
 +
 +
This chain converges but is not a limiting distribution as the rows are not the same and it doesn't converge to the stationary distribution.<br />
 +
<br />
 +
Double Stichastic Matrix: a double stichastic matrix is a matrix whose all colums sum to 1 and all rows sum to 1.<br />
 +
If a given transition matrix is a double stichastic matrix with n colums and n rows, then the stationary distribution matrix has all<br/>
 +
elements equals to 1/n.<br/>
 +
<br/>
 +
Example:<br/>
 +
For a stansition matrix <math> P= \left [ \begin{matrix}
 +
0 & \frac{1}{2} & \frac{1}{2} \\[6pt]
 +
\frac{1}{2} & 0 & \frac{1}{2} \\[6pt]
 +
\frac{1}{2} & \frac{1}{2} & 0 \\[6pt]
 +
\end{matrix} \right] </math>,<br/>
 +
We have:<br/>
 +
<math>0\times \pi_0+\frac{1}{2}\times \pi_1+\frac{1}{2}\times \pi_2=\pi_0</math><br/>
 +
<math>\frac{1}{2}\times \pi_0+0\times \pi_1+\frac{1}{2}\times \pi_2=\pi_1</math><br/>
 +
<math>\frac{1}{2}\times \pi_0+\frac{1}{2}\times \pi_1+0\times \pi_2=\pi_2</math><br/>
 +
<math>\pi_0+\pi_1+\pi_2=1</math><br/>
 +
The stationary distribution is <math>\pi = \left [ \begin{matrix}
 +
\frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\[6pt]
 +
\end{matrix} \right] </math> <br/>
 +
 +
 +
<span style="font-size:20px;color:red">The following contents are problematic. Please correct it if possible.</span><br />
 +
Suppose we're given that the limiting distribution <math> \pi </math> exists for  stochastic matrix P, that is, <math> \pi = \pi \times P </math> <br>
 +
 +
WLOG assume P is diagonalizable, (if not we can always consider the Jordan form and the computation below is exactly the same. <br>
 +
 +
Let <math> P = U  \Sigma  U^{-1} </math> be the eigenvalue decomposition of <math> P </math>, where <math>\Sigma = diag(\lambda_1,\ldots,\lambda_n) ; |\lambda_i| > |\lambda_j|, \forall i < j </math><br>
 +
 +
Suppose <math> \pi^T = \sum a_i u_i </math> where <math> a_i \in \mathcal{R} </math> and <math> u_i </math> are eigenvectors of <math> P </math> for <math> i = 1\ldots n </math> <br>
 +
 +
By definition: <math> \pi^k = \pi P = \pi P^k \implies \pi = \pi(U  \Sigma  U^{-1}) (U  \Sigma  U^{-1} ) \ldots (U  \Sigma  U^{-1}) </math> <br>
 +
 +
Therefore <math> \pi^k = \sum a_i  \lambda_i^k  u_i </math> since <math> <u_i , u_j> = 0, \forall i\neq j </math>. <br>
 +
 +
Therefore <math> \lim_{k \rightarrow \infty} \pi^k = \lim_{k \rightarrow \infty}  \lambda_i^k  a_1  u_1 = u_1 </math>
 +
 +
=== MatLab Code ===
 +
<pre style='font-size:14px'>
 +
>> P=[1/3, 1/3, 1/3; 1/4, 3/4, 0; 1/2, 0, 1/2]      % We input a matrix P. This is the same matrix as last class. 
 +
 +
P =
 +
 +
    0.3333    0.3333    0.3333
 +
    0.2500    0.7500        0
 +
    0.5000        0    0.5000
 +
 +
>> P^2
 +
 +
ans =
 +
 +
    0.3611    0.3611    0.2778
 +
    0.2708    0.6458    0.0833
 +
    0.4167    0.1667    0.4167
 +
 +
>> P^3
 +
 +
ans =
 +
 +
    0.3495    0.3912    0.2593
 +
    0.2934    0.5747    0.1319
 +
    0.3889    0.2639    0.3472
 +
 +
>> P^10
 +
 +
The example of code and an example of stand distribution, then the all the pi probability in the matrix are the same.
 +
 +
ans =
 +
 +
    0.3341    0.4419    0.2240
 +
    0.3314    0.4507    0.2179
 +
    0.3360    0.4358    0.2282
 +
 +
>> P^100                                  % The stationary distribution is [0.3333 0.4444 0.2222]  since values keep unchanged.
 +
 +
ans =
 +
 +
    0.3333    0.4444    0.2222
 +
    0.3333    0.4444    0.2222
 +
    0.3333    0.4444    0.2222
 +
 +
 +
>> [vec val]=eigs(P')                    % We can find the eigenvalues and eigenvectors from the transpose of matrix P.
 +
 +
vec =
 +
 +
  -0.5571    0.2447    0.8121
 +
  -0.7428  -0.7969  -0.3324
 +
  -0.3714    0.5523  -0.4797
 +
 +
 +
val =
 +
 +
    1.0000        0        0
 +
        0    0.6477        0
 +
        0        0  -0.0643
 +
 +
>> a=-vec(:,1)                            % The eigenvectors can be mutiplied by (-1) since  λV=AV  can be written as  λ(-V)=A(-V)
 +
 +
a =
 +
 +
    0.5571
 +
    0.7428
 +
    0.3714
 +
 +
>> sum(a)
 +
 +
ans =
 +
 +
    1.6713
 +
 +
>> a/sum(a)
 +
 +
ans =
 +
 +
    0.3333
 +
    0.4444
 +
    0.2222
 +
</pre>
 +
 +
This is <math>\pi_j = lim[p^n]_(ij)</math> exist and is independent of i
 +
 +
Another example:
 +
 +
 +
Find the stationary distribution of P= <math>\left[ {\begin{array}{ccc}
 +
0.5 & 0 & 0 \\
 +
1 & 0 & 0.5 \\
 +
0 & 1 & 0.5 \end{array} } \right]</math>
 +
 +
<math>\pi=\pi~P</math><br>
 +
 +
<math>\pi=</math> [<math>\pi</math><sub>0</sub>, <math>\pi</math><sub>1</sub>, <math>\pi</math><sub>2</sub>]<br>
 +
 +
The system of equations is:
 +
 +
<math>0.5\pi_0+1\pi_1+0\pi_2= \pi_0=> 2\pi_1 = \pi_0</math><br>
 +
<math>0\pi_0+0\pi_1+1\pi_2= \pi_1 => \pi_1=\pi_2</math><br>
 +
<math>0\pi_0+0.5\pi_1+0.5\pi_2 = \pi_2 => \pi_1 = \pi_2</math><br>
 +
<math>\pi_0+\pi_1+\pi_2 = 1</math><br>
 +
 +
<math>2\pi_1+\pi_1+\pi_1 = 4\pi_1 = 1</math>, which gives <math>\pi_1=\frac {1}{4}</math> <br>
 +
Also, <math>\pi_1 = \pi_2 = \frac {1}{4}</math> <br>
 +
So, <math>\pi = [\frac{1}{2}, \frac{1}{4}, \frac{1}{4}]</math> <br>
 +
 +
=== Ergodic Chain ===
 +
 +
A Markov chain is called an ergodic chain if it is possible to go from every state to every state (not necessarily in one move). For instance, note that we can claim a Markov chain is ergodic if it is possible to somehow start at any state i and end at any state j in the matrix. We could have a chain with states 0, 1, 2, 3, 4 where it is not possible to go from state 0 to state 4 in just one step. However, it may be possible to go from 0 to 1, then from 1 to 2, then from 2 to 3, and finally 3 to 4 so we can claim that it is possible to go from 0 to 4 and this would satisfy a requirement of an ergodic chain. The example below will further explain this concept.
 +
 +
'''Note:'''if there's a finite number N then every other state can be reached in N steps.
 +
'''Note:'''Also note that a Ergodic chain is irreducible (all states communicate) and aperiodic (d = 1). An Ergodic chain is promised to have a stationary and limiting distribution.<br/>
 +
'''Ergodicity:''' A state i is said to be ergodic if it is aperiodic and positive recurrent. In other words, a state i is ergodic if it is recurrent, has a period of 1 and it has finite mean recurrence time. If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic.<br/>
 +
'''Some more:'''It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state. A model has the ergodic property if there's a finite number N such that any state can be reached from any other state in exactly N steps. In case of a fully connected transition matrix where all transitions have a non-zero probability, this condition is fulfilled with N=1.<br/>
 +
 +
 +
====Example====
 +
<math> P= \left[ \begin{matrix}
 +
\frac{1}{3} \; & \frac{1}{3} \; & \frac{1}{3} \\ \\
 +
\frac{1}{4} \; & \frac{3}{4} \; & 0 \\ \\
 +
\frac{1}{2} \; & 0 \; & \frac{1}{2}
 +
\end{matrix} \right] </math><br />
 +
 +
 +
<math> \pi=\left[ \begin{matrix}
 +
\frac{1}{3} & \frac{4}{9} & \frac{2}{9}
 +
\end{matrix} \right] </math><br />
 +
 +
 +
There are three states in this example.
 +
 +
[[File:ab.png]]
 +
 +
In this case, state a can go to state a, b, or c; state b can go to state a, b, or c; and state c can go to state a, b, or c so it is possible to go from every state to every state. (Although state b cannot directly go into c in one move, it must go to a, and then to c.).
 +
 +
A k-by-k matrix indicates that the chain has k states.
 +
 +
- Ergodic Markov chains are irreducible.
 +
 +
- A Markov chain is called a '''regular''' chain if some power of the transition matrix has only positive elements.<br />
 +
*Any transition matrix that has no zeros determines a regular Markov chain
 +
*However, it is possible for a regular Markov chain to have a transition matrix that has zeros.
 +
<br />
 +
For example, recall the matrix of the Land of Oz
 +
 +
<math>P = \left[ \begin{matrix}
 +
& R & N & S \\
 +
R & 1/2 & 1/4 & 1/4 \\
 +
N & 1/2 & 0 & 1/2 \\
 +
S & 1/4 & 1/4 & 1/2 \\
 +
\end{matrix} \right]</math><br />
 +
 +
=== Theorem ===
 +
An ergodic Markov chain has a unique stationary distribution <math>\pi</math>. The limiting distribution exists and is equal to <math>\pi</math><br/>
 +
Note: Ergodic Markov Chain is irreducible, aperiodic and positive recurrent.
 +
 +
Example: Consider the markov chain of <math>\left[\begin{matrix}0 & 1 \\ 1 & 0\end{matrix}\right]</math>, the stationary distribution is obtained by solving <math>\pi P = \pi</math>, getting <math>\pi=[0.5, 0.5]</math>, but from the assignment we know that it does not converge, ie. there is no limiting distribution, because the Markov chain is not aperiodic and cycle repeats <math>P^2=\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]</math> and <math>P^3=\left[\begin{matrix}0 & 1 \\ 1 & 0\end{matrix}\right]</math>
 +
 +
'''Another Example'''
 +
 +
<math>P=\left[ {\begin{array}{ccc}
 +
\frac{1}{4} & \frac{3}{4} \\[6pt]
 +
\frac{1}{5} & \frac{4}{5} \end{array} } \right]</math> <br>
 +
 +
 +
 +
[[File:Untitled*.jpg]]
 +
 +
This matrix means that there are two points in the space, let's call them a and b<br/>
 +
Starting from a, the probability of staying in a is 1/4 <br/>
 +
Starting from a, the probability of going from a to b is 3/4 <br/>
 +
Starting from b, the probability of going from b to a is 1/5 <br/>
 +
Starting from b, the probability of staying in b is 4/5 <br/>
 +
 +
Solve the equation <math> \pi = \pi P </math> <br>
 +
<math> \pi_0 = .25 \pi_0 + .2 \pi_1 </math> <br>
 +
<math> \pi_1 = .75 \pi_0 + .8 \pi_1 </math> <br>
 +
<math> \pi_0 + \pi_1 = 1 </math> <br>
 +
Solving this system of equations we get: <br>
 +
<math> \pi_0 = \frac{4}{15} \pi_1 </math> <br>
 +
<math> \pi_1 = \frac{15}{19} </math> <br>
 +
<math> \pi_0 = \frac{4}{19} </math> <br>
 +
<math> \pi = [\frac{4}{19}, \frac{15}{19}] </math> <br>
 +
<math> \pi </math> is the long run distribution, and this is also a limiting distribution.
 +
 +
We can use the stationary distribution to compute the expected waiting time to return to state 'a' <br/>
 +
given that we start at state 'a' and so on.. Formula for this will be : <math> E[T_{i,i}]=\frac{1}{\pi_i}</math><br/>
 +
In the example above this will mean that that expected waiting time for the markov process to return to<br/>
 +
state 'a' given that we start at state 'a' is 19/4.<br/>
 +
 +
definition of limiting distribution: when the stationary distribution is convergent, it is a limiting distribution.<br/>
 +
 +
remark:satisfied balance of <math>\pi_i P_{ij} = P_{ji} \pi_j</math>, so there is other way to calculate the step probability.
 +
 +
=== MatLab Code ===
 +
In the following, P is the transition matrix. eye(n) refers to the n by n Identity matrix. L is the Laplacian matrix, L = (I - P). The Laplacian matrix will have at least 1 zero Eigenvalue. For every 0 in the diagonal, there is a component. If there is exactly 1 zero Eigenvalue, then the matrix is connected and has only 1 component. The number of zeros in the Laplacian matrix is the number of parts in your graph/process. If there is more than one zero on the diagonal of this matrix, means there is a disconnect in the graph.
