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== '''Computer Simulation of Complex Systems (Stat 340 - Spring 2013)''' ==
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<div style = "align:left; background:#00ffff; font-size: 150%">
 +
If you
 +
use ideas, plots, text, code and other intellectual property developed by someone else
 +
in your `wikicoursenote' contribution , you have to cite the
 +
original source. If you copy a sentence or a paragraph from work done by someone
 +
else, in addition to citing the original source you have to use quotation marks to
 +
identify the scope of the copied material. Evidence of copying or plagiarism will
 +
cause a failing mark in the course.
  
 +
Example of citing the original source
  
== Introduction ==
+
Assumptions Underlying Principal Component Analysis can be found here<ref>http://support.sas.com/publishing/pubcat/chaps/55129.pdf</ref>
  
 +
</div>
  
Four Fundamental Problems:
+
==Important Notes==
 +
<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>
  
1. Classification: Given an input object X, we have a function which will take in this input X and identity which 'class (Y)' it belongs to. (Discrete Case)
+
<div style = "align:left; background:#F5F5DC; font-size: 120%">
2.Regression: Same as classification but in the continuous case
+
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>.
3.Clustering: Use common features of objects in same class or group to form clusters.
 
4. Dimensionality Reduction
 
  
 +
Assumptions Underlying Principal Component Analysis can be found here<ref>http://support.sas.com/publishing/pubcat/chaps/55129.pdf</ref>
  
== Applications ==
+
</div>
  
Most useful when structure of the task is not well understood but can be characterized by a dataset with strong statistical regularity
+
== Introduction, Class 1 - Tuesday, May 7 ==
Examples
 
Computer Vision, Computer Graphics, Finance (fraud detection), Machine Learning
 
Search and recommendation (eg. Google)
 
Automatic speech recognition, speaker verification
 
Text parsing
 
Face identification
 
Tracking objects in video
 
Financial prediction, fraud detection
 
  
  
== Course Information ==
 
  
No required textbook, recommended: "Simulation" by Sheldon M. Ross
+
=== Course Instructor: Ali Ghodsi ===
Computing parts of the course will be done in Matlab, but prior knowledge of Matlab is not essential (will have a tutorial on it)
+
<!-- br tag for spacing-->
Learn for announcements, assignments, and emails.
+
Lecture: <br />
Other course material on: <<wikicoursenote.com>>  
+
001: T/Th 8:30-9:50am MC1085 <br />
Log on to both Learn and wikicoursenote frequently.
+
002: T/Th 1:00-2:20pm DC1351 <br />
 +
Tutorial: <br />
 +
2:30-3:20pm Mon M3 1006 <br />
 +
Office Hours: <br />
 +
Friday at 10am, M3 4208 <br />
  
<s>Must do both</s>''All primary contributions are now considered general contributions you must contribute to 50% of lectures for full marks''
+
=== Midterm ===
 +
Monday June 17,2013 from 2:30pm-3:20pm
 +
 
 +
=== Final ===
 +
Saturday August 10,2013 from 7:30pm-10:00pm
 +
 
 +
=== TA(s):  ===
 +
<!-- br tag for spacing-->
 +
{| class="wikitable"
 +
|-
 +
! TA
 +
! Day
 +
! Time
 +
! Location
 +
|-
 +
| Lu Cheng
 +
| Monday
 +
|  3:30-5:30 pm
 +
| M3 3108, space 2
 +
|-
 +
| Han ShengSun
 +
| Tuesday
 +
|  4:00-6:00 pm
 +
| M3 3108, space 2
 +
|-
 +
| Yizhou Fang
 +
| Wednesday
 +
|  1:00-3:00 pm
 +
| M3 3108, space 1
 +
|-
 +
| Huan Cheng
 +
| Thursday
 +
|  3:00-5:00 pm
 +
| M3 3111, space 1
 +
|-
 +
| Wu Lin
 +
| Friday
 +
| 11:00-1:00 pm
 +
| M3 3108, space 1
 +
|}
 +
 
 +
=== Four Fundamental Problems ===
 +
<!-- br tag for spacing-->
 +
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 />
 +
  <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 />
 +
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 />
 +
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 (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 ===
 +
 
 +
Most useful when structure of the task is not well understood but can be characterized by a dataset with strong statistical regularity<br />
 +
Examples: <br />
 +
*Computer Vision, Computer Graphics, Finance (fraud detection), Machine Learning <br />
 +
*Search and recommendation (eg. Google, Amazon) <br />
 +
*Automatic speech recognition, speaker verification <br />
 +
*Text parsing <br />
 +
*Face identification <br />
 +
*Tracking objects in video<br />
 +
*Financial prediction(e.g. credit cards) <br />
 +
*Fraud detection  <br />
 +
*Medical diagnosis <br />
 +
 
 +
=== Course Information ===
 +
 
 +
'''Prerequisite:''' (One of CS 116, 126/124, 134, 136, 138, 145, SYDE 221/322) and (STAT 230 with a grade of at least 60% or STAT 240) and (STAT 231 or 241)
 +
 
 +
'''Antirequisite:''' CM 361/STAT 341, CS 437, 457 <!--- Moved these down to declutter table of contents --->
 +
 
 +
'''General Information'''
 +
*No required textbook
 +
*Recommended: "Simulation" by Sheldon M. Ross
 +
*Computing parts of the course will be done in Matlab, but prior knowledge of Matlab is not essential (will have a tutorial on it)
 +
*First midterm will be held on Monday, June 17 from 2:30 to 3:30
 +
*Announcements and assignments will be posted on Learn.
 +
*Other course material on: http://wikicoursenote.com/wiki/
 +
*Log on to both Learn and wikicoursenote frequently.
 +
*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 (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.
 +
Example:<br/>
 +
User: questid<br/>
 +
Email: questid@uwaterloo.ca<br/>
 +
After the student has made the account request, do wait for several hours before students can login into the account using the passwords stated in the email. During the first login, students will be ask to create a new password for their account.
 
 
'''Primary contributor''': Put a summary of the lecture up within 48 hours.
+
'''As a technical/editorial contributor''': Make contributions within 1 week and do not copy the notes on the blackboard.
+
 
'''General contributor''': Elaborate on concepts, add example, add code, add pictures, reference, corrections etc… withing 2 weeks
+
''All contributions are now considered general contributions you must contribute to 50% of lectures for full marks''
  
:A general contribution can be correctinal (fixing mistakes) or technical (expanding content, adding examples, etc) but at least half of your contributions should be technical for full marks
+
*A general contribution can be correctional (fixing mistakes) or technical (expanding content, adding examples, etc.) but at least half of your contributions should be technical for full marks.
  
 
Do not submit copyrighted work without permission, cite original sources.
 
Do not submit copyrighted work without permission, cite original sources.
Each time you make a contribution, check mark the table. Marks are calculated on honour system, although there will be random verifications. If you are caught claiming to contribute but didn't, you will ''lose'' marks.
+
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
 +
 
 +
- you can submit your contributions multiple times.<br />
 +
- you will be able to edit the response right after submitting<br />
 +
- send email  to make changes to an old response : uwstat340@gmail.com<br />
 +
 
 +
=== Tentative Topics ===
 +
- Random variable and stochastic process generation<br />
 +
- Discrete-Event Systems<br />
 +
- Variance reduction<br />
 +
- Markov Chain Monte Carlo
 +
 
 +
==Class 2 - Thursday, May 9==
 +
===Generating Random Numbers===
 +
==== Introduction ====
 +
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 should:
 +
<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: 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>
 +
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>
 +
<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></font>
 +
 
 +
(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.
 +
[[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<br>
 +
 
 +
'''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.
 +
 
 +
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 ====
 +
Let <math>n \in \N</math> and <math>m \in \N^+</math>, then by Division Algorithm,
 +
<math>\exists q, \, r \in \N \;\text{with}\; 0\leq r < m, \; \text{s.t.}\; n = mq+r</math>,
 +
where <math>q</math> is called the quotient and <math>r</math> the remainder. Hence we can define a binary function
 +
<math>\mod : \N \times \N^+ \rightarrow \N </math> given by <math>r:=n \mod m</math> which returns the remainder after division by m. 
 +
<br />
 +
Generally, mod means taking the reminder after division by m.
 +
<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 />
 +
 
 +
'''Example 1:'''<br />
 +
 
 +
<math>30 = 4 \cdot  7 + 2</math><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 />
 +
 
 +
<math>2:=-3\mod 5</math><br />
 +
 
 +
<br />
 +
'''Example 2:'''<br />
 +
 
 +
If <math>23 = 3 \cdot  6 + 5</math> <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 />
 +
 
 +
'''Example 3:'''<br />
 +
<math>77 = 3 \cdot  25 + 2</math><br />
 +
 
 +
<math>2 := 77\mod 3</math><br />
 +
<br />
 +
<math>25 = 25 \cdot  1 + 0</math><br />
 +
 
 +
<math>0: = 25\mod 25</math><br />
 +
<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.
 +
 
 +
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 />
 +
 
 +
 
 +
 
 +
<math>x_{k+1}=(a \cdot x_{k} + b) \mod m  </math>(a little tip: <math>(a \cdot b)\mod c = (a\mod c)\cdot(b\mod c))</math><br/>
 +
 
 +
'''Example'''<br />
 +
<math>\text{Let }a=2,\, b=1, \, m=3, \, x_{0} = 10</math><br />
 +
<math>\begin{align}
 +
\text{Step 1: } 0&{}=(2\cdot 10 + 1) &{}\mod 3 \\
 +
\text{Step 2: } 1&{}=(2\cdot 0 + 1) &{}\mod 3 \\
 +
\text{Step 3: } 0&{}=(2\cdot 1 + 1) &{}\mod 3 \\
 +
\end{align}</math><br />
 +
<math>\ldots</math><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. 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 />
 +
 
 +
 
 +
 +
'''MatLab Instruction for Multiplicative Congruential Algorithm:'''<br />
 +
Before you start, you need to clear all existing defined variables and operations:<br />
 +
<pre style="font-size:16px">
 +
>>clear all
 +
>>close all
 +
</pre>
 +
 
 +
<pre style="font-size:16px">
 +
>>a=17
 +
>>b=3
 +
>>m=31
 +
>>x=5
 +
>>mod(a*x+b,m)
 +
ans=26
 +
>>x=mod(a*x+b,m)
 +
</pre>
 +
 
 +
''(Note: <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 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 />
 +
 
 +
''(Note on MATLAB commands: <br />
 +
1. clear all: clears all variables.<br />
 +
2. close all: closes all figures.<br />
 +
3. who: displays all defined variables.<br />
 +
4. clc: clears screen.<br />
 +
5. ; : prevents the results from printing.<br />
 +
6. disstool: displays a graphing tool.<br /><br />
 +
 
 +
<pre style="font-size:16px">
 +
>>a=13
 +
>>b=0
 +
>>m=31
 +
>>x(1)=1
 +
>>for ii=2:1000
 +
x(ii)=mod(a*x(ii-1)+b,m);
 +
end
 +
>>size(x)
 +
ans=1    1000
 +
>>hist(x)
 +
</pre>
 +
[[File:MCA_Example.jpg|300px]]
 +
 
 +
''(Note: The semicolon after the x(ii)=mod(a*x(ii-1)+b,m) ensures that Matlab will not print the entire vector of x. It will instead calculate it internally and you will be able to work with it. Adding the semicolon to the end of this line reduces the run time significantly.) ''
 +
 