 +
 +
 +
<pre style='font-size:14px'>
 +
>> P=[1/3, 1/3, 1/3; 1/4, 3/4, 0; 1/2, 0, 1/2]
 +
 +
P =
 +
 +
    0.3333    0.3333    0.3333
 +
    0.2500    0.7500        0
 +
    0.5000        0    0.5000
 +
 +
>> eye(3) %%returns 3x3 identity matrix
 +
 +
ans =
 +
 +
    1    0    0
 +
    0    1    0
 +
    0    0    1
 +
 +
>> L=(eye(3)-P) 
 +
 +
L =
 +
 +
    0.6667  -0.3333  -0.3333
 +
  -0.2500    0.2500        0
 +
  -0.5000        0    0.5000
 +
 +
>> [vec val]=eigs(L)
 +
 +
vec =
 +
 +
  -0.7295    0.2329    0.5774
 +
    0.2239  -0.5690    0.5774
 +
    0.6463    0.7887    0.5774
 +
 +
 +
val =
 +
 +
    1.0643        0        0
 +
        0    0.3523        0
 +
        0        0  -0.0000
 +
 +
%% Only one value of zero on the diagonal means the chain is connected
 +
 +
>> P=[0.8, 0.2, 0, 0;0.2, 0.8, 0, 0; 0, 0, 0.8, 0.2; 0, 0, 0.1, 0.9]
 +
 +
P =
 +
 +
    0.8000    0.2000        0        0
 +
    0.2000    0.8000        0        0
 +
        0        0    0.8000    0.2000
 +
        0        0    0.1000    0.9000
 +
 +
>> eye(4)
 +
 +
ans =
 +
 +
    1    0    0    0
 +
    0    1    0    0
 +
    0    0    1    0
 +
    0    0    0    1
 +
 +
>> L=(eye(4)-P)
 +
 +
L =
 +
 +
    0.2000  -0.2000        0        0
 +
  -0.2000    0.2000        0        0
 +
        0        0    0.2000  -0.2000
 +
        0        0  -0.1000    0.1000
 +
 +
>> [vec val]=eigs(L)
 +
 +
vec =
 +
 +
    0.7071        0    0.7071        0
 +
  -0.7071        0    0.7071        0
 +
        0    0.8944        0    0.7071
 +
        0  -0.4472        0    0.7071
 +
 +
 +
val =
 +
 +
    0.4000        0        0        0
 +
        0    0.3000        0        0
 +
        0        0  -0.0000        0
 +
        0        0        0  -0.0000
 +
 +
%% Two values of zero on the diagonal means there are two 'islands' of chains
 +
 +
</pre>
 +
 +
<math>\Pi</math> satisfies detailed balance if <math>\Pi_i P_{ij}=P_{ji} \Pi_j</math>. Detailed balance guarantees that <math>\Pi</math> is stationary distribution.<br />
 +
 +
'''Adjacency matrix''' - a matrix <math>A</math> that dictates which states are connected and way of portraying which vertices in the matrix are adjacent. Two vertices are adjacent if there exists a path between them of length 1.If we compute <math>A^2</math>, we can know which states are connected with paths of length 2.<br />
 +
 +
A '''Markov chain''' is called an irreducible chain if it is possible to go from every state to every state (not necessary in one more).<br />
 +
Theorem: An '''ergodic''' Markov chain has a unique stationary distribution <math>\pi</math>. The limiting distribution exists and is equal to <math>\pi</math>. <br />
 +
 +
 +
Markov process satisfies detailed balance  if and only if it is a '''reversible''' Markov process
 +
where P is the matrix of  Markov transition.<br />
 +
 +
Satisfying the detailed balance condition guarantees that <math>\pi</math> is stationary distributed.
 +
 +
<math> \pi </math> satisfies detailed balance if <math> \pi_i P_{ij} = P_{ji} \pi_j </math> <br>
 +
which is the same as the Markov process equation.
 +
 +
Example in the class:
 +
<math>P= \left[ {\begin{array}{ccc}
 +
\frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\[6pt]
 +
\frac{1}{4} & \frac{3}{4} & 0 \\[6pt]
 +
\frac{1}{2} & 0 & \frac{1}{2} \end{array} } \right]</math>
 +
 +
and  <math>\pi=(\frac{1}{3},\frac{4}{9}, \frac{2}{9})</math>
 +
 +
<math>\pi_1 P_{1,2} = 1/3 \times 1/3 = 1/9,\, P_{2,1} \pi_2 = 1/4 \times 4/9 = 1/9 \Rightarrow \pi_1 P_{1,2} = P_{2,1} \pi_2 </math><br>
 +
 +
<math>\pi_2 P_{2,3} = 4/9 \times 0 = 0,\, P_{3,2} \pi_3 = 0 \times 2/9 = 0 \Rightarrow \pi_2 P_{2,3} = P_{3,2} \pi_3</math><br>
 +
Remark:Detailed balance of <math> \pi_i \times Pij = Pji \times \pi_j</math> , so there is other way to calculate the step probability<br />
 +
<math>\pi</math> is stationary but is not limiting.
 +
Detailed balance implies that <math>\pi</math> = <math>\pi</math> * P as shown in the proof and guarantees that <math>\pi</math> is stationary distribution.
 +
 +
== Class 15 - Tuesday June 25th 2013 ==
 +
=== Announcement ===
 +
Note to all students, the first half of today's lecture will cover the midterm's solution; however please do not post the solution on the Wikicoursenote.<br />
 +
 +
====Detailed balance====
 +
<div style="border:2px solid black">
 +
<b>Definition (from wikipedia)</b>
 +
The principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions): At equilibrium, each elementary process should be equilibrated by its reverse process.
 +
</div>
 +
Let <math>P</math> be the transition probability matrix of a Markov chain. If there exists a distribution vector <math>\pi</math> such that <math>\pi_i \cdot P_{ij}=P_{ji} \cdot \pi_j, \; \forall i,j</math>, then the Markov chain is said to have '''detailed balance'''. A detailed balanced Markov chain must have <math>\pi</math> given above as a stationary distribution, that is <math>\pi=\pi P</math>, where <math>\pi</math> is a 1 by n matrix and P is a n by n matrix.<br>
 +
 +
 +
need to remember:
 +
'''Proof:''' <br>
 +
<math>\; [\pi P]_j = \sum_i \pi_i P_{ij} =\sum_i P_{ji}\pi_j =\pi_j\sum_i P_{ji} =\pi_j  ,\forall j</math>
 +
 +
:Note: Since <math>\pi_j</math> is a sum of column j and we can do this proof for every element in matrix P; in general, we can prove <math>\pi=\pi P</math> <br>
 +
 +
Hence <math>\pi</math> is always a stationary distribution of <math>P(X_{n+1}=j|X_n=i)</math>, for every n.
 +
 +
In other terms, <math> P_{ij} = P(X_n = j| X_{n-1} = i) </math>, where <math>\pi_j</math> is the equilibrium probability of being in state j and <math>\pi_i</math> is the equilibrium probability of being in state i. <math>P(X_{n-1} = i) = \pi_i</math> is equivalent to <math>P(X_{n-1} = i,  Xn = j)</math> being symmetric in i and j.
 +
 +
Keep in mind that the detailed balance is a sufficient but not required condition for a distribution to be stationary.
 +
i.e. A distribution satisfying the detailed balance is stationary, but a stationary distribution does not necessarily satisfy the detailed balance.
 +
 +
In the stationary distribution <math>\pi=\pi P</math>, in the proof the sum of the p is equal 1 so the <math>\pi P=\pi</math>.
 +
 +
=== PageRank (http://en.wikipedia.org/wiki/PageRank) ===
 +
 +
*PageRank is a probability distribution used to represent the likelihood that a person randomly clicking on links will arrive at any particular page. PageRank can be calculated for collections of documents of any size.
 +
*PageRank is a link-analysis algorithm developed by and named after Larry Page from Google; used for measuring a website's importance, relevance and popularity.
 +
*PageRank is a graph containing web pages and their links to each other.
 +
*Many social media sites use this (such as Facebook and Twitter)
 +
*It can also be used to find criminals (ie. theives, hackers, terrorists, etc.) by finding out the links.
 +
This is what made Google the search engine of choice over Yahoo, Bing, etc.- What made Google's search engine a huge success is not its search function, but rather the algorithm it used to rank the pages. (Ex. If we come up with 100 million search results, how do you list them by relevance and importance so the users can easily find what they are looking for. Most users will not go past the first 3 or so search pages to find what they are looking for. It is this ability to rank pages that allow Google to remain more popular than Yahoo, Bink, AskJeeves, etc.). It should be noted that after using the PageRank algorithm, Google uses other processes to filter results.<br/>
 +
 +
<br />'''The order of importance'''<br />
 +
1. A web page is more important if many other pages point to it<br />
 +
2. The more important a web page is, the more weight should be assigned to its outgoing links<br/ >
 +
3. If a webpage has many outgoing links, then its links have less value (ex: if a page links to everyone, like 411, it is not as important as pages that have incoming links)<br />
 +
 +
<br />
 +
[[File:diagram.jpg]]
 +
<math>L=
 +
\left[ {\begin{matrix}
 +
0 & 0 & 0 & 0 & 0 \\
 +
1 & 0 & 0 & 0 & 0 \\
 +
1 & 1 & 0 & 1 & 0 \\
 +
0 & 1 & 0 & 0 & 1 \\
 +
0 & 0 & 0 & 0 & 0 \end{matrix} } \right]</math>
 +
 +
The first row indicates who gives a link to 1. As shown in the diagram, nothing gives a link to 1, and thus it is all zero.
 +
The second row indicates who gives a link to 2. As shown in the diagram, only 1 gives a link to 2, and thus column 1 is a 1 for row 2, and the rest are all zero.
 +
 +
ie: According to the above example <br/ >
 +
Page 3 is the most important since it has the most links pointing to it, therefore more weigh should be placed on its outgoing links.<br/ >
 +
Page 4 comes after page 3 since it has the second most links pointing to it<br/ >
 +
Page 2 comes after page 4 since it has the third most links pointing to it<br/>
 +
Page 1 and page 5 are the least important since no links point to them<br/ >
 +
As page 1 and page 2 have the most outgoing links, then their links have less value compared to the other pages. <br/ >
 +
 +
:<math>
 +
Lij = \begin{cases}
 +
1, & \text{if j has a link to i} \\
 +
0, & \text{otherwise}
 +
\end{cases}</math>
 +
 +
<br />
 +
<math>C_j=</math> The number of outgoing links of page <math>j</math>:
 +
<math>C_j=\sum_i L_{ij}</math>
 +
(i.e. sum of entries in column j)<br />
 +
<br />
 +
<math>P_j</math> is the rank of page <math>j</math>.<br />
 +
Suppose we have <math>N</math> pages, <math>P</math> is a vector containing ranks of all pages.<br />
 +
- <math>P</math> is a <math>N \times 1</math> vector.
 +
 +
- <math>P_i</math> counts the number of incoming links of page <math>i</math>
 +
<math>P_i=\sum_j L_{ij}</math> <br />(i.e. sum of entries in row i)
 +
 +
For each row of <math>L</math>, if there is a 1 in the third column, it means page three point to that page.
 +
 +
However, we should not define the rank of the page this way because links shouldn't be treated the same. The weight of the link is based on different factors. One of the factors is the importance of the page that link is coming from. For example, in this case, there are two links going to Page 4: one from Page 2 and one from Page 5. So far, both links have been treated equally with the same weight 1. But we must rerate the two links based on the importance of the pages they are coming from.
 +
 +
A PageRank results from a mathematical algorithm based on the webgraph, created by all World Wide Web pages as nodes and hyperlinks as edges, taking into consideration authority hubs such as cnn.com or usa.gov. The rank value indicates an importance of a particular page. A hyperlink to a page counts as a vote of support. (This would be represented in our diagram as an arrow pointing towards the page. Hence in our example, Page 3 is the most important, since it has the most 'votes of support). The PageRank of a page is defined recursively and depends on the number and PageRank metric of all pages that link to it ("incoming links"). A page that is linked to by many pages with high PageRank receives a high rank itself. If there are no links to a web page, then there is no support for that page (In our example, this would be Page 1 and Page 5).
 +
(source:http://en.wikipedia.org/wiki/PageRank#Description)
 +
 +
For those interested in PageRank, here is the original paper by Google co-founders Brin and Page: http://infolab.stanford.edu/pub/papers/google.pdf
 +
 +
=== Example of Page Rank Application in Real Life ===
 +
 +
'''Page Rank checker'''
 +
- This is a free service to check Google™ page rank instantly via online PR checker or by adding a PageRank checking button to the web pages.
 +
  <font size="3">(http://www.prchecker.info/check_page_rank.php)</font>
 +
 +
 +
GoogleMatrix G = d * [ (Hyperlink Matrix H) + (Dangling Nodes Matrix A) ] + ((1-d)/N) * (NxN Matrix U of all 1's)
 +
 +
 +
[[File:Google matrix.png]]
 +
 +
 +
(source: https://googledrive.com/host/0B2GQktu-wcTiaWw5OFVqT1k3bDA/)
 +
 +
== Class 16 - Thursday June 27th 2013 ==
 +
 +
=== Page Rank ===
 +
*<math>
 +
L_{ij} = \begin{cases}
 +
1, & \text{if j has a link to i }  \\
 +
0, & \text{otherwise} \end{cases} </math> <br/>
 +
 +
*<math>C_j</math>: number of outgoing links for page j, where <math>c_j=\sum_i L_{ij}</math>
 +
 +
P is N by 1 vector contains rank of all N pages; for page i, the rank is <math>P_i</math>
 +
 +
<math>P_i= (1-d) + d\cdot \sum_j \frac {L_{ji}P_j}{c_j}</math>
 +
pi is the rank of a new created page(that no one knows about) is 0 since <math>L_ij</math> is 0 <br/>
 +
where 0 < d < 1 is constant (in original page rank algorithm d = 0.8), and <math>L_{ij}</math> is 1 if j has link to i, 0 otherwise.
 +
 +
Note that the rank of a page is proportional to the number of its incoming links and inversely proportional to the number of its outgoing links.
 +
 +
Interpretation of the formula:<br/>
 +
1) sum of L<sub>ij</sub> is the total number of incoming links<br/>
 +
2) the above sum is weighted by page rank of the pages that contain the link to i (P<sub>j</sub>) i.e. if a high-rank page points to page i, then this link carries more weight than links from lower-rank pages.<br/>
 +
3) the sum is then weighted by the inverse of the number of outgoing links from the pages that contain links to i (c<sub>j</sub>). i.e. if a page has more outgoing links than other pages then its links carry less weight.<br/>
 +
4) finally, we take a linear combination of the page rank obtained from above and a constant 1. This ensures that every page has a rank greater than zero.<br/>
 +
5) d is the damping factor.  It represents the probability a user, at any page, will continue clicking to another page.<br/>
 +
If there is no damping (i.e. d=1), then there are no assumed outgoing links for nodes with no links. However, if there is damping (e.g. d=0.8), then these nodes are assumed to have links to all pages in the web.