 +
 
 +
This algorithm involves three integer parameters <math>a, b,</math> and <math>m</math> and an initial value, <math>x_0</math> called the '''seed'''. A sequence of numbers is defined by <math>x_{k+1} = ax_k+ b \mod m</math>. <br />
 +
 
 +
Note: For some bad <math>a</math> and <math>b</math>, the histogram may not look uniformly distributed.<br />
 +
 
 +
Note: In MATLAB, hist(x) will generate a graph representing the distribution. Use this function after you run the code to check the real sample distribution.
 +
 
 +
'''Example''': <math>a=13, b=0, m=31</math><br />
 +
The first 30 numbers in the sequence are a permutation of integers from 1 to 30, and then the sequence repeats itself so '''it is important to  choose <math>m</math> large''' to decrease the probability of each number repeating itself too early. Values are between <math>0</math> and <math>m-1</math>. If the values are normalized by dividing by <math>m-1</math>, then the results are '''approximately''' numbers uniformly distributed in the interval [0,1]. There is only a finite number of values (30 possible values in this case). In MATLAB, you can use function "hist(x)" to see if it looks uniformly distributed. We saw that the values between 0-30 had the same frequency in the histogram, so we can conclude that they are uniformly distributed. <br />
 +
 
 +
If <math>x_0=1</math>, then <br />
 +
:<math>x_{k+1} = 13x_{k}\mod{31}</math>
 +
 
 +
So,
 +
 
 +
:<math>\begin{align} x_{0} &{}= 1 \\
 +
 
 +
x_{1} &{}= 13 \times 1  + 0 &{}\mod{31} = 13 \\
 +
 
 +
x_{2} &{}= 13 \times 13 + 0 &{}\mod{31} = 14 \\
 +
 
 +
x_{3} &{}= 13 \times 14 + 0 &{}\mod{31} =27 \\
 +
\end{align}</math>
 +
 
 +
etc.
 +
 
 +
For example, with <math>a = 3, b = 2, m = 4, x_0 = 1</math>, we have:
 +
 
 +
:<math>x_{k+1} = (3x_{k} + 2)\mod{4}</math>
 +
 
 +
So,
 +
 
 +
:<math>\begin{align}
 +
x_{0} &{}= 1 \\
 +
x_{1} &{}= 3 \times 1 + 2 \mod{4} = 1 \\
 +
x_{2} &{}= 3 \times 1 + 2 \mod{4} = 1 \\
 +
\end{align}</math><br />
 +
Another Example, a =3, b =2, m = 5, x_0=1
 +
etc.
 +
<hr/>
 +
<p style="color:red;font-size:16px;">FAQ:</P>
 +
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>
 +
Alternatively, {0} and {1,2,...,30} are two orbits of the left multiplication by 13 in the group <math>\Z_{31}</math>.<br>
 +
2.Will the number 31 ever appear?Is there a probability that a number never appears? <br>
 +
''The number 31 will never appear. When you perform the operation <math>\mod m</math>, the largest possible answer that you could receive is <math>m-1</math>. Whether or not a particular number in the range from 0 to <math>m - 1</math> appears in the above algorithm will be dependent on the values chosen for <math>a, b</math> and <math>m</math>. ''
 +
<hr/>
 +
 
 +
'''Examples:[From Textbook]'''<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 />
 +
<math>\begin{align}
 +
x_1 &{}= (5 \times 3+7) &{}\mod{200} &{}= 22 \\
 +
x_2 &{}= 117 &{}\mod{200} &{}= 117 \\
 +
x_3 &{}= 592 &{}\mod{200} &{}= 192 \\
 +
x_4 &{}= 2967 &{}\mod{200} &{}= 167 \\
 +
x_5 &{}= 14842 &{}\mod{200} &{}= 42 \\
 +
x_6 &{}= 74217 &{}\mod{200} &{}= 17 \\
 +
x_7 &{}= 371092 &{}\mod{200} &{}= 92 \\
 +
x_8 &{}= 1855467 &{}\mod{200} &{}= 67 \\
 +
x_9 &{}= 9277342 &{}\mod{200} &{}= 142 \\
 +
x_{10} &{}= 46386717 &{}\mod{200} &{}= 117 \\
 +
\end{align}</math><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 />
 +
 
 +
The computed values are between 0 and <math>m-1</math>. If the values are normalized by dividing by '''<math>m-1</math>''', their result is numbers uniformly distributed on the interval <math>\left[0,1\right]</math> (similar to computing from uniform distribution).<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 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 />
 +
Let c be a non-zero constant. Then for any seed x0, and LCG will have largest max. period if and only if<br />
 +
(i) m and c are coprime;<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 />
 +
 
 +
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 />
 +
<font size="3">Xn=(15Xn-1 + 4) mod 7</font><br />
 +
(i) m=7 c=4 -> coprime;<br />
 +
(ii) a-1=14 and a-1 is divisible by 7;<br />
 +
(iii) dose not apply.<br />
 +
(The extra knowledge stops here)
 +
 
 +
 
 +
 
 +
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.
 +
<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;">
 +
<h4 style="text-align:center;">Summary of Multiplicative Congruential Algorithm</h4>
 +
<p><b>Problem:</b> generate Pseudo Random Numbers.</p>
 +
<b>Plan:</b>
 +
<ol>
 +
<li>find integer: <i>a b m</i>(large prime) <i>x<sub>0</sub></i>(the seed) .</li>
 +
<li><math>x_{k+1}=(ax_{k}+b)</math>mod m</li>
 +
</ol>
 +
<b>Matlab Instruction:</b>
 +
<pre style="font-size:16px">&gt;&gt;clear all
 +
&gt;&gt;close all
 +
&gt;&gt;a=17
 +
&gt;&gt;b=3
 +
&gt;&gt;m=31
 +
&gt;&gt;x=5
 +
&gt;&gt;mod(a*x+b,m)
 +
ans=26
 +
&gt;&gt;x=mod(a*x+b,m)
 +
</pre>
 +
</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 ===
 +
Now that we know how to generate random numbers, we use these values to sample form distributions such as exponential. However, to easily use this method, the probability distribution consumed must have a cumulative distribution function (cdf) <math>F</math> with a tractable (that is, easily found) inverse <math>F^{-1}</math>.<br />
 +
 
 +
'''Theorem''': <br />
 +
If we want to generate the value of a discrete random variable X, we must generate a random number U, uniformly distributed over (0,1).
 +
Let <math>F:\R \rightarrow \left[0,1\right]</math> be a cdf. If <math>U \sim U\left[0,1\right]</math>, then the random variable given by <math>X:=F^{-1}\left(U\right)</math>
 +
follows the distribution function <math>F\left(\cdot\right)</math>,
 +
where <math>F^{-1}\left(u\right):=\inf F^{-1}\big(\left[u,+\infty\right)\big) = \inf\{x\in\R | F\left(x\right) \geq u\}</math> is the generalized inverse.<br />
 +
'''Note''': <math>F</math> need not be invertible everywhere on the real line, but if it is, then the generalized inverse is the same as the inverse in the usual case. We only need it to be invertible on the range of F(x), [0,1].
 +
 
 +
'''Proof of the theorem:'''<br />
 +
The generalized inverse satisfies the following: <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>,<br />
 +
<math>= F(x) , \text{ where } 0 \leq F(x) \leq 1 </math>  <br />
 +
 
 +
This is the c.d.f. of X.  <br />
 +
<br />
 +
 
 +
That is <math>F^{-1}\left(u\right) \leq x \Leftrightarrow u \leq F\left(x\right)</math><br />
 +
 
 +
Finally, <math>P(X \leq x) = P(F^{-1}(U) \leq x) = P(U \leq F(x)) = F(x)</math>, since <math>U</math> is uniform on the unit interval.<br />
 +
 
 +
This completes the proof.<br />
 +
 
 +
Therefore, in order to generate a random variable X~F, it can generate U according to U(0,1) and then make the transformation x=<math> F^{-1}(U) </math> <br />
 +
 
 +
Note that we can apply the inverse on both sides in the proof of the inverse transform only if the pdf of X is monotonic. A monotonic function is one that is either increasing for all x, or decreasing for all x. Of course, this holds true for all CDFs, since they are monotonic by definition. <br />
 +
 
 +
In short, what the theorem tells us is that we can use a random number <math> U from U(0,1) </math> to randomly sample a point on the CDF of X, then apply the inverse of the CDF to map the given probability to its domain, which gives us the random variable X.<br/>
 +
 
 +
 
 +
'''Example 1 - Exponential''': <math> f(x) = \lambda e^{-\lambda x}</math><br/>
 +
Calculate the CDF:<br />
 +
<math> F(x)= \int_0^x f(t) dt = \int_0^x \lambda e ^{-\lambda t}\ dt</math>
 +
<math> = \frac{\lambda}{-\lambda}\, e^{-\lambda t}\, | \underset{0}{x} </math>
 +
<math> = -e^{-\lambda x} + e^0 =1 - e^{- \lambda x} </math><br />
 +
Solve the inverse:<br />
 +
<math> y=1-e^{- \lambda x}  \Rightarrow  1-y=e^{- \lambda x} \Rightarrow  x=-\frac {ln(1-y)}{\lambda}</math><br />
 +
<math> y=-\frac {ln(1-x)}{\lambda}  \Rightarrow  F^{-1}(x)=-\frac {ln(1-x)}{\lambda}</math><br />
 +
Note that 1 − U is also uniform on (0, 1) and thus −log(1 − U) has the same distribution as −logU. <br />
 +
Steps: <br />
 +
Step 1: Draw U ~U[0,1];<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 />
 +
 
 +
<pre style="font-size:16px">
 +
>>u=rand(1,1000);
 +
>>hist(u)      # this will generate a fairly uniform diagram
 +
</pre>
 +
[[File:ITM_example_hist(u).jpg|300px]]
 +
<pre style="font-size:16px">
 +
#let λ=2 in this example; however, you can make another value for λ
 +
>>x=(-log(1-u))/2;
 +
>>size(x)      #1000 in size
 +
>>figure
 +
>>hist(x)      #exponential
 +
</pre>
 +
[[File:ITM_example_hist(x).jpg|300px]]
 +
 
 +
'''Example 2 - Continuous Distribution''':<br />
 +
 
 +
<math> f(x) = \dfrac {\lambda } {2}e^{-\lambda \left| x-\theta \right| } for -\infty < X < \infty , \lambda >0 </math><br/>
 +
 
 +
Calculate the CDF:<br />
 +
 
 +
<math> F(x)= \frac{1}{2} e^{-\lambda (\theta - x)} , for \ x \le \theta </math><br/>
 +
<math> F(x) = 1 - \frac{1}{2} e^{-\lambda (x - \theta)}, for \ x > \theta </math><br/>
 +
 
 +
Solve for the inverse:<br />
 +
 
 +
<math>F^{-1}(x)= \theta + ln(2y)/\lambda, for \ 0 \le y \le 0.5</math><br/>
 +
<math>F^{-1}(x)= \theta - ln(2(1-y))/\lambda, for \ 0.5 < y \le 1</math><br/>
 +
 
 +
Algorithm:<br />
 +
Steps: <br />
 +
Step 1: Draw U ~ U[0, 1];<br />
 +
Step 2: Compute <math>X = F^-1(U)</math> i.e. <math>X = \theta  + \frac {1}{\lambda} ln(2U)</math> for U < 0.5 else <math>X = \theta -\frac {1}{\lambda} ln(2(1-U))</math>
 +
 
 +
 
 +
'''Example 3 - <math>F(x) = x^5</math>''':<br/>
 +
Given a CDF of X: <math>F(x) = x^5</math>, transform U~U[0,1]. <br />
 +
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 />
 +
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 />
 +
 
 +
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/>
 +
Given u~U[0,1], generate x from BETA(1,β)<br />
 +
Solution:
 +
<math>F(x)= 1-(1-x)^\beta</math>,
 +
<math>u= 1-(1-x)^\beta</math><br />
 +
Solve for x:
 +
<math>(1-x)^\beta = 1-u</math>,
 +
<math>1-x = (1-u)^\frac {1}{\beta}</math>,
 +
<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 />
 +
'''MatLab Code''':<br />
 +
 
 +
<pre style="font-size:16px">
 +
>> u = rand(1,1000);
 +
x = 1-(1-u)^(1/3);
 +
>> hist(x,50)
 +
>> mean(x)
 +
</pre>
 +
 
 +
'''Example 5 - Estimating <math>\pi</math>''':<br/>
 +
Let's use rand() and Monte Carlo Method to estimate <math>\pi</math> <br />
 +
N= total number of points <br />
 +
N<sub>c</sub> = total number of points inside the circle<br />
 +
Prob[(x,y) lies in the circle=<math>\frac {Area(circle)}{Area(square)}</math><br />
 +
If we take square of size 2, circle will have area =<math>\pi (\frac {2}{2})^2 =\pi</math>.<br />
 +
Thus <math>\pi= 4(\frac {N_c}{N})</math><br />
 +
 
 +
  <font size="3">For example, '''UNIF(a,b)'''<br />
 +
  <math>y = F(x) = (x - a)/ (b - a) </math>
 +
  <math>x = (b - a ) * y + a</math>
 +
  <math>X = a + ( b - a) * U</math><br />
 +
  where U is UNIF(0,1)</font>
 +
 
 +
'''Limitations:'''<br />
 +
1. This method is flawed since not all functions are invertible or monotonic: generalized inverse is hard to work on.<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 transformation from cdf to inverse cdf,and use the uniform distribution to obtain a value of x from F(x).
 +
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. Then, we can look at the graph to determine what kind of distribution the graph resembles.
 +
 
 +
==== Probability Distribution Function Tool in MATLAB ====
 +
<pre style="font-size:16px">
 +
disttool        #shows different distributions
 +
</pre>
 +
 
 +
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.
  
== Tentative Marking Scheme ==
+
 
 +
[[File:Disttool.jpg|450px]]
 +
change the value of mu and sigma can change the graph skew side.
 +
 
 +
== Class 3 - Tuesday, May 14 ==
 +
=== 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 />
 +
 
 +
<math>1) U~ \sim~ Unif [0,1] </math>
 +
'''2) X = F<sup>-1</sup>(u)  '''<br />
 +
 
 +
 
 +
 
 +
 
 +
 
 +
<br />
 +
 
 +
'''Note''': CDF of a U(a,b) random variable is:
 +
:<math>
 +
  F(x)= \begin{cases}
 +
  0 & \text{for }x < a \\[8pt]
 +
  \frac{x-a}{b-a} & \text{for }a \le x < b \\[8pt]
 +
  1 & \text{for }x \ge b
 +
  \end{cases}
 +
</math>
 +
 
 +
Thus, for <math> U </math> ~ <math>U(0,1) </math>, we have <math>P(U\leq 1) = 1</math> and <math>P(U\leq 1/2) = 1/2</math>.<br />
 +
More generally, we see that <math>P(U\leq a) = a</math>.<br />
 +
For this reason, we had <math>P(U\leq F(x)) = F(x)</math>.<br />
 +
 
 +
'''Reminder:  '''  <br />
 +
'''This is only for uniform distribution <math> U~ \sim~ Unif [0,1] </math> '''<br />
 +
<math> P (U \le 1) = 1 </math>  <br />
 +
<math> P (U \le 0.5) = 0.5 </math>  <br />
 +
<math> P (U \le a) = a </math>  <br />
 +
 
 +
[[File:2.jpg]]            <math>P(U\leq a)=a</math>
 +
 
 +
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
 +
 
 +
====Limitations of the Inverse Transform Method====
 +
 
 +
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 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.
 +
 
 +
In conclusion, we need to find another way of sampling from more complicated distributions
 +
 
 +
=== Discrete Case ===
 +
The same technique can be used for discrete case. We want to generate a discrete random variable x, that has probability mass function: <br/>
 +
 
 +
:<math>\begin{align}P(X = x_i) &{}= p_i \end{align}</math>
 +
:<math>x_0 \leq x_1 \leq x_2 \dots \leq x_n</math>
 +
:<math>\sum p_i = 1</math>
 +
 
 +
Algorithm for applying Inverse Transformation Method in Discrete Case (Procedure):<br>
 +
1. Define a probability mass function for <math>x_{i}</math> where i = 1,....,k. Note: k could grow infinitely. <br>
 +
2. Generate a uniform random number U, <math> U~ \sim~ Unif [0,1] </math><br>
 +
3. If <math>U\leq p_{o}</math>, deliver <math>X = x_{o}</math><br>
 +
4. Else, if <math>U\leq p_{o} + p_{1} </math>, deliver <math>X = x_{1}</math><br>
 +
5. Repeat the process again till we reached to <math>U\leq p_{o} + p_{1} + ......+ p_{k}</math>, deliver <math>X = x_{k}</math><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 />
 +
 
 +
 
 +
 
 +
'''Example 3.0:''' <br />
 +
Generate a random variable from the following probability function:<br />
 
{| class="wikitable"
 
{| class="wikitable"
 
|-
 
|-
! Item
 
! Value
 
 
|-
 
|-
| Assignments (~6)
+
| x
| 30%
+
| -2
 +
| -1
 +
| 0
 +
| 1
 +
| 2
 
|-
 
|-
| WikiCourseNote
+
| f(x)
| 10%
+
| 0.1
 +
| 0.5
 +
| 0.07
 +
| 0.03
 +
| 0.3
 +
|}
 +
 
 +
Answer:<br />
 +
1. Gen U~U(0,1)<br />
 +
2. If U < 0.5 then output -1<br />
 +
else if U < 0.8 then output 2<br />
 +
else if U < 0.9 then output -2<br />
 +
else if U < 0.97 then output 0 else output 1<br />
 +
 
 +
'''Example 3.1 (from class):''' (Coin Flipping Example)<br />
 +
We want to simulate a coin flip. We have U~U(0,1) and X = 0 or X = 1.
 +
 
 +
We can define the U function so that:
 +
 
 +
If <math>U\leq 0.5</math>, then X = 0
 +
 
 +
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.
 +
 
 +
<math> U~ \sim~ Unif [0,1] </math>
 +
:<math>\begin{align}
 +
P(X = 0) &{}= 0.5\\
 +
P(X = 1) &{}= 0.5\\
 +
\end{align}</math>
 +
The answer is:
 +
:<math>
 +
x = \begin{cases}
 +
0, & \text{if } U\leq 0.5 \\
 +
1, & \text{if } 0.5 < U \leq 1
 +
\end{cases}</math>
 +
 
 +
 
 +
* '''Code'''<br />
 +
<pre style="font-size:16px">
 +
>>for ii=1:1000
 +
    u=rand;
 +
      if u<0.5
 +
        x(ii)=0;
 +
      else
 +
        x(ii)=1;
 +
      end
 +
  end
 +
>>hist(x)
 +
</pre>
 +
[[File:Coin_example.jpg|300px]]
 +
 
 +
Note: The role of semi-colon in Matlab: Matlab will not print out the results if the line ends in a semi-colon and vice versa.
 +
 
 +
'''Example 3.2 (From class):'''
 +
 
 +
Suppose we have the following discrete distribution:
 +
 
 +
:<math>\begin{align}
 +
P(X = 0) &{}= 0.3 \\
 +
P(X = 1) &{}= 0.2 \\
 +
P(X = 2) &{}= 0.5
 +
\end{align}</math>
 +
[[File:33.jpg]]
 +
 
 +
The cumulative distribution function (cdf) for this distribution is then:
 +
 
 +
:<math>
 +
F(x) = \begin{cases}
 +
0, & \text{if } x < 0 \\
 +
0.3, & \text{if }  x < 1 \\
 +
0.5, & \text{if }  x < 2 \\
 +
1, & \text{if }  x \ge 2
 +
\end{cases}</math>
 +
 
 +
Then we can generate numbers from this distribution like this, given <math>U \sim~ Unif[0, 1]</math>:
 +
 
 +
:<math>
 +
x = \begin{cases}
 +
0, & \text{if } U\leq 0.3 \\
 +
1, & \text{if } 0.3 < U \leq 0.5 \\
 +
2, & \text{if } 0.5 <U\leq 1
 +
\end{cases}</math>
 +
 
 +
"Procedure"<br />
 +
1. Draw U~u (0,1)<br />
 +
2. if U<=0.3 deliver x=0<br />
 +
3. else if 0.3<U<=0.5 deliver x=1<br />
 +
4. else 0.5<U<=1 deliver x=2
 +
 
 +
Can you find a faster way to run this algorithm? Consider:
 +
 
 +
:<math>
 +
x = \begin{cases}
 +
2, & \text{if } U\leq 0.5 \\
 +
1, & \text{if } 0.5 < U \leq 0.7 \\
 +
0, & \text{if } 0.7 <U\leq 1
 +
\end{cases}</math>
 +
 
 +
The logic for this is that U is most likely to fall into the largest range. Thus by putting the largest range (in this case x >= 0.5) we can improve the run time of this algorithm. Could this algorithm be improved further using the same logic?
 +
 
 +
* '''Code''' (as shown in class)<br />
 +
Use Editor window to edit the code  <br />
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>for ii=1:1000
 +
    u=rand;
 +
      if u<=0.3
 +
          x(ii)=0;
 +
      elseif u<=0.5
 +
          x(ii)=1;
 +
      else
 +
          x(ii)=2;
 +
      end
 +
    end
 +
>>size(x)
 +
>>hist(x)
 +
</pre>
 +
[[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>
 +
:<math>
 +
f_{x}(x) = \begin{cases}
 +
2x, & \text{if } 0\leq x \leq 1 \\
 +
0, & \text{if }  otherwise
 +
\end{cases}</math>
 +
 
 +
:<math>
 +
F_{x}(x) = \begin{cases}
 +
0, & \text{if } x < 0 \\
 +
\int_{0}^{x}2sds = x^{2}, & \text{if } 0\leq x \leq 1 \\
 +
1, & \text{if }  x > 1
 +
\end{cases}</math>
 +
 
 +
:<math>\begin{align} U = x^{2}, X = F^{-1}x(U)= U^{\frac{1}{2}}\end{align}</math>
 +
 
 +
'''Example 3.4''': Generating a Bernoulli random variable <br>
 +
:<math>\begin{align} P(X = 1) = p,  P(X = 0) = 1 - p\end{align}</math>
 +
:<math>
 +
F(x) = \begin{cases}
 +
1-p, & \text{if } x < 1 \\
 +
1, & \text{if }  x \ge 1
 +
\end{cases}</math>
 +
1. Draw <math> U~ \sim~ Unif [0,1] </math><br>
 +
2. <math>
 +
X = \begin{cases}
 +
0, & \text{if } 0 < U < 1-p \\
 +
1, & \text{if } 1-p \le U < 1
 +
\end{cases}</math>
 +
 