 +
 +
Note that this is a system of N equations with N unknowns.<br/>
 +
 +
<math>c_j</math> is the number of outgoing links, less outgoing links means more important.<br/>
 +
 +
 +
Let D be a diagonal N by N matrix such that <math> D_{ii}</math> = <math>c_i</math>
 +
 +
Note: Ranks are arbitrary, all we want to know is the order. That is, we want to know how important the page rank relative to the other pages and are not interested in the value of the page rank.
 +
 +
<math>D=
 +
\left[ {\begin{matrix}
 +
c_1 & 0 & ... & 0  \\
 +
0 & c_2 & ...  &  0  \\
 +
0 & 0 & ... &  0 \\
 +
0 & 0 & ... & c_N \end{matrix} } \right]</math>
 +
 +
Then <math>P=~(1-d)e+dLD^{-1}P</math>, P is an iegenvector of matrix A corresponding to an eigenvalue equal to 1.<br/> where e =[1 1 ....]<sup>T</sup> , i.e. a N by 1 vector.<br/>
 +
We assume that rank of all N pages sums to N. The sum of rank of all N pages can be any number, as long as the ranks have certain propotion. <br/>
 +
i.e. e<sup>T</sup> P = N, then <math>~\frac{e^{T}P}{N} = 1</math>
 +
 +
 +
D<sup>-1</sup> will be: 
 +
 +
D<sup>-1</sup><math>=
 +
\left[ {\begin{matrix}
 +
\frac {1}{c_1} & 0 & ... & 0  \\
 +
0 & \frac {1}{c_2} & ...  &  0  \\
 +
0 & 0 & ... &  0 \\
 +
0 & 0 & ... & \frac {1}{c_N} \end{matrix} } \right]</math>
 +
 +
<math>P=~(1-d)e+dLD^{-1}P</math>  where <math>e=\begin{bmatrix}
 +
1\\
 +
1\\
 +
...\\
 +
1
 +
\end{bmatrix}</math>
 +
 +
<math>P=(1-d)~\frac{ee^{T}P}{N}+dLD^{-1}P</math>
 +
 +
<math>P=[(1-d)~\frac{ee^T}{N}+dLD^{-1}]P</math>
 +
 +
<math>=> P=A*P</math>
 +
 +
'''Explanation of an eigenvector'''
 +
 +
An eigenvector is a non-zero vector '''v''' such that when multiplied by a square matrix, A, the result is a scalar times the vector '''v''' itself. <br>
 +
That is, A*v = c*v. Where c is the eigenvalue of A corresponding to the eigenvector v. In our case of Page Rank, the eigenvalue c=1. <br>
 +
 +
We obtain that <math>P=AP</math> where <math>A=(1-d)~\frac{ee^T}{N}+dLD^{-1}</math><br/>
 +
Thus, <math>P</math> is an eigenvector of <math>P</math> correspond to an eigen value equals 1.<br/>
 +
 +
 +
Since,
 +
L is a N*N matrix,
 +
D<sup>-1</sup> is a N*N matrix,
 +
P is a N*1 matrix <br/>
 +
Then as a result, <math>LD^{-1}P</math> is a N*1 matrix. <br/>
 +
 +
N is a N*N matrix, d is a constant between 0 and 1.
 +
 +
'''P=AP'''<br />
 +
P is an eigenvector of A with corresponding eigenvalue equal to 1.<br>
 +
'''P<sup>T</sup>=P<sup>T</sup>A<sup>T</sup><br>'''
 +
Notice that all entries in A are non-negative and each row sums to 1. Hence A satisfies the definition of a transition probability matrix.<br>
 +
P<sup>T</sup> is the stationary distribution of a Markov Chain with transition probability matrix A<sup>T</sup>.
 +
 +
We can consider A to be the matrix describing all possible movements following links on the internet, and P<sup>t</sup> as the probability of being on any given webpage if we were on the internet long enough.
 +
 +
Definition of rank page and proof it steps by steps, it shows with 3 n*n matrix and and one n*1 matrix and a constant d between 0 to 1.
 +
p is the stationary distribution so p=Ap.
 +
 +
=== Damping Factor "d" ===
 +
 +
The PageRank assumes that any imaginary user who is randomly clicking on links will eventually stop clicking. The probability, at any step, that the person will keep on clicking is a damping factor, <math>d</math>. After many studies, the approximation of <math>d</math> is 0.85. Other values for <math>d</math> have been used in class and may appear on assignments and exams.<br/>
 +
 +
In addition, <math>d</math> is a vector of ranks that are arbitrary. For example the rank can be [1 3 2], or [10 30 20], or [0.1 0.3 0.2]. All three of these examples are relative/equivalent since they are ranks, we could even have [1 10 3]. Therefore, <math>d</math> must have a relative rank.<br/>
 +
 +
So <math>P_1 + P_2 + \cdots + P_n=N</math> <br/>
 +
Which is equivalent to:
 +
<math>e^{T}P= [1 \cdots 1] [P_1 \cdots P_n]^T </math> <br/>
 +
Where <math>[1 \cdots 1]</math> is a 1 scalar vector and <math>[P_1 \cdots P_n]^T</math> is a rank vector. <br/>
 +
So <math>e^{T}P=N -> (e^{T}P)/N = 1 </math>
 +
 +
===Examples===
 +
<span style="background:#F5F5DC">
 +
 +
==== Example 1 ====
 +
 +
 +
[[File:eg1.jpg]]
 +
<br />
 +
</span>
 +
<math>L=
 +
\left[ {\begin{matrix}
 +
0 & 0 & 1  \\
 +
1 & 0 & 0  \\
 +
0 & 1 & 0 \end{matrix} } \right]\;c=
 +
\left[ {\begin{matrix}
 +
1 & 1 & 1 \end{matrix} } \right]\;D=
 +
\left[ {\begin{matrix}
 +
1 & 0 & 0  \\
 +
0 & 1 & 0  \\
 +
0 & 0 & 1 \end{matrix} } \right]</math>
 +
 +
<pre style='font-size:14px'>
 +
 +
MATLAB Code
 +
 +
d=0.8
 +
N=3
 +
A=(1-d)*ones(N)/N+d*L*pinv(D) #pinv: Moore-Penrose inverse (pseudoinverse) of symbolic matrix
 +
We use the pinv(D) function [pseudo-inverse] instead of the inv(D) function because in
 +
the case of a non-invertible matrix, it would not crash the program. 
 +
[vec val]=eigs(A) (eigen-decomposition)
 +
a=-vec(:,1) (find the eigenvector equals to 1)
 +
a=a/sum(a) (normalize a)
 +
or to show that A transpose is a stationary transition matrix
 +
(transpose(A))^200 will be the same as a=a/sum(a)
 +
</pre>
 +
 +
'''NOTE:''' Changing the value of d, does not change the ranking order of the pages.
 +
 +
By looking at each entry after normalizing a, we can tell the ranking order of each page.<br>
 +
<span style="background:#F5F5DC">
 +
 +
c = [1 1 1] since there are 3 pages, each page is one way recurrent to each other and there is only one outgoing for each page. Hence, D is a 3x3 standard diagonal matrix.
 +
 +
==== Example 2 ====
 +
 +
[[File:Screen_shot_2013-07-02_at_3.43.04_AM.png]]
 +
 +
 +
<math>L=
 +
\left[ {\begin{matrix}
 +
0 & 0 & 1  \\
 +
1 & 0 & 1  \\
 +
0 & 1 & 0 \end{matrix} } \right]\;
 +
c=
 +
\left[ {\begin{matrix}
 +
1 & 1 & 2 \end{matrix} } \right]\;
 +
D=
 +
\left[ {\begin{matrix}
 +
1 & 0 & 0  \\
 +
0 & 1 & 0  \\
 +
0 & 0 & 2 \end{matrix} } \right]</math>
 +
 +
<pre style='font-size:14px'>
 +
 +
Matlab code
 +
 +
>> L=[0 0 1;1 0 1;0 1 0];
 +
>> C=sum(L);
 +
>> D=diag(C);
 +
>> d=0.8;
 +
>> N=3;
 +
>> A=(1-d)*ones(N)/N+d*L*pinv(D);
 +
>> [vec val]=eigs(A)
 +
 +
vec =
 +
 +
  -0.3707            -0.3536 + 0.3536i  -0.3536 - 0.3536i
 +
  -0.6672            -0.3536 - 0.3536i  -0.3536 + 0.3536i
 +
  -0.6461            0.7071            0.7071         
 +
 +
 +
val =
 +
 +
  1.0000                  0                  0         
 +
        0            -0.4000 - 0.4000i        0         
 +
        0                  0            -0.4000 + 0.4000i
 +
 +
>> a=-vec(:,1)
 +
 +
a =
 +
 +
    0.3707
 +
    0.6672
 +
    0.6461
 +
 +
>> a=a/sum(a)
 +
 +
a =
 +
 +
    0.2201
 +
    0.3962
 +
    0.3836
 +
</pre>
 +
'''NOTE:''' Page 2 is the most important page because it has 2 incomings. Similarly, page 3 is more important than page 1 because page 3 has the incoming result from page 2.
 +
 +
This example is similar to the first example, but here, page 3 can go back to page 2, so the matrix of the outgoing matrix, the third column of the D matrix is 3 in the third row. And we use the code to calculate the p=Ap. Therefore 2, 3, 1 is the order of importance.
 +
 +
==== Example 3 ====
 +
 +
[[File:eg 3.jpg]]<br>
 +
 +
<math>L=
 +
\left[ {\begin{matrix}
 +
0 & 1 & 0  \\
 +
1 & 0 & 1  \\
 +
0 & 1 & 0 \end{matrix} } \right]\;
 +
c=
 +
\left[ {\begin{matrix}
 +
1 & 2 & 1 \end{matrix} } \right]\;
 +
D=
 +
\left[ {\begin{matrix}
 +
1 & 0 & 0  \\
 +
0 & 2 & 0  \\
 +
0 & 0 & 1 \end{matrix} } \right]</math>
 +
 +
<math>d=0.8</math><br>
 +
<math>N=3</math><br>
 +
 +
 +
</span>
 +
this example is the second page have 2 outgoings.
 +
 +
 +
 +
Another Example:
 +
 +
Consider: 1 -> ,<-2 ->3
 +
 +
<math>L=
 +
\left[ {\begin{matrix}
 +
0 & 1 & 0  \\
 +
1 & 0 & 0  \\
 +
0 & 1 & 0 \end{matrix} } \right]\;
 +
c=
 +
\left[ {\begin{matrix}
 +
1 & 1 & 1 \end{matrix} } \right]\;
 +
D=
 +
\left[ {\begin{matrix}
 +
1 & 0 & 0  \\
 +
0 & 1 & 0  \\
 +
0 & 0 & 1 \end{matrix} } \right]</math>
 +
 +
==== Example 4 ====
 +
 +
<math>1 \leftrightarrow 2 \rightarrow 3 \leftrightarrow 4 </math>
 +
<br />
 +
<br />
 +
<math>L=
 +
\left[ {\begin{matrix}
 +
0 & 1 & 0 & 0 \\
 +
1 & 0 & 0 & 0 \\
 +
0 & 1 & 0 & 1 \\
 +
0 & 0 & 1 & 0 \end{matrix} } \right]\;</math><br />
 +
 +
'''Matlab Code:'''<br>
 +
<pre style='font-size:16px'>
 +
>> L=L= [0 1 0 0;1 0 0 0;0 1 0 1;0 0 1 0];
 +
>> C=sum(L);
 +
>> D=diag(C);
 +
>> d=0.8;
 +
>> N=4;
 +
>> A=(1-d)*ones(N)/N+d*L*pinv(D);
 +
>> [vec val]=eigs(A);
 +
>> a=vec(:,1);
 +
>> a=a/sum(a)
 +
    a =
 +
        0.1029 <- Page 1
 +
        0.1324 <- Page 2
 +
        0.3971 <- Page 3
 +
        0.3676 <- Page 4
 +
 +
        % Therefore the PageRank for this matrix is: 3,4,2,1
 +
</pre>
 +
<br>
 +
 +
==== Example 5 ====
 +
 +
<math>L=
 +
\left[ {\begin{matrix}
 +
0 & 1 & 0 & 1 \\
 +
1 & 0 & 1 & 1 \\
 +
1 & 0 & 0 & 1 \\
 +
1 & 0 & 0 & 0 \end{matrix} } \right]</math>
 +
 +
<math>c=
 +
\left[ {\begin{matrix}
 +
3 & 1 & 1 & 3 \end{matrix} } \right]</math>
 +
 +
<math>D=
 +
\left[ {\begin{matrix}
 +
3 & 0 & 0 & 0 \\
 +
0 & 1 & 0 & 0 \\
 +
0 & 0 & 1 & 0  \\
 +
0 & 0 & 0 & 3 \end{matrix} } \right]</math>
 +
 +
<pre style='font-size:14px'>
 +
 +
Matlab code
 +
 +
>> L= [0 1 0 1; 1 0 1 1; 1 0 0 1;1 0 0 0];
 +
>> d = 0.8;
 +
>> N = 4;
 +
>> C = sum(L);
 +
>> D = diag(C);
 +
>> A=(1-d)*ones(N)/N+d*L*pinv(D);
 +
>> [vec val]=eigs(A);
 +
>> a=vec(:,1);
 +
>> a=a/sum(a)
 +
 +
a =
 +
 +
    0.3492
 +
    0.3263
 +
    0.1813
 +
    0.1431
 +
</pre>
 +
 +
==== Example 6 ====
 +
<math>L=
 +
\left[ {\begin{matrix}
 +
0 & 1 & 0 & 0 & 1\\
 +
1 & 0 & 0 & 0 & 0\\
 +
0 & 1 & 0 & 0 & 0\\
 +
0 & 1 & 1 & 0 & 1\\
 +
0 & 0 & 0 & 1 & 0 \end{matrix} } \right]</math>
 +
<br />
 +
 +
'''Matlab Code:'''<br />
 +
<pre style="font-size:16px">
 +
>> d=0.8;
 +
>> L=[0 1 0 0 1;1 0 0 0 0;0 1 0 0 0;0 1 1 0 1;0 0 0 1 0];
 +
>> c=sum(L);
 +
>> D=diag(c);
 +
>> N=5;
 +
>> A=(1-d)*ones(N)/N+d*L*pinv(D);
 +
>> [vec val]=eigs(A);
 +
>> a=-vec(:,1);
 +
>> a=a/sum(a) 
 +
    a =
 +
        0.1933 <- Page 1
 +
        0.1946 <- Page 2
 +
        0.0919 <- Page 3
 +
        0.2668 <- Page 4
 +
        0.2534 <- Page 5
 +
 +
        % Therefore the PageRank for this matrix is: 4,5,2,1,3
 +
</pre>
 +
<br>
 +
 +
== Class 17 - Tuesday July 2nd 2013 ==
 +
=== Markov Chain Monte Carlo (MCMC) ===
 +
 +
===Introduction===
 +
It is, in general, very difficult to simulate the value of a random vector X whose component random variables are dependent. We will present a powerful approach for generating a vector whose distribution is approximately that of X. This approach, called the Markov Chain Monte Carlo Methods, has the added significance of only requiring that the mass(or density) function of X be specified up to a multiplicative constant, and this, we will see, is of great importance in applications.