 +
 
 +
'''Example 3.5''': Generating Binomial(n,p) Random Variable<br>
 +
<math> use p\left( x=i+1\right) =\dfrac {n-i} {i+1}\dfrac {p} {1-p}p\left( x=i\right) </math>
 +
 
 +
Step 1: Generate a random number <math>U</math>.<br>
 +
Step 2: <math>c = \frac {p}{(1-p)}</math>, <math>i = 0</math>, <math>pr = (1-p)^n</math>, <math>F = pr</math><br>
 +
Step 3: If U<F, set X = i and stop,<br>
 +
Step 4: <math> pr = \, {\frac {c(n-i)}{(i+1)}} {pr}, F = F +pr, i = i+1</math><br>
 +
Step 5: Go to step 3<br>
 +
*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>
 +
 
 +
"Let X ~ Poi(u). Write an algorithm to generate X.
 +
The PDF of a poisson is:
 +
:<math>\begin{align} f(x) = \frac {\, e^{-u} u^x}{x!} \end{align}</math>
 +
We know that
 +
:<math>\begin{align} P_{x+1} = \frac {\, e^{-u} u^{x+1}}{(x+1)!} \end{align}</math>
 +
The ratio is <math>\begin{align} \frac {P_{x+1}}{P_x} = ... = \frac {u}{{x+1}} \end{align}</math>
 +
Therefore, <math>\begin{align} P_{x+1} = \, {\frac {u}{x+1}} P_x\end{align}</math>
 +
 
 +
Algorithm: <br>
 +
1) Generate U ~ U(0,1) <br>
 +
2) <math>\begin{align} X = 0 \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>
 +
  <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} x = x + 1 \end{align}</math> <br>
 +
4) Go to 1" <br>
 +
 +
Acknowledgements: This is an example from Stat 340 Winter 2013
 +
 
 +
 
 +
'''Example 3.7''': Generating Geometric Distribution:
 +
 
 +
Consider Geo(p) where p is the probability of success, and define random variable X such that X is the total number of trials required to achieve the first success. x=1,2,3..... We have pmf:
 +
<math>P(X=x_i) = \, p (1-p)^{x_{i}-1}</math>
 +
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 we observe the first success.
 +
Now consider the inverse transform:
 +
:<math>
 +
x = \begin{cases}
 +
1, & \text{if } U\leq p \\
 +
2, & \text{if } p < U \leq 1-(1-p)^2 \\
 +
3, & \text{if } 1-(1-p)^2 <U\leq 1-(1-p)^3 \\
 +
....
 +
k, & \text{if } 1-(1-p)^{k-1} <U\leq 1-(1-p)^k
 +
....
 +
\end{cases}</math>
 +
 
 +
 
 +
'''Note''': Unlike the continuous case, the discrete inverse-transform method can always be used for any discrete distribution (but it may not be the most efficient approach) <br>
 +
 
 +
 
 +
 
 +
'''General Procedure'''<br />
 +
1. Draw U ~ U(0,1)<br />
 +
2. If <math>U \leq P_{0}</math> deliver <math>x = x_{0}</math><br />
 +
3. Else if <math>U \leq P_{0} + P_{1}</math> deliver <math>x = x_{1}</math><br />
 +
4. Else if <math>U \leq P_{0} + P_{1} + P_{2} </math> deliver <math>x = x_{2}</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 />
 +
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).
 +
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
 +
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 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;">
 +
<h2 style="text-align:center;">Summary of Inverse Transform Method</h2>
 +
<p><b>Problem:</b>generate types of distribution.</p>
 +
<p><b>Plan:</b></p>
 +
<b style='color:lightblue;'>Continuous case:</b>
 +
<ol>
 +
<li>find CDF F</li>
 +
<li>find the inverse F<sup>-1</sup></li>
 +
<li>Generate a list of uniformly distributed number {x}</li>
 +
<li>{F<sup>-1</sup>(x)} is what we want</li>
 +
</ol>
 +
<b>Matlab Instruction</b>
 +
<pre style="font-size:16px">&gt;&gt;u=rand(1,1000);
 +
&gt;&gt;hist(u)
 +
&gt;&gt;x=(-log(1-u))/2;
 +
&gt;&gt;size(x)
 +
&gt;&gt;figure
 +
&gt;&gt;hist(x)
 +
</pre>
 +
<br>
 +
<b style='color:lightblue'>Discrete case:</b>
 +
<ol>
 +
<li>generate a list of uniformly distributed number {u}</li>
 +
<li>d<sub>i</sub>=x<sub>i</sub> if<math> X=x_i, </math> if <math> F(x_{i-1})<U\leq F(x_i) </math></li>
 +
<li>{d<sub>i</sub>=x<sub>i</sub>} is what we want</li>
 +
</ol>
 +
<b>Matlab Instruction</b>
 +
<pre style="font-size:16px">&gt;&gt;for ii=1:1000
 +
    u=rand;
 +
      if u&lt;0.5
 +
        x(ii)=0;
 +
      else
 +
        x(ii)=1;
 +
      end
 +
  end
 +
&gt;&gt;hist(x)
 +
</pre>
 +
</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===
 +
 
 +
Although the inverse transformation method does allow us to change our uniform distribution, it has two limits;
 +
# Not all functions have inverse functions (ie, the range of x and y have limit and do not fix the inverse functions)
 +
# For some distributions, such as Gaussian, it is too difficult to find the inverse
 +
 
 +
To generate random samples for these functions, we will use different methods, such as the '''Acceptance-Rejection Method'''. This method is more efficient than the inverse transform method. The basic idea is to find an alternative probability distribution with density function f(x);
 +
 
 +
Suppose we want to draw random sample from a target density function ''f(x)'', ''x∈S<sub>x</sub>'', where ''S<sub>x</sub>'' is the support of ''f(x)''. If we can find some constant ''c''(≥1) (In practice, we prefer c as close to 1 as possible) and a density function ''g(x)'' having the same support ''S<sub>x</sub>'' so that ''f(x)≤cg(x), ∀x∈S<sub>x</sub>'', then we can apply the procedure for Acceptance-Rejection Method. Typically we choose a density function that we already know how to sample from for ''g(x)''.
 +
 
 +
[[File:AR_Method.png]]
 +
 
 +
 
 +
The main logic behind the Acceptance-Rejection Method is that:<br>
 +
1. We want to generate sample points from an unknown distribution, say f(x).<br>
 +
2. We use <math>\,cg(x)</math> to generate points so that we have more points than f(x) could ever generate for all x. (where c is a constant, and g(x) is a known distribution)<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>
 +
 
 +
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>
 +
 
 +
c must be chosen so that <math>f(x)\leqslant c g(x)</math> for all value of x. c can only equal 1 when f and g have the same distribution. Otherwise:<br>
 +
Either use a software package to test if <math>f(x)\leqslant c g(x)</math>  for an arbitrarily chosen c > 0, or:<br>
 +
1. Find first and second derivatives of f(x) and g(x).<br>
 +
2. Identify and classify all local and absolute maximums and minimums, using the First and Second Derivative Tests, as well as all inflection points.<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>
 +
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.
 +
 
 +
The expected number of iterations of the algorithm required with an X is c.
 +
<br>
 +
'''Note:''' <br>
 +
1. Value around x<sub>1</sub> will be sampled more often under cg(x) than under f(x).There will be more samples than we actually need, if <math>\frac{f(y)}{\, c g(y)}</math> is small, the acceptance-rejection technique will need to be done to these points to get the accurate amount.In the region above x<sub>1</sub>, we should accept less and reject more. <br>
 +
2. Value around x<sub>2</sub>: number of sample that are drawn and the number we need are much closer. So in the region above x<sub>2</sub>, we accept more. As a result, g(x) and f(x) are comparable.<br>
 +
3. The constant c is needed because we need to adjust the height of g(x) to ensure that it is above f(x). Besides that, it is best to keep the number of rejected varieties small for maximum efficiency. <br>
 +
 
 +
Another way to understand why the the acceptance probability is <math>\frac{f(y)}{\, c g(y)}</math>, is by thinking of areas. From the graph above, we see that the target function in under the proposed function c g(y). Therefore, <math>\frac{f(y)}{\, c g(y)}</math> is the proportion or the area under c g(y) that also contains f(y). Therefore we say we accept sample points for which u is less then <math>\frac{f(y)}{\, c g(y)}</math> because then the sample points are guaranteed to fall under the area of c g(y) that contains f(y). <br>
 +
<br>
 +
 
 +
'''There are 2 cases that are possible:''' <br>
 +
-Sample of points is more than enough, <math>c g(x) \geq f(x) </math> <br>
 +
-Similar or the same amount of points, <math>c g(x) \geq f(x) </math> <br>
 +
'''There is 1 case that is not possible:''' <br>
 +
-Less than enough points, such that <math> g(x) </math> is greater than <math> f </math>, <math>g(x) \geq f(x)</math> <br>
 +
<br>
 +
 
 +
'''Procedure'''
 +
 
 +
#Draw Y~g(.)
 +
#Draw U~u(0,1) (Note: U and Y are independent)
 +
#If <math>u\leq \frac{f(y)}{cg(y)}</math> (which is <math>P(accepted|y)</math>) then x=y, else return to Step 1<br>
 +
 
 +
 
 +
Note: Recall <math>P(U\leq a)=a</math>. Thus by comparing u and <math>\frac{f(y)}{\, c g(y)}</math>, we can get a probability of accepting y at these points. For instance, at some points that cg(x) is much larger than f(x), the probability of accepting x=y is quite small.<br>
 +
ie. At X<sub>1</sub>, low probability to accept the point since f(x) is much smaller than cg(x).<br>
 +
At X<sub>2</sub>, high probability to accept the point.  <math>P(U\leq a)=a</math> in Uniform Distribution.
 +
 
 +
Note: Since U is the variable for uniform distribution between 0 and 1. It equals to 1 for all. The condition depends on the constant c. so the condition changes to <math>c\leq \frac{f(y)}{g(y)}</math>
 +
 
 +
 
 +
introduce the relationship of cg(x)and f(x),and prove why they have that relationship and where we can use this rule to reject some cases.
 +
and learn how to see the graph to find the accurate point to reject or accept the ragion above the random variable x.
 +
for the example, x1 is bad point and x2 is good point to estimate the rejection and acceptance
 +
 
 +
'''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 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 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 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 .
 +
 
 +
=== Theorem ===
 +
 
 +
Let <math>f: \R \rightarrow  [0,+\infty]</math> be a well-defined pdf, and <math>\displaystyle Y</math> be a random variable with pdf <math>g: \R \rightarrow [0,+\infty]</math> such that <math>\exists c \in \R^+</math> with <math>f \leq c \cdot g</math>. If <math>\displaystyle U \sim~ U(0,1)</math> is independent of <math>\displaystyle Y</math>, then the random variable defined as <math>X := Y \vert U \leq \frac{f(Y)}{c \cdot g(Y)}</math> has pdf <math>\displaystyle f</math>, and the condition <math>U \leq \frac{f(Y)}{c \cdot g(Y)}</math> is denoted by "Accepted".
 +
 
 +
=== Proof ===
 +
Recall the conditional probability formulas:<br />
 +
<math>\begin{align}
 +
P(A|B)=\frac{P(A \cap B)}{P(B)}, \text{ or }P(A|B)=\frac{P(B|A)P(A)}{P(B)} \text{ for pmf}
 +
\end{align}</math><br />
 +
 
 +
<math>P(y|accepted)=f(y)=\frac{P(accepted|y)P(y)}{P(accepted)}</math><br />       
 +
<br />based on the concept from '''procedure-step1''':<br />
 +
<math>P(y)=g(y)</math><br />
 +
 
 +
<math>P(accepted|y)=\frac{f(y)}{cg(y)}</math> <br />
 +
(the larger the value is, the larger the chance it will be selected) <br /><br />
 +
 
 +
 
 +
<math>
 +
\begin{align}
 +
P(accepted)&=\int_y\ P(accepted|y)P(y)\\
 +
          &=\int_y\ \frac{f(s)}{cg(s)}g(s)ds\\
 +
          &=\frac{1}{c} \int_y\  f(s) ds\\
 +
          &=\frac{1}{c}
 +
\end{align}</math><br />
 +
 
 +
Therefore:<br />
 +
<math>\begin{align}
 +
P(x)&=P(y|accepted)\\
 +
&=\frac{\frac{f(y)}{cg(y)}g(y)}{1/c}\\
 +
&=\frac{\frac{f(y)}{c}}{1/c}\\
 +
&=f(y)\end{align}</math><br /><br /><br />
 +
 
 +
'''''Here is an alternative introduction of Acceptance-Rejection Method'''''
 +
 
 +
'''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 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.
 +
 
 +
'''Acceptance-Rejection Method'''<br/>
 +
'''Example 1''' (discrete case)<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/>
 +
<math>f(x)=Pr(X=x)=2Cx×(0.5)^2\,</math><br/>
 +
 
 +
{| class=wikitable  align=left
 +
|<math>x</math>||0||1||2 
 +
|-
 +
|<math>f(x)</math>||1/4||1/2||1/4
 +
|-
 +
|<math>g(x)</math>||1/3||1/3||1/3 
 
|-
 
|-
| Midterm
+
|<math>c=f(x)/g(x)</math>||3/4||3/2||3/4
| 20%
 
 
|-
 
|-
| Final
+
|<math>f(x)/(cg(x))</math>||1/2||1||1/2
| 40%
 
 
|}
 
|}
  
==Class 2 - Thursday, May 9==
+
 
 +
Since we need <math>c \geq f(x)/g(x)</math><br/>
 +
We need <math>c=3/2</math><br/>
 +
 
 +
Therefore, the algorithm is:<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/>
 +
3. If <math>(y=0)</math> and <math>(v<\tfrac{1}{2}), output=0</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/>
 +
 
 +
 
 +
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 < \tfrac{f(x)}{cg(x)}</math> is not satisfied, we need to go over the code again.<br/>
 +
 
 +
Proof<br/>
 +
 
 +
Let <math>f(x)</math> be the function we wish to generate from, but we cannot use inverse transform method to generate directly.<br/>
 +
Let <math>g(x)</math> be the helper function <br/>
 +
Let <math>kg(x)>=f(x)</math><br/>
 +
Since we need to generate y from <math>g(x)</math>,<br/>
 +
<math>Pr(select y)=g(y)</math><br/>
 +
<math>Pr(output y|selected y)=Pr(u<f(y)/(cg(y)))= f(y)/(cg(y))</math> (Since u~Unif(0,1))<br/>
 +
<math>Pr(output y)=Pr(output y1|selected y1)Pr(select y1)+ Pr(output y2|selected y2)Pr(select y2)+…+ Pr(output yn|selected yn)Pr(select yn)=1/c</math> <br/>
 +
Consider that we are asking for expected time for the first success, it is a geometric distribution with probability of success=1/c<br/>
 +
Therefore, <math>E(X)=1/(1/c))=c</math> <br/>
 +
 
 +
Acknowledgements: Some materials have been borrowed from notes from Stat340 in Winter 2013.
 +
 
 +
Use the conditional probability to proof if the probability is accepted, then the result is closed pdf of the original one.
 +
the example shows how to choose the c for the two function <math>g(x)</math> and <math>f(x)</math>.
 +
 
 +
=== Example of Acceptance-Rejection Method===
 +
 
 +
Generating a random variable having p.d.f. <br />
 +
<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<br />
 +
<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<br />
 +
<math>\displaystyle f(x)/g(x) = 20x(1 - x)^3 </math><br />
 +
Differentiation of this quantity yields <br />                             
 +
<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,
 +
and thus, <br />
 +
<math>\displaystyle f(x)/g(x)<= 20*(1/4)*(3/4)^3=135/64=c </math><br />
 +
Hence,<br />
 +
<math>\displaystyle f(x)/cg(x)=(256/27)*(x*(1-x)^3)</math><br />
 +
and thus the simulation procedure is as follows:
 +
 
 +
1)      Generate two random numbers U1 and U2 .
 +
 
 +
2)      If U<sub>2</sub><(256/27)*U<sub>1</sub>*(1-U<sub>1</sub>)<sup>3</sup>, set X=U<sub>1</sub>, and stop
 +
Otherwise return to step 1).
 +
The average number of times that step 1)  will be performed is  c = 135/64.
 +
 
 +
(The above example is from http://www.cs.bgu.ac.il/~mps042/acceptance.htm, example 2.)
 +
 
 +
use the derivative to proof the accepetance-rejection method,
 +
find the local maximum of f(x)/g(x).
 +
and we can calculate the best constant c.
 +
 
 +
===Another Example of Acceptance-Rejection Method===
 +
Generate a random variable from:<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 />
 +
Therefore:<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.
 +
because g(.) is uniform so the g(x) is 1. max(g(x)) is 1<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
 +
 
 +
== Class 4 - Thursday, May 16 ==
 +
 
 +
'''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 />
 +
*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 cannot be a negative number.<br />
 +
 
 +
 
 +
'''How to find C''':<br />
 +
 
 +
<math>\begin{align}
 +
&c \cdot g(x) \geq f(x)\\
 +
&c\geq \frac{f(x)}{g(x)}  \\
 +
&c= \max \left(\frac{f(x)}{g(x)}\right)
 +
\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/>
 +
<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/>
 +
 
 +
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/>.
 +
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>
 +
:<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>
 +
*So far, the only distribution we know how to sample from is the '''UNIFORM''' distribution. <br>
 +
 
 +
 
 +
'''Procedure''': <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 <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.
 +
 
 +
'''Recall the general procedure of Acceptance-Rejection Method'''
 +
#Let <math>Y \sim~ g(y)</math>
 +
#Let <math>U \sim~ Unif [0,1] </math>
 +
#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: <br>
 +
 
 +
Generate a random variable from the pdf</b><br>
 +
<math> f(x) =
 +
\begin{cases}
 +
2x,  & \mbox{if }0 \leqslant x \leqslant 1 \\
 +
0, & \mbox{otherwise}
 +
\end{cases} </math>
 +
 
 +
We can note that this is a special case of Beta(2,1), where,
 +
<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
 +
 
 +
<math>Gamma(z)=\int _{0}^{\infty }t^{z-1}e^{-t}dt</math>
 +
 
 +
Aside: Beta function
 +
 
 +
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by
 +
<math>B(x,y)=\int_0^1 \! {t^{(x-1)}}{(1-t)^{(y-1)}}\,dt</math><br>
 +
 
 +
 
 +
<math>beta(2,1)= \frac{\Gamma(3)}{(\Gamma(2)\Gamma(1))}x^1 (1-x)^0 = 2x</math><br>
 +
 
 +
<hr>
 +
<math>g=u(0,1)</math><br>
 +
<math>y=g</math><br>
 +
<math>f(x)\leq c\cdot g(x)</math><br>
 +
<math>c\geq \frac{f(x)}{g(x)}</math><br>
 +
<math>c = \max \frac{f(x)}{g(x)} </math><br>
 +
<br><math>c = \max \frac{2x}{1}, 0 \leq x \leq 1</math><br>
 +
Taking x = 1 gives the highest possible c, which is c=2
 +
<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]]
 +
 
 +
'''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:'''<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).
 +
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>
 +
<ol>
 +
<li>Draw y~U(0,1)</li>
 +
<li>Draw u~U(0,1)</li>
 +
<li>if <math>u \leq \frac{(2\cdot y)}{(2\cdot 1)},  u \leq y,</math> then <math> x=y</math><br>
 +
<li>Else go to Step 1</li>
 +
</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.
 +
 
 +
<span style="font-weight:bold;color:green;">Matlab Code</span>
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>ii=1;            # ii:numbers that are accepted
 +
>>jj=1;            # jj:numbers that are generated
 +
>>while ii<1000
 +
    y=rand;
 +
    u=rand;
 +
    jj=jj+1;
 +
    if u<=y
 +
      x(ii)=y;
 +
      ii=ii+1;
 +
    end
 +
  end
 +
>>hist(x)          # It is a histogram
 +
>>jj
 +
  jj = 2024        # should be around 2000
 +
</pre>
 +
[[File:ARM_Example.jpg|300px]]
 +
 
 +
:'''*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.
 +
 
 +
:'''*Note3:''' We use '''while''' instead of '''for''' when looping because we do not know how many iterations are required to generate 1000 successful samples. We can view this as a negative binomial distribution so while the expected number of iterations required is n * c, it will likely deviate from this amount. We expect 2000 in this case.
 +
 
 +
:'''*Note4:''' If c=1, we will accept all points, which is the ideal situation. However, this is essentially impossible because if c = 1 then our distributions f(x) and g(x) must be identical, so we will have to be satisfied with as close to 1 as possible.
 +
 
 +
'''Use Inverse Method for this Example'''<br>
 +
:<math>F(x)=\int_0^x \! 2s\,ds={x^2}-0={x^2}</math><br>
 +
:<math>y=x^2</math><br>
 +
:<math>x=\sqrt y</math>
 +
:<math> F^{-1}\left (\, x \, \right) =\sqrt x</math>
 +
 
 +
:*'''Procedure'''
 +
:1: Draw <math> U~ \sim~ Unif [0,1] </math><br>
 +
:2: <math> x=F^{-1}\left (\, u\, \right) =\sqrt u</math>
 +
 
 +
<span style="font-weight:bold;color:green;">Matlab Code</span>
 +
<pre style="font-size:16px">
 +
>>u=rand(1,1000);
 +
>>x=u.^0.5;
 +
>>hist(x)
 +
</pre>
 +
[[File:ARM(IFM)_Example.jpg|300px]]
 +
 
 +
<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 the matrix dimensions must agree.
 +
 
 +
'''
 +
'''Example for A-R method:'''
 +
 
 +
Given <math> f(x)= \frac{3}{4} (1-x^2),  -1 \leq x \leq 1 </math>,  use A-R method to generate random number
 +
 
 +
 
 +
[[Solution:]]
 +
 
 +
Let g=U(-1,1) and g(x)=1/2
 +
 
 +
let y ~ f,
 +
<math> cg(x)\geq f(x),
 +
c\frac{1}{2} \geq \frac{3}{4} (1-x^2) /1,
 +
c=max 2\cdot\frac{3}{4} (1-x^2) = 3/2 </math>
 +
 
 +
The process:
 +
 
 +
:1: Draw U1 ~ U(0,1) <br>
 +
:2: Draw U2 ~ U(0,1)  <br>
 +
:3: let <math> y = U1*2 - 1 </math>
 +
:4: if <math>U2 \leq \frac { \frac{3}{4} * (1-y^2)} { \frac{3}{4}} = {1-y^2}</math>, then x=y,  '''note that''' (3/4(1-y^2)/(3/4) is getting from f(y) / (cg(y)) )
 +
:5: else: return to '''step 1'''
 +
 