 +
(referenced by Sheldon M.Ross,Simulation)
 +
The basic idea used here is to generate a Markov Chain whose stationary distribution is the same as the target distribution.
 +
 +
====Definition:====
 +
Markov Chain
 +
A Markov Chain is a special form of stochastic process in which <math>\displaystyle X_t</math> depends only on <math> \displaystyle X_{t-1}</math>.
 +
 +
For example,
 +
:<math>\displaystyle f(X_1,...X_n)= f(X_1)f(X_2|X_1)f(X_3|X_2)...f(X_n|X_{n-1})</math>
 +
A random Walk is the best example  of a Markov process
 +
 +
<br>'''Transition Probability:'''<br>
 +
The probability of going from one state to another state.
 +
:<math>p_{ij} = \Pr(X_{n}=j\mid X_{n-1}= i). \,</math>
 +
 +
<br>'''Transition Matrix:'''<br>
 +
For n states, transition matrix P is an <math>N \times N</math> matrix with entries <math>\displaystyle P_{ij}</math> as below:
 +
Markov Chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. The state of the chain after a large number of steps is then used as a sample of the desired distribution. The quality of the sample improves as a function of the number of steps. (http://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo)</span>
 +
 +
<a style="color:red" href="http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-165.pdf">some notes form UCb</a>
 +
 +
'''One of the main purposes of MCMC''' : to simulate samples from a joint distribution where the joint random variables are dependent. In general, this is not easily sampled from. Other methods learned in class allow us to simulate i.i.d random variables, but not dependent variables . In this case, we could sample non-independent random variables using a Markov Chain. Its Markov properties help to simplify the simulation process.
 +
 +
 +
<b>Basic idea:</b>  Given a probability distribution <math>\pi</math> on a set <math>\Omega</math>, we want to generate random elements of <math>\Omega</math> with distribution <math>\pi</math>. MCMC does that by constructing a Markov Chain with stationary distribution <math>\pi</math> and simulating the chain. After a large number of iterations, the Markov Chain will reach its stationary distribution. By sampling from the Markov chain for large amount of iterations, we are effectively sampling from the desired distribution as the Markov Chain would converge to its stationary distribution <br/>
 +
 +
Idea: generate a Markov chain whose stationary distribution is the same as target distribution. <br/>
 +
 +
 +
'''Notes'''
 +
 +
# Regardless of the chosen starting point, the Markov Chain will converge to its stationary distribution (if it exists). However, the time taken for the chain to converge depends on its chosen starting point. Typically, the burn-in period is longer if the chain is initialized with a value of low probability density.
 +
# Markov Chain Monte Carlo can be used for sampling from a distribution, estimating the distribution, and computing the mean and optimization (e.g. simulated annealing, more on that later).
 +
# Markov Chain Monte Carlo is used to sample using “local” information. It is used as a generic “problem solving technique” to solve decision/optimization/value problems, but is not necessarily very efficient.
 +
# MCMC methods do not suffer as badly from the "curse of dimensionality" that badly affects efficiency in the acceptance-rejection method. This is because a point is always generated at each time-step according to the Markov Chain regardless of how many dimensions are introduced.
 +
# The goal when simulating with a Markov Chain is to create a chain with the same stationary distribution as the target distribution.
 +
# The MCMC method is usually used in continuous cases but a discrete example is given below.
 +
 +
 +
'''Some properties of the stationary distribution <math>\pi</math>'''
 +
 +
<math>\pi</math> indicates the proportion of time the process spends in each of the states 1,2,...,n. Therefore <math>\pi</math> satisfies the following two inequalities: <br>
 +
 +
# <math>\pi_j = \sum_{i=1}^{n}\pi_i P_{ij}</math> <br /> This is because <math>\pi_i</math> is the proportion of time the process spends in state i, and <math>P_{ij}</math> is the probability the process transition out of state i into state j. Therefore, <math>\pi_i p_{ij}</math> is the proportion of time it takes for the process to enter state j. Therefore, <math>\pi_j</math> is the sum of this probability over overall states i.
 +
#<math> \sum_{i=1}^{n}\pi_i= 1 </math> as <math>\pi</math> shows the proportion of time the chain is in each state. If we view it as the probability of the chain being in state i at time t for t sufficiently large, then it should sum to one as the chain must be in one of the states.
 +
 +
====Motivation example====
 +
- Suppose we want to generate a random variable X according to distribution <math>\pi=(\pi_1, \pi_2,  ...  , \pi_m)</math> <br/>
 +
X can take m possible different values from <math>{1,2,3,\cdots, m}</math><br />
 +
- We want to generate <math>\{X_t: t=0, 1, \cdots\}</math> according to <math>\pi</math><br />
 +
 +
Suppose our example is of a bias die. <br/>
 +
Now we have m=6, <math>\pi=[0.1,0.1,0.1,0.2,0.3,0.2]</math>, <math>X \in [1,2,3,4,5,6]</math><br/>
 +
 +
Suppose <math>X_t=i</math>. Consider an arbitrary probability transition matrix Q with entry <math>q_{ij}</math> being the probability of moving to state j from state i. (<math>q_{ij}</math> can not be zero.) <br/>
 +
 +
<math> \mathbf{Q} =
 +
\begin{bmatrix}
 +
q_{11} & q_{12} & \cdots & q_{1m} \\
 +
q_{21} & q_{22} & \cdots & q_{2m} \\
 +
\vdots & \vdots & \ddots & \vdots \\
 +
q_{m1} & q_{m2} & \cdots & q_{mm}
 +
\end{bmatrix}
 +
</math> <br/>
 +
 +
 +
We generate Y = j according to the i-th row of Q. Note that the i-th row of Q is a probability vector that shows the probability of moving to any state j from the current state i, i.e.<math>P(Y=j)=q_{ij}</math><br />
 +
 +
In the following algorithm: <br>
 +
<math>q_{ij}</math> is the <math>ij^{th}</math> entry of matrix Q. It is the probability of Y=j given that <math>x_t = i</math>. <br/>
 +
<math>r_{ij}</math> is the probability of accepting Y as <math>x_{t+1}</math>. <br/>
 +
 +
 +
'''How to get the acceptance probability?'''
 +
 +
If <math>\pi </math> is the stationary distribution, then it must satisfy the detailed balance condition:<br/>
 +
If <math>\pi_i P_{ij}</math> = <math>\pi_j P_{ji}</math><br/>then <math>\pi </math> is the stationary distribution of the chain
 +
 +
Since <math>P_{ij}</math> = <math>q_{ij} r_{ij}</math>, we have <math>\pi_i q_{ij} r_{ij}</math> = <math>\pi_j q_{ji} r_{ji}</math>.<br/>
 +
We want to find a general solution: <math>r_{ij} = a(i,j) \pi_j q_{ji}</math>, where a(i,j) = a(j,i).<br/>
 +
 +
'''Recall'''
 +
<math>r_{ij}</math> is the probability of acceptance, thus it must be that <br/>
 +
 +
1.<math>r_{ij} = a(i,j)</math> <math>\pi_j q_{ji} </math>≤1, then we get: <math>a(i,j) </math>≤ <math>1/(\pi_j q_{ji})</math>
 +
 +
2. <math>r_{ji} = a(j,i) </math> <math>\pi_i q_{ij} </math> ≤ 1, then we get: <math>a(j,i)</math> ≤ <math>1/(\pi_i q_{ij})</math>
 +
 +
So we choose a(i,j) as large as possible, but it needs to satisfy the two conditions above.<br/>
 +
 +
<math>a(i,j) = \min \{\frac{1}{\pi_j q_{ji}},\frac{1}{\pi_i q_{ij}}\} </math><br/>
 +
 +
Thus, <math> r_{ij} = \min \{\frac{\pi_j q_{ji}}{\pi_i q_{ij}}, 1\} </math><br/>
 +
 +
'''Note''':
 +
1 is the upper bound to make r<sub>ij</sub> a probability
 +
 +
 +
'''Algorithm:'''  <br/>
 +
*<math> (*) P(Y=j) = q_{ij} </math>. <math>\frac{\pi_j q_{ji}}{\pi_i q_{ij}}</math> is a positive ratio.
 +
 +
*<math> r_{ij} = \min \{\frac{\pi_j q_{ji}}{\pi_i q_{ij}}, 1\} </math> <br/>
 +
*<math>
 +
x_{t+1} = \begin{cases}
 +
Y, & \text{with probability } r_{ij} \\
 +
x_t, & \text{otherwise} \end{cases} </math> <br/>
 +
* go back to the first step (*)  <br/>
 +
 +
We can compare this with the Acceptance-Rejection model we learned before. <br/>
 +
* <math>U</math> ~ <math>Uniform(0,1)</math> <br/>
 +
* If <math>U < r_{ij}</math>, then accept. <br/>
 +
EXCEPT that a point is always generated at each time-step. <br>
 +
 +
The algorithm generates a stochastic sequence that only depends on the last state, which is a Markov Chain.<br>
 +
 +
====Metropolis Algorithm====
 +
 +
'''Proposition: ''' Metropolis works:
 +
 +
The <math>P_{ij}</math>'s from Metropolis Algorithm satisfy detailed balance property w.r.t <math>\pi</math> . i.e. <math>\pi_i P_{ij} = \pi_j P_{ji}</math>. The new Markov Chain has a stationary distribution <math>\pi</math>. <br/>
 +
'''Remarks:''' <br/>
 +
1) We only need to know ratios of values of <math>\pi_i</math>'s.<br/>
 +
2) The MC might converge to <math>\pi</math> at varying speeds depending on the proposal distribution and the value the chain is initialized with<br/>
 +
 +
 +
This algorithm generates <math>\{x_t:  t=0,...,m\}</math>. <br/>
 +
In the long run, the marginal distribution of <math> x_t </math> is the stationary distribution <math>\underline{\Pi} </math><br>
 +
<math>\{x_t: t = 0, 1,...,m\}</math> is a Markov chain with probability transition matrix (PTM), P.<br>
 +
 +
This is a Markov Chain since <math> x_{t+1} </math> only depends on <math> x_t </math>, where <br>
 +
<math> P_{ij}= \begin{cases}
 +
q_{ij} r_{ij}, & \text{if }i \neq j  (q_{ij} \text{is the probability of generating j from i and } r_{ij} \text{ is the probiliity of accepting)}\\[6pt]
 +
1 - \displaystyle\sum_{k \neq i} q_{ik} r_{ik}, & \text{if }i = j \end{cases} </math><br />
 +
 +
<math>q_{ij}</math> is the probability of generating state j; <br/>
 +
<math> r_{ij}</math> is the probability of accepting state j as the next state. <br/>
 +
 +
Therefore, the final probability of moving from state i to j when i does not equal to j is <math>q_{ij}*r_{ij}</math>. <br/>
 +
For the probability of moving from state i to state i, we deduct all the probabilities of moving from state i to any j that are not equal to i, therefore, we get the second probability.
 +
 +
===Proof of the proposition:===
 +
 +
A good way to think of the detailed balance equation is that they balance the probability from state i to state j with that from state j to state i.