 +
----
 +
 
 +
 
 +
=====Example of Acceptance-Rejection Method=====
 +
 
 +
<math>\begin{align}
 +
& 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>\frac{f(x)}{c \cdot g(x)} = x^2</math>
 +
 
 +
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>\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>\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>.
 +
Use the inverse method to sample from <math>\begin{align}g(x)\end{align}</math>
 +
<math>\begin{align}G(x)=x^2\end{align}</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>
 +
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'''
 +
 
 +
-This method could be computationally inefficient depending on the rejection rate. We may have to sample many points before<br>
 +
we get the 1000 accepted points. In the example we did in class relating the <math>f(x)=2x</math>, <br>
 +
we had to sample around 2070 points before we finally accepted 1000 sample points.<br>
 +
-If the form of the proposal distribution, g, is very different from target distribution, f, then c is very large and the algorithm is not computationally efficient.<br>
 +
 
 +
'''Acceptance - Rejection Method Application on Normal Distribution''' <br>
 +
 
 +
<math>X \sim∼ N(\mu,\sigma^2), \text{ or } X = \sigma Z + \mu, Z \sim~ N(0,1) </math><br>
 +
<math>\vert Z \vert</math> has probability density function of <br>
 +
 
 +
f(x) = (2/<math>\sqrt{2\pi}</math>) e<sup>-x<sup>2</sup>/2</sup>
 +
 
 +
g(x) = e<sup>-x</sup>
 +
 
 +
Take h(x) = f(x)/g(x) and solve for h'(x) = 0 to find x so that h(x) is maximum.
 +
 
 +
Hence x=1 maximizes h(x) => c = <math>\sqrt{2e/\pi}</math>
 +
 
 +
Thus f(y)/cg(y) = e<sup>-(y-1)<sup>2</sup>/2</sup>
 +
 
 +
 
 +
learn how to use code to calculate the c between f(x) and g(x).
 +
 
 +
<p style="font-weight:bold;text-size:20px;">How to transform <math>U(0,1)</math> to <math>U(a, b)</math></p>
 +
 
 +
1. Draw U from <math>U(0,1)</math>
 +
 
 +
2. Take <math>Y=(b-a)U+a</math>
 +
 
 +
3. Now Y follows <math>U(a,b)</math>
 +
 
 +
'''Example''': Generate a random variable z from the Semicircular density <math>f(x)= \frac{2}{\pi R^2} \sqrt{R^2-x^2}, -R\leq x\leq R</math>.
 +
 
 +
-> Proposal distribution: UNIF(-R, R)
 +
 
 +
-> We know how to generate using <math> U \sim UNIF (0,1) </math> Let <math> Y= 2RU-R=R(2U-1)</math>, therefore Y follows <math>U(-R,R)</math>
 +
 
 +
-> In order to maximize the function we must maximize the top and minimize the bottom.
 +
 
 +
Now, we need to find c:
 +
Since c=max[f(x)/g(x)], where <br />
 +
<math>f(x)= \frac{2}{\pi R^2} \sqrt{R^2-x^2}</math>, <math>g(x)=\frac{1}{2R}</math>, <math>-R\leq x\leq R</math><br />
 +
Thus, we have to maximize R^2-x^2.
 +
=> When x=0, it will be maximized.
 +
Therefore, c=4/pi. * Note: This also means that the probability of accepting a point is <math>\pi/4</math>.
 +
 
 +
We will accept the points with limit f(x)/[cg(x)].
 +
Since <math>\frac{f(y)}{cg(y)}=\frac{\frac{2}{\pi R^{2}} \sqrt{R^{2}-y^{2}}}{\frac{4}{\pi} \frac{1}{2R}}=\frac{\frac{2}{\pi R^{2}} \sqrt{R^{2}-R^{2}(2U-1)^{2}}}{\frac{2}{\pi R}}</math>
 +
 
 +
* Note: Y= R(2U-1)
 +
We can also get Y= R(2U-1) by using the formula y = a+(b-a)*u, to transform U~(0,1) to U~(a,b). Letting a=-R and b=R, and substituting it in the formula y = a+(b-a)*u, we get Y= R(2U-1).
 +
 
 +
Thus, <math>\frac{f(y)}{cg(y)}=\sqrt{1-(2U-1)^{2}}</math> * this also means the probability we can accept points
 +
 
 +
The algorithm to generate random variable x is then:
 +
 
 +
1. Draw <Math>\ U</math> from <math>\ U(0,1)</math>
 +
 
 +
2. Draw <Math>\ U_{1}</math> from <math>\ U(0,1)</math>
 +
 
 +
3. If  <math>U_{1} \leq \sqrt{1-(2U-1)^2}, set x = U_{1}</math>
 +
  else return to step 1.
 +
 
 +
 
 +
 
 +
The condition is <br />
 +
<Math> U_{1} \leq \sqrt{(1-(2U-1)^2)}</Math><br>
 +
<Math>\ U_{1}^2 \leq 1 - (2U -1)^2</Math><br>
 +
<Math>\ U_{1}^2 - 1 \leq -(2U - 1)^2</Math><br>
 +
<Math>\ 1 - U_{1}^2 \geq (2U - 1)^2</Math>
 +
 
 +
 
 +
 
 +
 
 +
'''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)
 +
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/>
 +
<br/>
 +
Solution: First of all, we need to find c<br/>
 +
<math>cg(x)>=f(x)</math> <br/>
 +
<math>c>=\frac{f(x)}{g(x)}</math> <br/>
 +
<math>\frac{f(x)}{g(x)}=\frac{x}{a} * e^{-(1-a)x}</math> <br/>
 +
take derivative with respect to x, and set it to 0 to get the maximum, <br/>
 +
<math>\frac{1}{a} * e^{-(1-a)x} - \frac{x}{a} * e^{-(1-a)x} * (1-a) = 0 </math><br/>
 +
<math>x=\frac {1}{1-a}</math> <br/>
 +
 
 +
<math>\frac {f(x)}{g(x)} = \frac {e^{-1}}{a*(1-a)} </math><br/>
 +
<math>\frac {f(0)}{g(0)} = 0</math><br/>
 +
<math>\frac {f(\infty)}{g(\infty)} = 0</math><br/>
 +
<br/>
 +
therefore, <b><math>c= \frac {e^{-1}}{a*(1-a)}</math></b><br/>
 +
<br/>
 +
<b>In order to minimize c, we need to find the appropriate a</b> <br/>
 +
Take derivative with respect to a and set it to be zero, <br/>
 +
We could get <math>a= \frac {1}{2}</math> <br/>
 +
<b><math>c=\frac{4}{e}</math></b>
 +
<br/>
 +
Procedure: <br/>
 +
1. Generate u v ~unif(0,1) <br/>
 +
2. Generate y from g, since g is exponential with rate 2, let y=-0.5*ln(u) <br/>
 +
3. If <math>v<\frac{f(y)}{c\cdot g(y)}</math>, output y<br/>
 +
Else, go to 1<br/>
 +
 
 +
Acknowledgements: The example above is from Stat 340 Winter 2013 notes.
 +
 
 +
'''Summary of how to find the value of c''' <br/>
 +
Let <math>h(x) = \frac {f(x)}{g(x)}</math>, and then we have the following:<br />
 +
1. First, take derivative of h(x) with respect to x, get x<sub>1</sub>;<br />
 +
2. Plug x<sub>1</sub> into h(x) and get the value(or a function) of c, denote as c<sub>1</sub>;<br />
 +
3. Check the endpoints of x and sub the endpoints into h(x);<br />
 +
4. (if c<sub>1</sub> is a value, then we can ignore this step) Since we want the smallest value of c such that <math>f(x) \leq c\cdot g(x)</math> for all x, we want the unknown parameter that minimizes c. <br />So we take derivative of c<sub>1</sub> with respect to the unknown parameter (ie k=unknown parameter) to get the value of k. <br />Then we submit k to get the value of c<sub>1</sub>. (Double check that <math>c_1 \geq 1</math><br />
 +
5. Pick the maximum value of h(x) to be the value of c.<br />
 +
 
 +
For the two examples above, we need to generate the probability function to uniform distribution,
 +
and figure out  <math>c=max\frac {f(y)}{g(y)} </math>.
 +
If <math>v<\frac {f(y)}{c\cdot g(y)}</math>, output y.
 +
 
 +
 
 +
'''Summary of when to use the Accept Rejection Method''' <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/>
 +
3) A constant c where <math>f(x)\leq c\cdot g(x)</math><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%.
 +
 
 +
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.
 +
 
 +
 
 +
>> 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 ==
 +
Recall the example in the last lecture. The following code will generate a random variable required by the question.
 +
 
 +
* '''Code'''<br />
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>ii=1;
 +
>>R=1;        #Note: that R is a constant in which we can change
 +
                        i.e. if we changed R=4 then we would have a density between -4 and 4
 +
>>while ii<1000
 +
        u1 = rand;
 +
        u2 = rand;
 +
        y = R*(2*u2-1);
 +
        if (1-u1^2)>=(2*u2-1)^2
 +
          x(ii) = y;
 +
          ii = ii + 1;      #Note: for beginner programmers that this step increases
 +
                                the ii value for next time through the while loop
 +
        end
 +
  end
 +
>>hist(x,20)                  # 20 is the number of bars
 +
 
 +
>>hist(x,30)                #30 is the number of bars
 +
</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) plots a histogram of variable x, where y is the number of bars in the graph.
 +
 
 +
[[File:ARM_cont_example.jpg|300px]]
 +
 
 +
=== Discrete Examples ===
 +
* '''Example 1''' <br>
 +
Generate random variable <math>X</math> according to p.m.f<br/>
 +
<math>\begin{align}
 +
P(x &=1) &&=0.15 \\
 +
P(x &=2) &&=0.25 \\
 +
P(x &=3) &&=0.3 \\
 +
P(x &=4) &&=0.1 \\
 +
P(x &=5) &&=0.2 \\
 +
\end{align}</math><br/>
 +
 
 +
The discrete case is analogous to the continuous case. Suppose we want to generate an X that is a discrete random variable with pmf f(x)=P(X=x). Suppose also that we use the discrete uniform distribution as our target distribution, then <math> g(x)= P(X=x) =0.2 </math> for all 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 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/>
 +
Else return to Step 1.<br/>
 +
 
 +
C can be found by maximizing the ratio :<math> \frac{f(x)}{g(x)} </math>. To do this, we want to maximize <math> f(x) </math> and minimize <math> g(x) </math>. <br>
 +
:<math>c = max \frac{f(x)}{g(x)} = \frac {0.3}{0.2} = 1.5  </math> <br/>
 +
Note: In this case <math>f(x)=P(X=x)=0.3</math> (highest probability from the discrete probabilities in the question)
 +
:<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.
 +
~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 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.
 +
<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 />
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>p=[.15 .25 .3 .1 .2];    %This a vector holding the values
 +
>>ii=1;
 +
>>while ii < 1000
 +
    y=unidrnd(5);          %generates random numbers for the discrete uniform 
 +
    u=rand;                distribution with maximum 5.
 +
    if u<= p(y)/0.3
 +
      x(ii)=y;
 +
      ii=ii+1;
 +
    end
 +
  end
 +
>>hist(x)
 +
</pre>
 +
[[File:ARM_disc_example.jpg|300px]]
 +
 