 +
We need to show that the stationary distribition of the Markov Chain is <math>\underline{\Pi}</math>, i.e. <math>\displaystyle \underline{\Pi} = \underline{\Pi}P</math><br />
 +
<div style="text-size:20px">
 +
Recall<br/>
 +
If a Markov chain satisfies the detailed balance property, i.e. <math>\displaystyle \pi_i P_{ij} = \pi_j P_{ji} \, \forall i,j</math>, then <math>\underline{\Pi}</math> is the stationary distribution of the chain.<br /><br />
 +
</div>
 +
 +
'''Proof:'''
 +
 +
WLOG, we can assume that <math>\frac{\pi_j q_{ji}}{\pi_i q_{ij}}<1</math><br/>
 +
 +
LHS:<br />
 +
<math>\pi_i P_{ij} = \pi_i q_{ij} r_{ij} = \pi_i q_{ij} \cdot \min(\frac{\pi_j q_{ji}}{\pi_i q_{ij}},1) = \cancel{\pi_i q_{ij}} \cdot \frac{\pi_j q_{ji}}{\cancel{\pi_i q_{ij}}} = \pi_j q_{ji}</math><br />
 +
 +
RHS:<br />
 +
Note that by our assumption, since <math>\frac{\pi_j q_{ji}}{\pi_i q_{ij}}<1</math>, its reciprocal <math>\frac{\pi_i q_{ij}}{\pi_j q_{ji}} \geq 1</math><br />
 +
So <math>\displaystyle \pi_j P_{ji} = \pi_ j q_{ji} r_{ji} = \pi_ j q_{ji} \cdot \min(\frac{\pi_i q_{ij}}{\pi_j q_{ji}},1) =  \pi_j q_{ji} \cdot 1 = \pi_ j q_{ji}</math><br />
 +
 +
Hence LHS=RHS
 +
 +
If we assume that <math>\frac{\pi_j q_{ji}}{\pi_i q_{ij}}=1</math><br/> (essentially <math>\frac{\pi_j q_{ji}}{\pi_i q_{ij}}>=1</math>)<br/>
 +
 +
LHS:<br />
 +
<math>\pi_i P_{ij} = \pi_i q_{ij} r_{ij} = \pi_i q_{ij} \cdot \min(\frac{\pi_j q_{ji}}{\pi_i q_{ij}},1)  =\pi_i q_{ij} \cdot 1 = \pi_i q_{ij}</math><br />
 +
 +
RHS:<br />
 +
'''Note''' <br/>
 +
by our assumption, since <math>\frac{\pi_j q_{ji}}{\pi_i q_{ij}}\geq 1</math>, its reciprocal <math>\frac{\pi_i q_{ij}}{\pi_j q_{ji}} \leq 1 </math> <br />
 +
 +
So <math>\displaystyle \pi_j P_{ji} = \pi_ j q_{ji} r_{ji} = \pi_ j q_{ji} \cdot \min(\frac{\pi_i q_{ij}}{\pi_j q_{ji}},1) =  \cancel{\pi_j q_{ji}} \cdot \frac{\pi_i q_{ij}}{\cancel{\pi_j q_{ji}}} = \pi_i q_{ij}</math><br />
 +
 +
Hence LHS=RHS which indicates <math>pi_i*P_{ij} = pi_j*P_{ji}</math><math>\square</math><br /><br />
 +
 +
'''Note'''<br />
 +
1) If we instead assume <math>\displaystyle \frac{\pi_i q_{ij}}{\pi_j q_{ji}} \geq 1</math>, the proof is similar with LHS= RHS =  <math> \pi_i q_{ij} </math> <br />
 +
 +
2) If <math>\displaystyle i = j</math>, then detailed balance is satisfied trivially.<br />
 +
 +
since <math>{\pi_i q_{ij}}</math>, and <math>{\pi_j q_{ji}}</math> are smaller than one. so the above steps show the proof of  <math>\frac{\pi_i q_{ij}}{\pi_j q_{ji}}<1</math>.
 +
 +
== Class 18 - Thursday July 4th 2013 ==
 +
=== Last class ===
 +
 +
Recall: The Acceptance Probability,
 +
<math>r_{ij}=min(\frac {{\pi_j}q_{ji}}{{\pi_i}q_{ij}},1)</math> <br />
 +
 +
1)  <math>r_{ij}=\frac {{\pi_j}q_{ji}}{{\pi_i}q_{ij}}</math>, and <math>r_{ji}=1 </math>,    (<math>\frac {{\pi_j}q_{ji}}{{\pi_i}q_{ij}} < 1</math>) <br />
 +
 +
 +
2)  <math>r_{ji}=\frac {{\pi_i}q_{ij}}{{\pi_j}q_{ji}}</math>, and <math> r{ij}=1 </math>,    (<math>\frac {{\pi_j}q_{ji}}{{\pi_i}q_{ij}} \geq 1</math> ) <br />
 +
 +
===Example: Discrete Case===
 +
 +
 +
Consider a biased die,
 +
<math>\pi</math>= [0.1, 0.1, 0.2, 0.4, 0.1, 0.1]
 +
 +
We could use any <math>6 x 6 </math> matrix <math> \mathbf{Q} </math> as the proposal distribution <br>
 +
For the sake of simplicity ,using a discrete uniform distribution is the simplest. This is because all probabilities are equivalent, hence during the calculation of r, qxy and qyx will cancel each other out.
 +
 +
<math> \mathbf{Q} =
 +
\begin{bmatrix}
 +
1/6 & 1/6 & \cdots & 1/6 \\
 +
1/6 & 1/6 & \cdots & 1/6 \\
 +
\vdots & \vdots & \ddots & \vdots \\
 +
1/6 & 1/6 & \cdots & 1/6
 +
\end{bmatrix}
 +
</math> <br/>
 +
 +
 +
 +
'''Algorithm''' <br>
 +
1. <math>x_t=5</math> (sample from the 5th row, although we can initialize the chain from anywhere within the support)<br />
 +
2. Y~Unif[1,2,...,6]<br />
 +
3. <math> r_{ij} = \min \{\frac{\pi_j q_{ji}}{\pi_i q_{ij}}, 1\} = \min \{\frac{\pi_j  1/6}{\pi_i  1/6}, 1\} = \min \{\frac{\pi_j}{\pi_i}, 1\}</math><br>
 +
Note:  current state <math>i</math> is <math>X_t</math>,  the candidate state <math>j</math> is <math>Y</math>. <br>
 +
Note: since <math>q_{ij}= q_{ji}</math> for all i and j, that is, the proposal distribution is symmetric, we have <math> r_{ij} = \min \{\frac{\pi_j}{\pi_i }, 1\} </math><br/>
 +
4. U~Unif(0,1)<br/>
 +
if <math>u \leq r_{ij}</math>, X<sub>t+1</sub>=Y<br />
 +
else X<sub>t+1</sub>=X<sub>t</sub><br />
 +
go back to 2<br>
 +
 +
Notice how a point is always generated for X<sub>t+1</sub>, regardless of whether the candidate state Y is accepted <br>
 +
 +
'''Matlab'''
 +
<pre style="font-size:14px">
 +
pii=[.1,.1,.2,.4,.1,.1];
 +
x(1)=5;
 +
for ii=2:1000
 +
  Y=unidrnd(6);                %%% Unidrnd(x) is a built-in function which generates a number between (0) and (x)
 +
  r = min (pii(Y)/pii(x(ii-1)), 1);
 +
  u=rand;
 +
  if u<r
 +
    x(ii)=Y;
 +
  else
 +
    x(ii)=x(ii-1);
 +
  end
 +
end
 +
hist(x,6)    %generate histogram displaying all 1000 points
 +
xx = x(501,end);    %After 500, the chain will mix well and converge.
 +
hist(xx,6)                % The result should be better.
 +
</pre>
 +
[[File:MH_example1.jpg|300px]]
 +
 +
 +
'''NOTE:''' Generally, we generate a large number of points (say, 1500) and throw away some of the points that were first generated(say, 500). Those first points are called the [[burn-in period]]. A chain will converge to the limiting distribution eventually, but not immediately. The burn-in period is that beginning period before the chain has converged to the desired distribution. By discarding those 500 points, our data set will be more representative of the desired limiting distribution; once the burn-in period is over, we say that the chain "mixes well".
 +
 +
===Alternate Example: Discrete Case===
 +
 +
 +
Consider the weather. If it is sunny one day, there is a 5/7 chance it will be sunny the next. If it is rainy, there is a 5/8 chance it will be rainy the next.
 +
<math>\pi= [\pi_1 \ \pi_2] </math>
 +
 +
Use a discrete uniform distribution as the proposal distribution, because it is the simplest.
 +
 +
<math> \mathbf{Q} =
 +
\begin{bmatrix}
 +
5/7 & 2/7 \\
 +
3/8 & 5/8\\
 +
 +
\end{bmatrix}
 +
</math> <br/>
 +
 +
 +
 +
'''Algorithm''' <br>
 +
1. Set initial chain state: <math>X_t=1</math> (i.e. sample from the 1st row, although we could also choose the 2nd row)<br />
 +
2. Sample from proposal distribution: Y~q(y|x) = Unif[1,2]<br />
 +
3. <math> r_{ij} = \min \{\frac{\pi_j q_{ji}}{\pi_i q_{ij}}, 1\} = \min \{\frac{\pi_j  1/6}{\pi_i  1/6}, 1\} = \min \{\frac{\pi_j}{\pi_i}, 1\}</math><br>
 +
'''Note:'''  Current state <math>i</math> is <math>X_t</math>,  the candidate state <math>j</math> is <math>Y</math>. Since <math>q_{ij}= q_{ji}</math> for all i and j, that is, the proposal distribution is symmetric, we have <math> r_{ij} = \min \{\frac{\pi_j}{\pi_i }, 1\} </math>
 +
 +
4. U~Unif(0,1)<br>
 +
  If  <math>U \leq r_{ij}</math>, then<br>
 +
        <math>X_t=Y</math><br>
 +
  else<br />
 +
        <math>X_{t+1}=X_t</math><br>
 +
  end if<br />
 +
5. Go back to step 2<br>
 +
 +
 +
'''Generalization of the above framework to the continuous case'''<br>
 +
 +
In place of <math>\pi</math> use <math>f(x)</math>
 +
In place of r<sub>ij</sub> use <math>q(y|x)</math> <br>
 +
In place of r<sub>ij</sub> use <math>r(x,y)</math> <br>
 +
Here, q(y|x) is a friendly distribution that is easy to sample, usually a symmetric distribution will be preferable, such that <math>q(y|x) = q(x|y)</math> to simplify the computation for <math>r(x,y)</math>.
 +
 +
 +
'''Remarks'''<br>
 +
1. The chain may not get to a stationary distribution if the # of steps generated are small. That is it will take a very large amount of steps to step through the whole support<br>
 +
2. The algorithm can be performed with a <math>\pi</math> that is not even a probability mass function, it merely needs to be proportional to the probability mass function we wish to sample from. This is useful as we do not need to calculate the normalization factor. <br>
 +
 +
For example, if we are given <math>\pi^'=\pi\alpha=[5,10,11,2,100,1]</math>, we can normalize this vector by dividing the sum of all entries <math>s</math>.<br>
 +
However we notice that when calculating <math>r_{ij}</math>, <br>
 +
<math>\frac{\pi^'_j/s}{\pi^'_i/s}\times\frac{q_{ji}}{q_{ij}}=\frac{\pi^'_j}{\pi^'_i}\times\frac{q_{ji}}{q_{ij}}</math> <br>
 +
<math>s</math> cancels out in this case. Therefore it is not necessary to calculate the sum and normalize the vector.<br>
 +
 +
This also applies to the continuous case,where we merely need <math> f(x) </math> to be proportional to the pdf of the distribution we wish to sample from. <br>
 +
 +
===Metropolis–Hasting Algorithm===
 +
 +
'''Definition''': <br>
 +
Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. The Metropolis–Hastings algorithm can draw samples from any probability distribution P(x), provided you can compute the value of a function f(x) which is proportional to the density of P. <br>
 +
 +
 +
 +
'''Purpose''': <br>
 +
"The purpose of the Metropolis-Hastings Algorithm is to <b>generate a collection of states according to a desired distribution</b> <math>P(x)</math>. <math>P(x)</math> is chosen to be the stationary distribution of a Markov process, <math>\pi(x)</math>." <br>
 +
Source:(http://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm)<br>
 +
 +
 +
Metropolis-Hastings is an algorithm for constructing a Markov chain with a given limiting probability distribution. In particular, we consider what happens if we apply the Metropolis-Hastings algorithm repeatedly to a “proposal” distribution which has already been updated.<br>
 +
 +
 +
The algorithm was named after Nicholas Metropolis and W. K. Hastings who extended it to the more general case in 1970.<br>
 +
 +
<math>q(y|x)</math> is used instead of <math>qi,j</math>. In continuous case, we use these notation which means given state x, what's the probability of y.<br>
 +
 +
Note that the Metropolis-Hasting algorithm possess some advantageous properties. One of which is that this algorithm "can be used when \pi(x) is known up to the constant of proportionality". The second is that in this algorithm, "we do not require the conditional distribution, which, in contrast, is required for the Gibbs sampler. "
 +
Source:https://www.msu.edu/~blackj/Scan_2003_02_12/Chapter_11_Markov_Chain_Monte_Carlo_Methods.pdf
 +
 +
 +
 +
'''Differences between the discrete and continuous case of the Markov Chain''':<br/>
 +
 +
1. <math>q(y|x)</math> is used in continuous, instead of <math>q_{ij}</math> in discrete <br/>
 +
2. <math>r(x,y)</math> is used in continuous, instead of <math>r{ij}</math> in discrete <br/>
 +
3. <math>f</math> is used instead of <math>\pi</math> <br/>
 +
 +
 +
'''Build the Acceptance Ratio'''<br/>
 +
Before we consider the algorithm there are a couple general steps to follow to build the acceptance ratio:<br/>
 +
 +
a) Find the distribution you wish to use to generate samples from<br/>
 +
b) Find a candidate distribution that fits the desired distribution, q(y|x). (the proposed moves are independent of the current state)<br/>
 +
c) Build the acceptance ratio <math>\displaystyle \frac{f(y)q(x|y)}{f(x)q(y|x)}</math>
 +
 +
 +
 +
Assume that f(y) is the target distribution; Choose q(y|x) such that it is a friendly distribution and easy to sample from.<br />
 +
'''Algorithm:'''<br />
 +
 +
# Set <math>\displaystyle i = 0</math> and initialize the chain, i.e. <math>\displaystyle x_0 = s</math> where <math>\displaystyle s</math> is some state of the Markov Chain.
 +
# Sample <math>\displaystyle Y \sim q(y|x)</math>
 +
# Set <math>\displaystyle r(x,y) = min(\frac{f(y)q(x|y)}{f(x)q(y|x)},1)</math>
 +
# Sample <math>\displaystyle u \sim \text{UNIF}(0,1)</math>
 +
# If <math>\displaystyle u \leq r(x,y), x_{i+1} = Y</math><br /> Else <math>\displaystyle x_{i+1} = x_i</math>
 +
# Increment i by 1 and go to Step 2, i.e. <math>\displaystyle i=i+1</math>
 +
 +
<br> '''Note''': q(x|y) is moving from y to x and q(y|x) is moving from x to y.
 +
<br>We choose q(y|x) so that it is simple to sample from.
 +
<br>Usually, we choose a normal distribution.
 +
 +
NOTE2: The proposal q(y|x) y depends on x (is conditional on x)the current state, this makes sense ,because it's a necessary condition for MC. So the proposal should depend on x (also their supports should match) e.g q(y|x) ~ N( x, b<sup>2</sup>) here the proposal depends on x.
 +
If the next state is INDEPENDENT of the current state, then our proposal will not depend on x e.g. (A4 Q2, sampling from Beta(2,2) where the proposal was UNIF(0,1)which is independent of the current state. )
 +
 +
However, it is important to remember that even if generating the proposed/candidate state does not depend on the current state, the chain is still a markov chain.
 +
 +
<br />
 +
Comparing with previous sampling methods we have learned, samples generated from M-H algorithm are not independent of each other, since we accept future sample based on the current sample. Furthermore, unlike acceptance and rejection method, we are not going to reject any points in Metropolis-Hastings. In the equivalent of the "reject" case, we just leave the state unchanged. In other words, if we need a sample of 1000 points, we only need to generate the sample 1000 times.<br/>
 +
 +
<p style="font-size:20px;color:red;">
 +
Remarks
 +
</p>
 +
===='''Remark 1'''====
 +
<span style="text-shadow: 0px 2px 3px 3399CC;margin-right:1em;font-family: 'Nobile', Helvetica, Arial, sans-serif;font-size:16px;line-height:25px;color:3399CC">
 +
A common choice for <math>q(y|x)</math> is a normal distribution centered at x with standard deviation b. Y~<math>N(x,b^2)</math>
 +
 +
In this case, <math> q(y|x)</math> is symmetric.
 +
 +
i.e.
 +
<math>q(y|x)=q(x|y)</math><br>
 +
(we want to sample q centered at the current state.)<br>
 +
<math>q(y|x)=\frac{1}{\sqrt{2\pi}b}\,e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2b^2} (y-x)^2}</math>, (centered at x)<br>
 +
<math>q(x|y)=\frac{1}{\sqrt{2\pi}b}\,e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2b^2} (x-y)^2}</math>,(centered at y)<br>
 +
<math>\Rightarrow (y-x)^2=(x-y)^2</math><br>
 +
so <math>~q(y \mid x)=q(x \mid y)</math> <br>
 +
In this case <math>\frac{q(x \mid y)}{q(y \mid x)}=1</math> and therefore <math> r(x,y)=\min \{\frac{f(y)}{f(x)}, 1\} </math> <br/><br />
 +
This is true for any symmetric q. In general if q(y|x) is symmetric, then this algorithm is called Metropolis.<br/>
 +
When choosing function q, it makes sense to choose a distribution with the same support as the distribution you want to simulate. eg. If target is Beta, then can choose q ~ Uniform(0,1)<br>
 +
The chosen q is not necessarily symmetric. Depending on different target distribution, q can be uniform.</span>
 +
 +
===='''Remark 2'''====
 +
<span style="text-shadow: 0px 2px 3px 3399CC;margin-right:1em;font-family: 'Nobile', Helvetica, Arial, sans-serif;font-size:16px;line-height:25px;color:3399CC">
 +
The value y is accepted if u<=<math>min\{\frac{f(y)}{f(x)},1\}</math>, so it is accepted with the probability <math>min\{\frac{f(y)}{f(x)},1\}</math>.<br/>
 +
Thus, if <math>f(y)>=f(x)</math>, then y is always accepted.<br/>
 +
The higher that value of the pdf is in the vicinity of a point <math>y_1</math> , the more likely it is that a random variable will take on values around <math>y_1</math>.<br/>
 +
Therefore,we would want a high probability of acceptance for points generated near <math>y_1</math>.<br>
 +
[[File:Diag1.png‎]]<br>
 +
 +
'''Note''':<br/>
 +
If the proposal comes from a region with low density, we may or may not accept; however, we accept for sure if the proposal comes from a region with high density.<br>
 +
 +
===='''Remark 3'''====
 +
 +
One strength of the Metropolis-Hastings algorithm is that normalizing constants, which are often quite difficult to determine, can be cancelled out in the ratio <math> r </math>. For example, consider the case where we want to sample from the beta distribution, which has the pdf:<br>
 +
(also notice that Metropolis Hastings is just a special case of Metropolis algorithm)
 +
 +
<math>
 +
\begin{align}
 +
f(x;\alpha,\beta)& = \frac{1}{\mathrm{B}(\alpha,\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\end{align}
 +
</math>
 +
 +
The beta function, ''B'', appears as a normalizing constant but it can be simplified by construction of the method.
 +
 +
====='''Example'''=====
 +
 +
<math>\,f(x)=\frac{1}{\pi^{2}}\frac{1}{1+x^{2}}</math>, where <math>\frac{1}{\pi^{2}} </math> is normalization factor and <math>\frac{1}{1+x^{2}} </math> is target distribution. <br>
 +
Then, we have <math>\,f(x)\propto\frac{1}{1+x^{2}}</math>.<br>
 +
And let us take <math>\,q(x|y)=\frac{1}{\sqrt{2\pi}b}e^{-\frac{1}{2b^{2}}(y-x)^{2}}</math>.<br>
 +
Then <math>\,q(x|y)</math> is symmetric since <math>\,(y-x)^{2} = (x-y)^{2}</math>.<br>
 +
Therefore Y can be simplified.
 +
 +
 +
We get :
 +
 +
<math>\,\begin{align}
 +
\displaystyle r(x,y)
 +
& =min\left\{\frac{f(y)}{f(x)}\frac{q(x|y)}{q(y|x)},1\right\} \\
 +
& =min\left\{\frac{f(y)}{f(x)},1\right\} \\
 +
& =min\left\{ \frac{ \frac{1}{1+y^{2}} }{ \frac{1}{1+x^{2}} },1\right\}\\
 +
& =min\left\{ \frac{1+x^{2}}{1+y^{2}},1\right\}\\
 +
\end{align}
 +
</math>.
 +
 +
<br/>
 +
<math>\pi=[0.1\,0.1\,...] </math> stands for probility;<br/>
 +
<math>\pi \propto [3\,2\, 10\, 100\, 1.5] </math> is not brobility, so we take:<br/>
 +
<math>\Rightarrow \pi=1/c \times [3\, 2\, 10\, 100\, 1.5]</math> is probility where<br/>
 +
<math>\Rightarrow c=3+2+10+100+1.5 </math><br/>
 +
<br/>
 +
<br/>
 +
 +
In practice, if elements of <math>\pi</math> are functions or random variables, we need c to be the normalization factor, the summation/integration over all members of <math>\pi</math>. This is usually very difficult. Since we are taking ratios, with the Metropolis-Hasting algorithm, it is not necessary to do this.
 +
 +
<br>
 +
For example, to find the relationship between weather temperature and humidity, we only have a proportional function instead of a probability function. To make it into a probability function, we need to compute c, which is really difficult. However, we don't need to compute c as it will be cancelled out during calculation of r.<br>
 +
 +
======'''MATLAB'''======
 +
The Matlab code of the algorithm is the following :
 +
<pre style="font-size:12px">
 +
clear all
 +
close all
 +
clc
 +
b=2;
 +
x(1)=0;
 +
for i=2:10000
 +
    y=b*randn+x(i-1);
 +
    r=min((1+x(i-1)^2)/(1+y^2),1);
 +
    u=rand;
 +
    if u<r
 +
        x(i)=y;
 +
    else
 +
        x(i)=x(i-1);
 +
    end
 +
   
 +
end
 +
hist(x,100);
 +
%The Markov Chain usually takes some time to converge and this is known as the "burning time".
 +
</pre>
 +
[[File:MH_example2.jpg|300px]]
 +
 +
However, while the data does approximately fit the desired distribution, it takes some time until the chain gets to the stationary distribution. To generate a more accurate graph, we modify the code to ignore the initial points.<br>
 +
 +
'''MATLAB'''
 +
<pre style="font-size:16px">
 +
b=2;
 +
x(1)=0;
 +
for ii=2:10500
 +
y=b*randn+x(ii-1);
 +
r=min((1+x(ii-1)^2)/(1+y^2),1);
 +
u=rand;
 +
if u<=r
 +
x(ii)=y;
 +
else
 +
x(ii)=x(ii-1);
 +
end
 +
end
 +
xx=x(501:end) %we don't display the first 500 points because they don't show the limiting behaviour of the Markov Chain
 +
hist(xx,100)
 +
</pre>
 +
[[File:MH_Ex.jpg|300px]]
 +
<br>
 +
'''If a function f(x) can only take values from <math>[0,\infty)</math>, but we need to use normal distribution as the candidate distribution, then we can use <math>q=\frac{2}{\sqrt{2\pi}}*exp(\frac{-(y-x)^2}{2})</math>, where y is from <math>[0,\infty)</math>. <br>(This is essentially the pdf of the absolute value of a normal distribution centered around x)'''<br><br>
 +
 +
Example:<br>
 +
We want to sample from <math>exp(2), q(y|x)~\sim~N(x,b^2)</math><br>
 +
<math>r=\frac{f(y)}{f(x)}=\frac{2*exp^(-2y)}{2*exp^(-2x)}=exp(2*(x-y))</math><br>
 +
<math>r=min(exp(2*(x-y)),1)</math><br>
 +
 +
'''MATLAB'''
 +
<pre style="font-size:16px">
 +
x(1)=0;
 +
for ii=2:100
 +
y=2*(randn*b+abs(x(ii-1)))
 +
r=min(exp(2*(x-y)),1);
 +
u=rand;
 +
if u<=r
 +
x(ii)=y;
 +
else
 +
x(ii)=x(ii-1);
 +
end
 +
end
 +
</pre>
 +
<br>
 +
 +
'''Definition of Burn in:'''
 +
 +
Typically in a MH Algorithm, a set of values generated at at the beginning of the sequence are "burned" (discarded) after which the chain is assumed to have converged to its target distribution. In the first example listed above, we "burned" the first 500 observations because we believe the chain has not quite reached our target distribution in the first 500 observations. 500 is not a set threshold, there is no right or wrong answer as to what is the exact number required for burn-in. Theoretical calculation of the burn-in is rather difficult, in the above mentioned example, we chose 500 based on experience and quite arbitrarily. 
 +
 +
Burn-in time can also be thought of as the time it takes for the chain to reach its stationary distribution. Therefore, in this case you will disregard everything uptil the burn-in period because the chain is not stabilized yet.
 +
 +
The Metropolis–Hasting Algorithm is started from an arbitrary initial value <math>x_0</math> and the algorithm is run for many iterations until this initial state is "forgotten". These samples, which are discarded, are known as ''burn-in''. The remaining
 +
set of accepted values of <math>x</math> represent a sample from the distribution f(x).(http://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm)<br/>
 +
 +
Burn-in time can also be thought of as the time it takes for the process to reach the stationary distribution pi. Suppose it takes 5 samples after which you reach the stationary distribution. You should disregard the first five samples and consider the remaining samples as representing your target distribution f(x). <br>
 +
   
 +
Several extensions have been proposed in the literature to speed up the convergence and reduce the so called “burn-in” period.
 +
One common suggestion is to match the first few moments of q(y|x) to f(x).
 +
 +
'''Aside''': The algorithm works best if the candidate density q(y|x) matches the shape of the target distribution f(x). If a normal distribution is used as a candidate distribution, the variance parameter b<sup>2</sup> has to be tuned during the burn-in period. <br/>
 +
 +
1. If b is chosen to be too small, the chain will mix slowly (smaller proposed move, the acceptance rate will be high and the chain will converge only slowly the f(x)).
 +
 +
2. If b is chosen to be too large, the acceptance rate will be low (larger proposed move and the chain will converge only slowly the f(x)).
 +
 +
 +
 +
'''Note''':
 +
The histogram looks much nicer if we reject the points within the burning time.<br>
 +
 +
 +
Example: Use M-H method to generate sample from f(x)=2x
 +
0<x<1, 0 otherwise.
 +
 +
1) Initialize the chain with <math>x_i</math> and set <math>i=0</math>
 +
 +
2)<math>Y~\sim~q(y|x_i)</math>
 +
where our proposal function would be uniform [0,1] since it matches our original ones support.
 +
=><math>Y~\sim~Unif[0,1]</math>
 +
 +
3)consider <math>\frac{f(y)}{f(x)}=\frac{y}{x}</math>,
 +
<math>r(x,y)=min (\frac{y}{x},1)</math> since q(y|x<sub>i</sub>) and q(x<sub>i</sub>|y) can be cancelled together.
 +
 +
4)<math>X_{i+1}=Y</math> with prob <math>r(x,y)</math>,
 +
<math>X_{i+1}=X_i</math>, otherwise
 +
 +
5)<math>i=i+1</math>, go to 2
 +
 +
<br>
 +
 +
Example form wikipedia
 +
 +
===Step-by-step instructions===
 +
 +
Suppose the most recent value sampled is <math>x_t\,</math>. To follow the Metropolis–Hastings algorithm, we next draw a new proposal state <math>x'\,</math> with probability density <math>Q(x'\mid x_t)\,</math>, and calculate a value
 +
 +
:<math>
 +
a = a_1 a_2\,
 +
</math>
 +
 +
where
 +
 +
:<math>
 +
a_1 = \frac{P(x')}{P(x_t)} \,\!
 +
</math>
 +
 +
is the likelihood ratio between the proposed sample <math>x'\,</math> and the previous sample <math>x_t\,</math>, and
 +
 +
:<math>
 +
a_2 = \frac{Q(x_t \mid x')}{Q(x'\mid x_t)}
 +
</math>
 +
 +
is the ratio of the proposal density in two directions (from <math>x_t\,</math> to <math>x'\,</math> and ''vice versa'').
 +
This is equal to 1 if the proposal density is symmetric.
 +
Then the new state <math>\displaystyle x_{t+1}</math> is chosen according to the following rules.
 +
 +
:<math>
 +
\begin{matrix}
 +
\mbox{If } a \geq 1: &  \\
 +
& x_{t+1} = x',
 +
\end{matrix}
 +
</math>
 +
:<math>
 +
\begin{matrix}
 +
\mbox{else} & \\
 +
& x_{t+1} = \left\{
 +
                  \begin{array}{lr}
 +
                      x' & \mbox{ with probability }a \\
 +
                      x_t & \mbox{ with probability }1-a.
 +
                  \end{array}
 +
            \right.
 +
\end{matrix}
 +
</math>
 +
 +
The Markov chain is started from an arbitrary initial value <math>\displaystyle x_0</math> and the algorithm is run for many iterations until this initial state is "forgotten". 
 +
These samples, which are discarded, are known as ''burn-in''. The remaining set of accepted values of <math>x</math> represent a sample from the distribution <math>P(x)</math>.