 +
unidrnd(k) draws from the discrete uniform distribution of integers <math>1,2,3,...,k</math> If this function is not built in to your MATLAB then we can do simple transformation on the rand(k) function to make it work like the unidrnd(k) function.
 +
 
 +
The acceptance rate is <math>\frac {1}{c}</math>, so the lower the c, the more efficient the algorithm. Theoretically, c equals 1 is the best case because all samples would be accepted; however it would only be true when the proposal and target distributions are exactly the same, which would never happen in practice.
 +
 
 +
For example, if c = 1.5, the acceptance rate would be <math>\frac {1}{1.5}=\frac {2}{3}</math>. Thus, in order to generate 1000 random values, on average, a total of 1500 iterations would be required.
 +
 
 +
A histogram to show 1000 random values of f(x), more random value make the probability close to the express probability value.
 +
 
 +
 
 +
* '''Example 2'''<br>
 +
p(x=1)=0.1<br />p(x=2)=0.3<br />p(x=3)=0.6<br />
 +
Let g be the uniform distribution of 1, 2, or 3<br />
 +
g(x)= 1/3<br />
 +
<math>c=max(\tfrac{p_{x}}{g(x)})=0.6/(\tfrac{1}{3})=1.8</math><br />
 +
Hence <math>\tfrac{p(x)}{cg(x)} = p(x)/(1.8 (\tfrac{1}{3}))= \tfrac{p(x)}{0.6}</math>
 +
 
 +
1,y~g<br />
 +
2,u~U(0,1)<br />
 +
3, If <math>U \leq \frac{f(y)}{cg(y)}</math>, set x = y. Else go to 1.
 +
 
 +
* '''Code for example 2'''<br />
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>p=[.1 .3 .6];    %This a vector holding the values 
 +
>>ii=1;
 +
>>while ii < 1000
 +
    y=unidrnd(3);  %generates random numbers for the discrete uniform distribution with maximum 3
 +
    u=rand;           
 +
    if u<= p(y)/0.6
 +
      x(ii)=y;   
 +
      ii=ii+1;    %else ii=ii+1
 +
    end
 +
  end
 +
>>hist(x)
 +
</pre>
 +
 
 +
 
 +
* '''Example 3'''<br>
 +
 
 +
Suppose <math>\begin{align}p_{x} = e^{-3}3^{x}/x! , x\geq 0\end{align}</math> (Poisson distribution)
 +
 
 +
'''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>
 +
 
 +
'''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>
 +
 
 +
Suppose we are given f(x) such that it is hypergeometically distributed, given 10 white balls, 5 red balls, and select 3 balls, let X be the number of red ball selected, without replacement.
 +
 
 +
Choose g(x) such that it is binomial distribution, Bin(3, 1/3). Find the rejection constant, c
 +
 
 +
Solution:
 +
For hypergeometric: <math>P(X=0) =\binom{10}{3}/\binom{15}{3} =0.2637, P(x=1)=\binom{10}{2} * \binom{5}{1} /\binom{15}{3}=0.4945, P(X=2)=\binom{10}{1} * \binom{5}{2} /\binom{15}{3}=0.2198,</math><br><br>
 +
<math>P(X=3)=\binom{5}{3}/\binom{15}{3}= 0.02198</math>
 +
 
 +
 
 +
For Binomial g(x): P(X=0) = (2/3)^3=0.2963; P(X=1)= 3*(1/3)*(2/3)^2 = 0.4444, P(X=2)=3*(1/3)^2*(2/3)=0.2222, P(X=3)=(1/3)^3=0.03704
 +
 
 +
Find the value of f/g for each X
 +
 
 +
X=0: 0.8898;
 +
X=1: 1.1127;
 +
X=2: 0.9891;
 +
X=3: 0.5934
 +
 
 +
Choose the maximum which is [[c=1.1127]]
 +
 
 +
Looking for the max f(x) is 0.4945 and the max g(x) is 0.4444, so we can calculate the max c is 1.1127.
 +
But for the graph, this c is not the best because it does not cover all the point of f(x), so we need to move the c*g(x) graph to cover all f(x), and decreasing the rejection ratio.
 +
 
 +
Limitation: If the shape of the proposed distribution g is very different from the target distribution f, then the rejection rate will be high (High c value). Computationally, the algorithm is always right; however it is inefficient and requires many iterations. <br>
 +
Here is an example:
 +
[[File:ARM_Fail.jpg]]
 +
 
 +
In the above example, we need to move c*g(x) to the peak of f to cover the whole f. Thus c will be very large and 1/c will be small.
 +
The higher the rejection rate, more points will be rejected.<br>
 +
More on rejection/acceptance rate: 1/c is the acceptance rate. As c decreases (note: the minimum value of c is 1), the acceptance rate increases. In our last example, 1/c=1/1.5≈66.67%. Around 67% of points generated will be accepted.<br>
 +
<div style="margin-bottom:10px;border:10px solid red;background: yellow"> the example below provides a better understanding about the pros and cons of the AR method. The AR method is useless when dealing with sampling distribution with a higher peak since c will be large, hence making our algorithm inefficient<br>
 +
which brings the acceptance rate low which leads to very time consuming sampling </div>
 +
<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;">Acceptance-Rejection Method</h2>
 +
<p><b>Problem:</b> The CDF is not invertible or it is difficult to find the inverse.</p>
 +
<p><b>Plan:</b></p>
 +
<ol>
 +
<li>Draw y~g(.)</li>
 +
<li>Draw u~Unif(0,1)</li>
 +
<li>If <math>u\leq \frac{f(y)}{cg(y)}</math>then set x=y. Else return to Step 1</li>
 +
</ol>
 +
<p>x will have the desired distribution.</p>
 +
<b>Matlab Example</b>
 +
<pre style="font-size:16px">close all
 +
clear all
 +
ii=1;
 +
R=1;
 +
while ii&lt;1000
 +
  u1 = rand;
 +
  u2 = rand;
 +
  y = R*(2*u2-1);
 +
  if (1-u1^2)&gt;=(2*u2-1)^2
 +
    x(ii) = y;
 +
    ii = ii + 1;
 +
  end
 +
end
 +
hist(x,20)
 +
</pre>
 +
</div>
 +
 
 +
 
 +
Recall that,
 +
Suppose we have an efficient method for simulating a random variable having probability mass function {q(j),j>=0}. We can use this as the basis for simulating from the distribution having mass function {p(j),j>=0} by first simulating a random variable Y having mass function {q(j)} and then accepting this simulated value with a probability proportional to p(Y)/q(Y).
 +
Specifically, let c be a constant such that
 +
                p(j)/q(j)<=c for all j such that p(j)>0
 +
We now have the following technique, called the acceptance-rejection method, for simulating a random variable X having mass function p(j)=P{X=j}.
 +
 
 +
=== Sampling from commonly used distributions ===
 +
 
 +
Please note that this is not a general technique as is that of acceptance-rejection sampling. Later, we will generalize the distributions for multidimensional purposes.
 +
 
 +
* '''Gamma'''<br />
 +
 
 +
The CDF of the Gamma distribution <math>Gamma(t,\lambda)</math> is(t denotes the shape, <math>\lambda</math> denotes the scale:  <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.
 +
 
 +
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.
 +
 
 +
* '''Additive Property'''<br />
 +
If <math>X_1, \dots, X_t</math> are independent exponential distributions with hazard rate <math> \lambda </math> (in other words, <math> X_i\sim~ Exp (\lambda) </math><math>,  Exp (\lambda)= Gamma (1, \lambda)), then \Sigma_{i=1}^t X_i \sim~ Gamma (t, \lambda) </math>
 +
 
 +
 
 +
Side notes: if <math> X_i\sim~ Gamma(a,\lambda)</math> and <math> Y_i\sim~ Gamma(B,\lambda)</math> are independent gamma distributions, then <math>\frac{X}{X+Y}</math> has a distribution of <math> Beta(a,B). </math>
 +
 
 +
 
 +
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.<br />
 +
<math> x_1 \sim~Exp(\lambda)</math><br />
 +
<math>x_2 \sim~Exp(\lambda)</math><br />
 +
...<br />
 +
<math>x_t \sim~Exp(\lambda)</math><br />
 +
<math>x_1+x_2+...+x_t~</math>
 +
 
 +
<pre style="font-size:16px">
 +
>>l=1
 +
>>u-rand(1,1000);
 +
>>x=-(1/l)*log(u); 
 +
>>hist(x)
 +
>>rand
 +
</pre>
 +
 
 +
 
 +
* '''Procedure '''
 +
 
 +
:#Sample  independently from a uniform distribution <math>t</math> times, giving <math> U_1,\dots,U_t \sim~ U(0,1)</math>
 +
:#Use the Inverse Transform Method, <math> X_i = -\frac {1}{\lambda}\log(1-U_i)</math>, giving <math> X_1,\dots,X_t \sim~Exp(\lambda)</math>
 +
:#Use the additive property,<math> X = \Sigma_{i=1}^t X_i \sim~ Gamma (t, \lambda)  </math><br>
 +
 
 +
 
 +
* '''Note for Procedure '''
 +
:#If <math>U\sim~U(0,1)</math>, then <math>U</math> and <math>1-U</math> will have the same distribution (both follows <math>U(0,1)</math>)
 +
:#This is because the range for <math>1-U</math> is still <math>(0,1)</math>, and their densities are identical over this range.
 +
:#Let <math>Y=1-U</math>, <math>Pr(Y<=y)=Pr(1-U<=y)=Pr(U>=1-y)=1-Pr(U<=1-y)=1-(1-y)=y</math>, thus <math>1-U\sim~U(0,1)</math>
 +
 
 +
 
 +
 
 +
* '''Code'''<br />
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>lambda = 1;
 +
>>u = rand(20, 1000);            Note: this command generate a 20x1000 matrix
 +
                            (which means we generate 1000 number for each X_i with t=20);
 +
                            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)
 +
>>xx = sum(x)                    Note: sum(x) will sum all elements in the same column.
 +
                                                size(xx) can help you to verify
 +
>>size(sum(x))                  Note: see the size of x if we forget it
 +
                                      (the answer is 20 1000)
 +
>>hist(x(1:))                    Note: the graph of the first exponential distribution
 +
>>hist(xx)
 +
</pre>
 +
[[File:Gamma_example.jpg|300px]]
 +
 
 +
 
 +
 
 +
size(x) and size(u) are both 20*1000 matrix.
 +
Since if u~unif(0, 1), u and 1 - u have the same distribution, we can substitute 1-u with u to simply the equation.
 +
Alternatively, the following command will do the same thing with the previous commands.
 +
 
 +
* '''Code'''<br />
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>lambda = 1;
 +
>>xx = sum((-1/lambda)*log(rand(20, 1000))); ''This is simple way to put the code in one line.
 +
                                              Here we can use either log(u) or log(1-u) since U~U(0,1);
 +
>>hist(xx)
 +
</pre>
 +
 
 +
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.
 +
 
 +
=== Other Sampling Method: Box Muller ===
 +
[[File:Unnamed_QQ_Screenshot20130521203625.png‎]]
 +
* From cartesian to polar coordinates <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 />
 +
 +
*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''' ===
 +
 