 +
 +
The algorithm works best if the proposal density matches the shape of the target distribution <math>\displaystyle P(x)</math> from which direct sampling is difficult, that is <math>Q(x'\mid x_t) \approx P(x') \,\!</math>.
 +
If a Gaussian proposal density <math>\displaystyle Q</math> is used the variance parameter <math>\displaystyle \sigma^2</math> has to be tuned during the burn-in period.
 +
This is usually done by calculating the ''acceptance rate'', which is the fraction of proposed samples that is accepted in a window of the last <math>\displaystyle N</math> samples.
 +
The desired acceptance rate depends on the target distribution, however it has been shown theoretically that the ideal acceptance rate for a one dimensional Gaussian distribution is approx 50%, decreasing to approx 23% for an <math>\displaystyle N</math>-dimensional Gaussian target distribution.<ref name=Roberts/>
 +
 +
If <math>\displaystyle \sigma^2</math> is too small the chain will ''mix slowly'' (i.e., the acceptance rate will be high but successive samples will move around the space slowly and the chain will converge only slowly to <math>\displaystyle P(x)</math>).  On the other hand,
 +
if <math>\displaystyle \sigma^2</math> is too large the acceptance rate will be very low because the proposals are likely to land in regions of much lower probability density, so <math>\displaystyle a_1</math> will be very small and again the chain will converge very slowly.
 +
 +
== Class 19 - Tuesday July 9th 2013 ==
 +
'''Recall: Metropolis–Hasting Algorithm'''
 +
 +
1) <math>X_i</math> = State of chain at time i. Set <math>X_0</math> = 0<br>
 +
2) Generate proposal distribution: Y ~ q(y|x) <br>
 +
3) Set <math>\,r=min[\frac{f(y)}{f(x)}\,\frac{q(x|y)}{q(y|x)}\,,1]</math><br>
 +
4) Generate U ~ U(0,1)<br>
 +
  If <math>U<r</math>, then<br>
 +
        <math>X_{i+1} = Y</math> % i.e. we accept Y as the next point in the Markov Chain <br>
 +
  else <br>
 +
        <math>X_{i+1}</math> = <math>X_i</math><br>
 +
  End if<br>
 +
5) Set i = i + 1. Return to Step 2. <br>
 +
 +
 +
Why can we use this algorithm to generate a Markov Chain?<br>
 +
 +
<math>\,Y</math>~<math>\,q(y|x)</math> satisfies the Markov Property, as the current state does not depend on previous trials. Note that Y does not '''''have''''' to depend on X<sub>t-1</sub>; the Markov Property is satisfied as long as Y is not dependent on  X<sub>0</sub>, X<sub>1</sub>,..., X<sub>t-2</sub>. Thus, time t will not affect the choice of state.<br> 
 +
 +
 +
==='''Choosing b: 3 cases'''===
 +
If y and x have the same domain, say R, we could use normal distribution to model <math>q(y|x)</math>. <math>q(x|y)~normal(y,b^2), and q(y|x)~normal(x,b^2)</math>.
 +
In the continuous case of MCMC, <math>q(y|x)</math> is the probability of observing y, given you are observing x. We normally assume <math>q(y|x)</math> ~ N(x,b^2). A reasonable choice of b is important to ensure the MC does indeed converges to the target distribution f. If b is too small it is not possible to explore the whole support because the jumps are small. If b is large than the probability of accepting the proposed state y is small, and it is very likely that we reject the possibilities of leaving the current state, hence the chain will keep on producing the initial state of the Markov chain.
 +
 +
To be precise, we are discussing the choice of variance for the proposal distribution.Large b simply implies larger variance for our choice of proposal distribution (Gaussian) in this case. Therefore, many points will be rejected and we will generate same points many times since there are many points that have been rejected.<br>
 +
 +
In this example, <math>q(y|x)=N(x, b^2)</math><br>
 +
 +
Demonstrated as follows, the choice of b will be significant in determining the quality of the Metropolis algorithm. <br>
 +
 +
This parameter affects the probability of accepting the candidate states, and the algorithm will not perform well if the acceptance probability is too large or too small, it also affects the size of the "jump" between the sampled <math>Y</math> and the previous state x<sub>i+1</sub>, as a larger variance implies a larger such "jump".<br>
 +
 +
If the jump is too large, we will have to repeat the previous stage; thus, we will repeat the same point for many times.<br>
 +
 +
'''MATLAB b=2, b= 0.2, b=20 '''
 +
<pre style="font-size:12px">
 +
clear all
 +
close all
 +
clc
 +
b=2 % b=0.2 b=20;
 +
x(1)=0;
 +
for i=2:10000
 +
    y=b*randn+x(i-1);
 +
    r=min((1+x(i-1)^2)/(1+y^2),1);
 +
    u=rand;
 +
    if u<r
 +
        x(i)=y;
 +
    else
 +
        x(i)=x(i-1);
 +
    end
 +
   
 +
end
 +
figure(1);
 +
hist(x(5000:end,100));
 +
figure(2);
 +
plot(x(5000:end));
 +
%The Markov Chain usually takes some time to converge and this is known as the "burning time"
 +
%Therefore, we don't display the first 5000 points because they don't show the limiting behaviour of the Markov Chain
 +
 +
generate the Markov Chain with 10000 random variable, using a large b and a small  b.
 +
</pre>
 +
 +
b tells where the next point is going to be. The appropriate b is supposed to explore all the support area.
 +
 +
f(x) is the stationary distribution list of the chain in MH. We generating y using q(y|x) and accept it with respect to r.
 +
 +
===='''b too small====
 +
If <math>b = 0.02</math>, the chain takes small steps so the chain doesn't explore enough of sample space.
 +
 +
If <math>b = 20</math>, jumps are very unlikely to be accepted; i.e. <math> y </math> is rejected as <math> u> r </math> and <math> Xt+1 = Xt</math>.
 +
i.e <math>\frac {f(y)}{f(x)}</math> and consequent <math> r </math> is very small and very unlikely that <math> u < r </math>, so the current value will be repeated.
 +
 +
==='''Detailed Balance Holds for Metropolis-Hasting'''===
 +
 +
In metropolis-hasting, we generate y using q(y|x) and accept it with probability r, where <br>
 +
 +
<math>r(x,y) = min\left\{\frac{f(y)}{f(x)}\frac{q(x|y)}{q(y|x)},1\right\} = min\left\{\frac{f(y)}{f(x)},1\right\}</math><br>
 +
 +
Without loss of generality we assume <math>\frac{f(y)}{f(x)}\frac{q(x|y)}{q(y|x)} > 1</math><br>
 +
 +
Then r(x,y) (probability of accepting y given we are currently in x) is <br>
 +
 +
<math>r(x,y) = min\left\{\frac{f(y)}{f(x)}\frac{q(x|y)}{q(y|x)},1\right\} = \frac{f(y)}{f(x)}\frac{q(x|y)}{q(y|x)}</math><br>
 +
 +
Now suppose that the current state is y and we are generating x; the probability of accepting x given that we are currently in state y is <br>
 +
 +
<math>r(x,y) = min\left\{\frac{f(x)}{f(y)}\frac{q(y|x)}{q(x|y)},1\right\} = 1 </math><br>
 +
 +
This is because <math>\frac{f(y)}{f(x)}\frac{q(x|y)}{q(y|x)} < 1 </math> and its reverse <math>\frac{f(x)}{f(y)}\frac{q(y|x)}{q(x|y)} > 1 </math>. Then <math>r(x,y) = 1</math>.<br>
 +
We are interested in the probability of moving from from x to y in the Markov Chain generated by MH algorithm: <br>
 +
P(y|x) depends on two probabilities:
 +
1. Probability of generating y, and<br>
 +
2. Probability of accepting y. <br>
 +
 +
<math>P(y|x) = q(y|x)*r(x,y) = q(y|x)*{\frac{f(y)}{f(x)}\frac{q(x|y)}{q(y|x)}} = \frac{f(y)*q(x|y)}{f(x)} </math> <br>
 +
 +
The probability of moving to x given the current state is y:
 +
 +
<math>P(x|y) = q(x|y)*r(y,x) = q(x|y)</math><br>
 +
 +
So does detailed balance hold for MH? <br>
 +
 +
If it holds we should have <math>f(x)*P(y|x) = f(y)*P(x|y)</math>.<br>
 +
 +
Left-hand side: <br>
 +
 +
<math>f(x)*P(y|x) = f(x)*{\frac{f(y)*q(x|y)}{f(x)}} = f(y)*q(x|y)</math><br>
 +
 +
Right-hand side: <br>
 +
 +
<math>f(y)*P(x|y) = f(y)*q(x|y)</math><br>
 +
 +
Thus LHS and RHS are equal and the detailed balance holds for MH algorithm. <br>
 +
Therefore, f(x) is the stationary distribution of the chain.<br>
 +
 +
== Class 20 - Thursday July 11th 2013 ==
 +
=== Simulated annealing ===
 +
<br />
 +
'''Definition:''' Simulated annealing (SA) is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). <br />
 +
(http://en.wikipedia.org/wiki/Simulated_annealing) <br />
 +
"Simulated annealing is a popular algorithm in simulation for minimizing functions." (from textbook)<br />
 +
 +
Simulated annealing is developed to solve the traveling salesman problem: finding the optimal path to travel all the cities needed<br/>
 +
 +
It is called "Simulated annealing" because it mimics the process undergone by misplaced atoms in a metal when<br />
 +
its heated and then slowly cooled.<br />
 +
(http://mathworld.wolfram.com/SimulatedAnnealing.html)<br />
 +
 +
It is a probabilistic method proposed in Kirkpatrick, Gelett and Vecchi (1983) and Cerny (1985) for finding the global minimum of a function that may have multiple local minimums.<br />
 +
(http://www.mit.edu/~dbertsim/papers/Optimization/Simulated%20annealing.pdf)<br />
 +
 +
Simulated annealing was developed as an approach for finding the minimum of complex functions <br />
 +
with multiple peaks; where standard hill-climbing approaches may trap the algorithm at a less that optimal peak.<br />
 +
 +
Suppose we generated a point <math> x </math> by an existing algorithm, and we would like to get a "better" point. <br>
 +
(eg. If we have generated a local min of a function and we want the global min) <br>
 +
Then we would use simulated annealing as a method to "perturb" <math> x </math> to obtain a better solution. <br>
 +
 +
Suppose we would like to min <math> h(x)</math>, for any arbitrary constant <math> T > 0</math>, this problem is equivalent to  max <math>e^{-h(x)/T}</math><br />
 +
Note that the exponential function is monotonic. <br />
 +
Consider f proportional  to  e<sup>-h(x)/T</sup>, sample of this distribution when T is small and
 +
close to the optimal point of h(x). Based on this observation, SA algorithm is introduced as :<br />
 +
<b>1.</b> Set T to be a large number<br />
 +
<b>2.</b> Initialize the chain: set <math>\,X_{t}  (ie.  i=0, x_0=s)</math><br />
 +
<b>3.</b> <math>\,y</math>~<math>\,q(y|x)</math><br/>
 +
(q should be symmetric)<br />
 +
<b>4.</b> <math>r = \min\{\frac{f(y)}{f(x)},1\}</math><br />
 +
<b>5.</b> U ~ U(0,1)<br />
 +
<b>6.</b> If U < r, <math>X_{t+1}=y</math> <br/>
 +
else, <math>X_{t+1}=X_t</math><br/>
 +
<b>7.</b> end  decrease T, and let i=i+1. Go back to 3. (This is where the difference lies between SA and MH. <br />
 +
(repeat the procedure until T is very small)<br/>
 +
<br/>
 +
<b>Note</b>: q(y|x) does not have to be symmetric. If q is non-symmetric, then the original MH formula is used.<br />
 +
 +
The significance of T <br />
 +
Initially we set T to be large when initializing the chain so as to explore the entire sample space and to avoid the possibility of getting stuck/trapped in one region of the sample space. Then we gradually start decreasing T so as to get closer and closer to the actual solution. 
 +
 +
Notice that we have:
 +
    <math> r = \min\{\frac{f(y)}{f(x)},1\} </math><br/>
 +
    <math> = \min\{\frac{e^{\frac{-h(y)}{T}}}{e^{\frac{-h(x)}{T}}},1\} </math>  <br/>
 +
    <math> = \min\{e^{\frac{h(x)-h(y)}{T}},1\} </math><br/>
 +
 +
Reasons we start with a large T but not a small T at the beginning:<br />
 +
 +
<ul><li>A point in the tail when T is small would be rejected <br />
 +
</li><li>Chances that we reject points get larger as we move from large T to small T <br />
 +
</li><li>Large T helps get to the mode of maximum value<br />
 +
</li></ul>
 +
 +
Assume T is large <br />
 +
1. h(y) < h(x), e<sup>(h(x)-h(y))/T </sup> > 1, then r = 1, y will always be accepted.<br />
 +
2. h(y) > h(x), e<sup>(h(x)-h(y))/T </sup>< 1, then r < 1, y will be accepted with probability r.  '''Remark:'''this will help to scape from local minimum, because the algorithm prevents it from reaching and staying in the local minimum forever. <br />
 +
Assume T is small<br />
 +
1. h(y) < h(x), then r = 1, y will always be accepted.<br />
 +
2. h(y) > h(x), e<sup>(h(x)-h(y))/T </sup> approaches to 0, then r goes to 0 and y will almost never be accepted.
 +
 +
<p><br /> All in all, choose a large T to start off with in order for a higher chance that the points can explore. <br />
 +
 +
'''Note''': The variable T is known in practice as the "Temperature", thus the higher T is, the more variability there is in terms of the expansion and contraction of materials. The term "Annealing" follows from here, as annealing is the process of heating materials and allowing them to cool slowly.<br />
 +
 +
Asymptotically this algorithm is guaranteed to generate the global optimal answer, however in practice, we never sample forever and this may not happen.
 +
 +
</p><p><br />
 +
</p><p>Example: Consider <math>h(x)=3x^2</math>, 0&lt;x&lt;1
 +
</p><p><br />1) Set T to be large, for example, T=100<br />
 +
<br />2) Initialize the chain<br />
 +
<br />3) Set <math>q(y|x)~\sim~Unif[0,1]</math><br />
 +
<br />4) <math>r=min(exp(\frac{(3x^2-3y^2)}{100}),1)</math><br />
 +
<br />5) <math>U~\sim~U[0,1]</math><br />
 +
<br />6) If <i>U</i> &lt; <i>r</i> then <i>X</i><sub><i>t</i> + 1</sub> = <i>y</i> <br>
 +
<i>e</i><i>l</i><i>s</i><i>e</i>,<i>X</i><sub><i>t</i> + 1</sub> = <i>x</i><sub><i>t</i></sub><br />
 +
<br />7) Decrease T, go back to 3<br />
 +
</p>
 +
<div style="border:1px red solid">
 +
<p><b>MATLAB </b>
 +
</p>
 +
<pre style="font-size:12px">
 +
Syms x
 +
Ezplot('(x-3)^2',[-6,12])
 +
Ezplot('exp(-((x-3)^2))', [-6, 12])
 +
</pre>
 +
 +
[[File:Snip2013.png|350px]]
 +
 +
[[File:Snip20131.png|350px]]
 +
 +
[[File:STAT_340.JPG]]
 +
http://www.wolframalpha.com/input/?i=graph+exp%28-%28x-3%29%5E2%2F10%29
 +
<b>MATLAB </b>
 +
 +
Note that when T is small, the graph consists of a much higher bump; when T is large, the graph is flatter.
 +
 +
<pre style="font-size:14px">
 +
 +
clear all
 +
close all
 +
T=100;
 +
x(1)=randn;
 +
ii=1;
 +
b=1;
 +
while T&gt;0.001
 +
  y=b*randn+x(ii);
 +
  r=min(exp((H(x(ii))-H(y))/T),1);
 +
  u=rand;
 +
  if u&lt;r
 +
      x(ii+1)=y;
 +
  else
 +
      x(ii+1)=x(ii);
 +
  end
 +
 +
T=0.99*T;
 +
ii=ii+1;
 +
end
 +
plot(x)
 +
 +
</pre>
 +
[[File:SA_example.jpg|350px]]
 +
</div>
 +
<p>Helper function:
 +
</p><p>an example is for H(x)=(x-3)^2
 +
</p>
 +
<pre style="font-size:12px">
 +
function c=H(x)
 +
c=(x-3)^2;
 +
end
 +
</pre>
 +
<p><b>Another Example:</b>
 +
<span class="texhtml"><i>h</i>(<i>x</i>) = ((<i>x</i> &minus; 2)<sup>2</sup> &minus; 4)((<i>x</i> &minus; 4)<sup>2</sup> &minus; 8)</span>
 +
</p>
 +
<pre style="font-size:12px">
 +
&gt;&gt;syms x
 +
&gt;&gt;ezplot(((x-2)^2-4)*((x-4)^2-8),[-1,8])
 +
</pre>
 +
<pre style="font-size:12px">
 +
function c=H(x)
 +
c=((x-2)^2-4)*((x-4)^2-8);
 +
end
 +
</pre>
 +
[[File:SA_example2.jpg|350px]]
 +
<p>Run earlier code with the new H(x) function
 +
</p>
 +
<h3> <span class="mw-headline" id="Motivation:_Simulated_Annealing_and_the_Travelling_Salesman_Problem"> Motivation: Simulated Annealing and the Travelling Salesman Problem </span></h3>
 +
<p>The Travelling Salesman Problem asks:  <br />
 +
Given n numbers of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the original city? By letting two permutations if one results from an interchange of two of the coordinates of the other, we can use simulated annealing to approximate the best path.
 +
<p>[[File:Salesman n5.png|350px]]
 +
</p>
 +
<ul><li>An example of a solution of a travelling salesman problem on n=5. This is only one of many solutions, but we want to ensure we find the optimal solution.
 +
</li></ul>
 +
 +
<ul><li>Given n=5 cities, we search for the best route with the minimum distance to visit all cities and return to the starting city.
 +
</li></ul>
 +
<p><b>The idea of using Simulated Annealing algorithm</b>&nbsp;:
 +
Let Y (let Y be all possible combinations of route in terms of cities index) be generated by permutation of all cities. Let the target or objective distribution (f(x)) be the distance of the route given Y.
 +
Then use the Simulated Annealing algorithm to find the minimum value of f(x).<br />
 +
</p><p><b>Note</b>: in this case, Q is the permutation of the numbers. There will be may possible paths, especially when n is large. If n is very large, then it will take forever to check all the combination of routes.
 +
</p>
 +
<ul><li>This sort of knowledge would be very useful for those in a situation where they are on a limited budget or must visit many points in a short period of time. For example, a truck driver may have to visit multiple cities in southern Ontario and make it back to his original starting point within a 6-hour period. <br />
 +
</li></ul>
 +
 +
'''Disadvantages of Simulated Annealing:'''<br/>
 +
1. This method converges very slowly, and therefore very expensive.<br/>
 +
2. This algorithm cannot tell whether it has found the global minimum.<br/><ref>
 +
Reference: http://cs.adelaide.edu.au/~paulc/teaching/montecarlo/node140.html
 +
</ref>
 +
 +
== Class 21 - Tuesday July 16, 2013 ==
 +
=== Gibbs Sampling===
 +
'''Definition'''<br>
 +
In statistics and in statistical physics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximately from a specified multivariate probability distribution (i.e. from the joint probability distribution of two or more random variables), when direct sampling is difficult.<br/>
 +
(http://en.wikipedia.org/wiki/Gibbs_sampling)<br/>
 +
 +
The Gibbs sampling method was originally developed by Geman and Geman [1984]. It was later brought into mainstream statistics by Gelfand and Smith [1990] and Gelfand, et al. [1990]<br/>
 +
Source:  https://www.msu.edu/~blackj/Scan_2003_02_12/Chapter_11_Markov_Chain_Monte_Carlo_Methods.pdf<br/>
 +
 +
Gibbs sampling is a general method for probabilistic inference which is often used when dealing with incomplete information. However, generality comes at some computational cost, and for many applications including those involving missing information, there are often alternative methods that have been proven to be more efficient in practice. For example, say we want to sample from a joint distribution <math>p(x_1,...,x_k)</math> (i.e. a posterior distribution). If we knew the full conditional distributions for each parameter (i.e. <math>p(x_i|x_1,x_2,...,x_{i-1},x_{i+1},...,x_k)</math>), we can use the Gibbs sampler to sample from these conditional distributions. <br>
 +
 +
When utilizing the Gibbs sampler, the candidate state is always accepted as the next state of the chain.(from text book)<br/>
 +
 +
*Another Markov Chain Monte Carlo (MCMC) method (first MCMC method introduced in this course is the MH Algorithm) <br/>
 +
*a special case of Metropolis-Hastings sampling where the random value is always accepted, i.e. as long as a point is proposed, it is accepted. <br/>
 +
* useful and make it simple and easier for sampling a d-dimensional random vector <math>\vec{x} = (x_1, x_2,...,x_d)</math><br />
 +
* then the observations of d-dimensional random vectors <math>{\vec{x_1}, \vec{x_2}, ... , \vec{x_n}}</math> form a d-dimensional Markov Chain and the joint density <math>f(x_1, x_2, ... , x_d)</math> is an invariant distribution for the chain. i.e. for sampling multivariate distributions.<br />
 +
* useful if sampling from conditional pdf, since they are easier to sample, in comparison to the joint distribution.<br/>
 +
*Definition of univariate conditional distribution: all the random variables are fixed except for one; we need to use n such univariate conditional distributions to simulate n random variables.
 +
 +
'''Difference between Gibbs Sampling & MH'''<br>
 +
Gibbs Sampling generates new value based on the conditional distribution of other components (unlike MH, which does not require conditional distribution).<br/>
 +
eg. We are given the following about <math> f(x_1,x_2) , f(x_1|x_2),f(x_2|x_1) </math><br/>
 +
1. let <math>x^*_1 \sim f(x_1|x_2)</math><br/>
 +
2. <math>x^*_2 \sim f(x_2|x^*_1)</math><br/>
 +
3. substitute <math>x^*_2</math> back into first step and repeat the process. <br/>
 +
 +
Also, for Gibbs sampling, we will "always accept a candidate point", unlike MH<br/>
 +
Source:  https://www.msu.edu/~blackj/Scan_2003_02_12/Chapter_11_Markov_Chain_Monte_Carlo_Methods.pdf<br/>
 +
 +
<div style = "align:left; background:#F5F5DC; font-size: 120%">
 +
'''Gibbs Sampling as a special form of the Metropolis Hastings algorithm'''<br>
 +
 +
The Gibbs Sampler is simply a case of the Metropolis Hastings algorithm<br>
 +
 +
here, the proposal distribution is <math>q(Y|X)=f(X^j|X^*_i, i\neq j)=\frac{f(Y)}{f(X_i, i\neq j)}</math> for <math>X=(X_1,...,X_n)</math>, <br>
 +
which is simply the conditional distribution of each element conditional on all the other elements in the vector. <br>
 +
similarly <math>q(X|Y)=f(X|Y^*_i, i\neq j)=\frac{f(X)}{f(Y_i, i\neq j)}</math><br>
 +
notice that <math>(Y_i, i\neq j)</math> and <math>(X_i, i\neq j)</math> are identically distributed. <br>
 +
 +
the distribution we wish to simulate from is <math>p(X) = f(X) </math>
 +
also, <math>p(Y) = f(Y) </math>
 +
 +
Hence, the acceptance ratio in the Metropolis-Hastings algorithm is: <br>
 +
<math>r(x,y) = min\left\{\frac{f(x)}{f(y)}\frac{q(y|x)}{q(x|y)},1\right\} = min\left\{\frac{f(x)}{f(y)}\frac{f(y)}{f(x)},1\right\} = 1 </math><br>
 +
so the new point will always be accepted, and no points are rejected and the Gibbs Sampler is an efficient algorithm in that aspect. <br>
 +
</div>
 +
 +
<b>Advantages </b><ref>
 +
http://wikicoursenote.com/wiki/Stat341#Gibbs_Sampling_-_June_30.2C_2009
 +
</ref>
 +
 +
*The algorithm has an acceptance rate of 1. Thus, it is efficient because we keep all the points that we sample from.
 +
*It is simple and straightforward if and only if we know the conditional pdf. 
 +
*It is useful for high-dimensional distributions. (ie. for sampling multivariate PDF)
 +
*It is useful if sampling from conditional PDF are easier than sampling from the joint.
 +
 +
<br />
 +
<b>Disadvantages</b><ref>
 +
http://wikicoursenote.com/wiki/Stat341#Gibbs_Sampling_-_June_30.2C_2009
 +
</ref>
 +
 +
*We rarely know how to sample from the conditional distributions.
 +
*The probability functions of the conditional probability are usually unknown or hard to sample from.
 +
*The algorithm can be extremely slow to converge.
 +
*It is often difficult to know when convergence has occurred.
 +
*The method is not practical when there are relatively small correlations between the random variables.
 +
 +
'''Gibbs Sampler Steps:'''<br\><ref>
 +
http://www.people.fas.harvard.edu/~plam/teaching/methods/mcmc/mcmc.pdf
 +
</ref>
 +
Let's suppose that we are interested in sampling from the posterior p(x|y), where x is a vector of three parameters, x1, x2, x3. <br\>
 +
The steps to a Gibbs Sampler are:<br\>
 +
1. Pick a vector of starting value x(0). Any x(0) will converge eventually, but it can be chosen to take fewer iterations<br\>
 +
2. Start with any x(order does not matter, but I will start with x1 for convenience). Draw a value x1(1)from the full conditional p(x1|x2(0),x3(0),y)<br\>
 +
3. Draw a value x2(1) from the full conditional p(x2|x1(1),x3(0),y). Note that we must use the updated value of x1(1).<br\>
 +
4. Draw a value x3(1) from the full conditional p(x3|x1(1),x2(1),y) using both updated values.<br\>
 +
5. Draw x2 using x1 and continually using the most updated values. <br\>
 +
6. Repeat until we get M draws, we each draw being a vector x(t).<br\>
 +
7. Optional burn-in or thinning.<br\>
 +
Our result is a Markov chain with a bunch of draws of x that are approximately from our posterior.
 +
 +
'''The Basic idea:''' <br>
 +
The distinguishing feature of Gibbs sampling is that the underlying Markov chain is constructed from a sequence of conditional distributions. The essential idea is updating one part of the previous element while keeping the other parts fixed - it is useful in many instances where the state variable is a random variable taking values in a general space, not just in R<sup>n</sup>. (Simulation and the Monte Carlo Method, Reuven Y. Rubinstein)
 +
 +
'''Note:''' <br>
 +
1.Other optimizing algorithms introduced such as Simulated Annealing settles on a minimum eventually,which means that if we generate enough observations and plot them in a time series plot, the plot will eventually flatten at the optimal value.<br\> 
 +
2.For Gibbs Sampling however, when convergence is achieved, instead of staying at the optimal value, the Gibbs Sampler continues to wonder through the target distribution (i.e. will not stay at the optimal point) forever.<br\> 
 +
'''Special Example'''<br\>
 +
<pre>
 +
function gibbs2(n, thin)
 +
  x_samp = zeros(n,1)
 +
  y_samp = zeros(n,1)
 +
  x=0.0
 +
  y=0.0
 +
  for i=1:n
 +
      for j=1:thin
 +
        x=(y^2+4)*randg(3)
 +
        y=1/(1+x)+randn()/sqrt(2*x+2)
 +
      end
 +
      x_samp[i] = x
 +
      y_samp[i] = y
 +
  end
 +
  return x_samp, y_samp
 +
end
 +
1
 +
2
 +
julia> @elapsed gibbs2(50000,1000)
 +
7.6084020137786865
 +
</pre>
 +
 +
'''Theoretical Example''' <br/>
 +
 +
Gibbs Sampler Application (Inspired by Example 10b in the Ross Simulation (4th Edition Textbook))
 +
 +
Suppose we are a truck driver who randomly puts n basketballs into a 3D storage cube sized so that each edge of the cube is 300cm in length. The basket balls are spherical and have a radius of 25cm each.
 +
 +
Because the basketballs have a radius of 25cm, the centre of each basketball must be at least 50cm away from the centre of another basketball. That is to say