 +
If X is a matrix,
 +
* ''X(1,:)'' returns the first row
 +
* ''X(:,1)'' returns the first column
 +
* ''X(i,j)'' returns the (i,j)th entry
 +
* ''sum(X,'''1''')'' or ''sum(X)'' is a summation of the '''rows''' of X. The output is a row vector of the sums of each column.
 +
* ''sum(X,'''2''')'' is a summation of the '''columns''' of X, returning a vector.
 +
* ''rand(r,c)'' will generate uniformly distributed random numbers in r rows and c columns.
 +
* The dot operator (.), when placed before a function, such as +,-,^, *, and many others specifies to apply that function to every element of a vector or a matrix. For example, to add a constant c to elements of a matrix A, do A.+c as opposed to simply A+c. The dot operator is not required for functions that can only take a number as their input (such as log).
 +
* Matlab processes loops very slow, while it is fast with matrices and vectors, so it is preferable to use the dot operator to and matrices of random numbers than loops if it is possible.
 +
 
 +
== Class 6 - Thursday, May 23 ==
 +
 
 +
=== Announcement ===
 +
1. On the day of each lecture, students from the morning section can only contribute the first half of the lecture (i.e. 8:30 - 9:10 am), so that the second half can be saved for the ones from the afternoon section. After the day of lecture, students are free to contribute anything.
 +
 
 +
=== Standard Normal distribution ===
 +
If X ~ N(0,1) i.e. Standard Normal Distribution - then its p.d.f. is of the form
 +
:<math>f(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}</math>
 +
 
 +
*Warning : the General Normal distribution is:
 +
<table>
 +
<tr>
 +
<td><div onmouseover="document.getElementById('woyun').style.visibility='visible'"
 +
onmouseout="document.getElementById('woyun').style.visibility='hidden'">
 +
<math>
 +
f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }
 +
</math>
 +
</div>
 +
</td>
 +
<td>
 +
<div id="woyun" style="
 +
 
 +
visibility:hidden;
 +
width:100px;
 +
height:100px;
 +
background:#FFFFAD;
 +
position:relative;
 +
animation:movement infinite;
 +
animation-duration:2s;
 +
animation-direction:alternate;
 +
 
 +
 
 +
/* Safari and Chrome */
 +
-webkit-animation:movement infinite;
 +
-webkit-animation-duration:2s;
 +
-webkit-animation-direction:alternate;
 +
 
 +
 
 +
@keyframes movement
 +
{
 +
from {left:0px;}
 +
to {left:200px;}
 +
}
 +
 
 +
@-webkit-keyframes movement /* Safari and Chrome */
 +
{
 +
from {left:0px;}
 +
to {left:200px;}
 +
}"
 +
>which is almost useless in this course</div>
 +
</td>
 +
</tr>
 +
</table>
 +
where <math> \mu </math> is the mean or expectation of the distribution and <math> \sigma </math> is standard deviation <br />
 +
<br />
 +
*N(0,1) is standard normal. <math> \mu </math> =0 and <math> \sigma </math>=1 <br />
 +
<br />
 +
 
 +
Let X and Y be independent standard normal.
 +
 
 +
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]]
 +
 
 +
Note: R must satisfy two properties:
 +
 
 +
:1. Be a positive number (as it is a length)
 +
 
 +
:2. It must be from a distribution that has more data points closer to the origin so that as we go further from the origin, less points are generated (the two options are Chi-squared and Exponential distribution)
 +
 
 +
The form of the joint distribution of R and <math>\theta</math> will show that the best choice for distribution of R<sup>2</sup> is exponential.
 +
 
 +
 
 +
We cannot use the Inverse Transformation Method since F(x) does not have a closed form solution. So we will use joint probability function of two independent standard normal random variables and polar coordinates to simulate the distribution:
 +
 
 +
We know that
 +
 
 +
<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(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. 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 />
 +
 
 +
<math>x= \sqrt {d}\cos \theta </math>
 +
<math>y= \sqrt {d}\sin \theta </math>
 +
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>
 +
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>
 +
 
 +
 
 +
 
 +
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{align} R^2 = d = x^2 + y^2 \end{align} </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>
 +
<br>
 +
<math>\begin{align} f(\theta) =\frac{1}{2\pi}\ \end{align}</math>
 +
<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 />
 +
2) Transforming back to rectangular (Cartesian) coordinates.<br />
 +
 
 +
 
 +
'''Alternative Method of Generating Standard Normal Random Variables'''<br /> 
 +
 
 +
Step 1: Generate <math>u_{1}</math> ~<math>Unif(0,1)</math><br />
 +
Step 2: Generate <math>Y_{1}</math> ~<math>Exp(1)</math>,<math>Y_{2}</math>~<math>Exp(2)</math><br />
 +
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>\phi(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}.</math>
 +
:<math>=\;\int_{-\infty}^{\infty} x \phi(x), dx.</math>
 +
:Since the first derivative ''ϕ''′(''x'') is −''xϕ''(''x'')
 +
:<math>=\;\ - \int_{-\infty}^{\infty} \phi'(x), dx.</math>
 +
:<math>= - \left[\phi(x)\right]_{-\infty}^{\infty}</math>
 +
:<math>= 0</math><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>
 +
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>
 +
 
 +
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>
 +
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>
 +
 
 +
 
 +
1) Generate random number <math> \begin{align} U_1,U_2 \sim~ \mathrm{Unif}(0, 1) \end{align} </math> <br />
 +
2) Generate polar coordinates using the exponential distribution of d and uniform distribution of θ,
 +
 
 +
 
 +
 
 +
<math> \begin{align} R^2 = d = -2\log(U_1), & \quad r = \sqrt{d} \\ & \quad \theta = 2\pi    U_2  \end{align} </math>
 +
 
 +
 
 +
<math> \begin{matrix} \ R^2 \sim~ Exp(1/2),  \theta \sim~ Unif[0,2\pi] \end{matrix} </math> <br />
 +
 
 +
Note: If U~unif(0,1), then ln(1-U)=ln(U)
 +
 
 +
3) Transform polar coordinates (i.e. R and θ) back to Cartesian coordinates (i.e. X and Y), <br> <math> \begin{align} x = R\cos(\theta) \\ y = R\sin(\theta) \end{align} </math> <br />.
 +
 
 +
Alternatively,<br> <math> x =\cos(2\pi U_2)\sqrt{-2\ln U_1}\, </math> and<br> <math> y =\sin(2\pi U_2)\sqrt{-2\ln U_1}\, </math><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 />
 +
 
 +
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 />
 +
If you want 8 independent standard normal distributed numbers, then run the Box-Muller methods 4 times (8/2 times). <br />
 +
If you want 9 independent standard normal distributed numbers, then run the Box-Muller methods 5 times (10/2 times), and then delete one. <br />
 +
 
 +
 
 +
'''Matlab Code'''<br />
 +
 
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>u1=rand(1,1000);
 +
>>u2=rand(1,1000);
 +
>>d=-2*log(u1);
 +
>>tet=2*pi*u2;
 +
>>x=d.^0.5.*cos(tet);
 +
>>y=d.^0.5.*sin(tet);
 +
>>hist(tet)       
 +
>>hist(d)
 +
>>hist(x)
 +
>>hist(y)
 +
</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>
 +
 
 +
'''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>
 +
 +
 
 +
[[File:Normal_theta.jpg|300px]][[File:Normal_d.jpg|300px]]
 +
[[File:normal_x.jpg|300x300px]][[File:normal_y.jpg|300x300px]]
 +
 
 +
As seen in the histograms above, X and Y generated from this procedure have a standard normal distribution.
 +
 
 +
* '''Code'''<br />
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>x=randn(1,1000);
 +
>>hist(x)
 +
>>hist(x+2)
 +
>>hist(x*2+2)<br>
 +
</pre>
 +
<br>
 +
'''Note:'''<br>
 +
1. randn is random sample from a standard normal distribution.<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]]
 +
<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 />
 +
 
 +
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 />
 +
3. '''Ziggurat algorithm''' which is known to be faster than Box-Muller transformation and a version of this algorithm is used for the randn function in matlab.<br />
 +
 
 +
If Z~N(0,1) and X= μ +Zσ then X~<math> N(\mu, \sigma^2)</math>
 +
 
 +
If Z<sub>1</sub>, Z<sub>2</sub>... Z<sub>d</sub> are independent identically distributed N(0,1),
 +
then Z=(Z<sub>1</sub>,Z<sub>2</sub>...Z<sub>d</sub>)<sup>T</sup> ~N(0, I<sub>d</sub>), where 0 is the zero vector and I<sub>d</sub> is the identity matrix.
 +
 
 +
For the histogram, the constant is the parameter that affect the center of the graph.
 +
 
 +
=== Proof of Box Muller Transformation ===
 +
 
 +
'''Definition:'''<br />
 +
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
 +
<math>X_{1} = ((-2lnU_{1})^.5)*cos(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.
 +
 
 +
This is a standard transformation problem. The joint distribution is given by
 +
                  f(x1 ,x2) = f<sub>u1</sub>, <sub>u2</sub>(g1^− 1(x1,x2),g2^− 1(x1,x2)) * | J |
 +
 
 +
where J is the Jacobian of the transformation,
 +
                   
 +
                  J = |∂u<sub>1</sub>/∂x<sub>1</sub>,∂u<sub>1</sub>/∂x<sub>2</sub>|
 +
                      |∂u<sub>2</sub>/∂x<sub>1</sub>,∂u<sub>2</sub>/∂x<sub>2</sub>|
 +
where
 +
      u<sub>1</sub> = g<sub>1</sub> ^-1(x1,x2)
 +
      u<sub>2</sub> = g<sub>2</sub> ^-1(x1,x2)
 +
 
 +
Inverting the above transformation, we have
 +
    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>)
 +
 
 +
Finally we get
 +
  f(x1,x2) = {exp^(-(x1^2+x2^2)/2)}/2pi
 +
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 distribution is a special version of the standard normal distribution. The domain of the general normal distribution is affected by the standard deviation and  translated by the mean value.
 +
*The pdf of the general normal distribution is
 +
:
 +
<table>
 +
<tr>
 +
<td><div onmouseover="document.getElementById('woyun').style.visibility='visible'"
 +
onmouseout="document.getElementById('woyun').style.visibility='hidden'">
 +
<math>
 +
f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }
 +
</math>
 +
</div>
 +
</td>
 +
<td>
 +
<div id="woyun" style="visibility:hidden">which is almost useless in this course</div>
 +
</td>
 +
</tr>
 +
</table>
 +
where <math> \mu </math> is the mean or expectation of the distribution and <math> \sigma </math> is standard deviation <br />
 +
 
 +
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 <math>Z_1,...Z_d</math> ~ N(0,1) and are independent then <math>Z = (Z_1,..Z_d)^{T} </math>~ <math>N(0,I_d)</math>
 +
ie.
 +
* '''Code'''<br />
 +
<pre style="font-size:16px">
 +
>>close all
 +
>>clear all
 +
>>z1=randn(1,1000);    <-generate variable from standard normal distribution
 +
>>z2=randn(1,1000);
 +
>>z=[z1;z2];          <-produce a vector
 +
>>plot(z(1,:),z(2,:),'.')
 +
</pre>
 +
[[File:Nonstdnormal_example.jpg|300px]]
 +
 
 +
If Z~N(0,Id) and X= <math>\underline{\mu} +  \Sigma^{\frac{1}{2}} \,Z </math> then <math>\underline{X}</math> ~<math>N(\underline{\mu},\Sigma)</math>
 +
 
 +
=====Non-Standard Normal Distributions=====
 +
 
 +
'''Example 1: Single-variate Normal'''
 +
 
 +
If X ~ Norm(0, 1) then (a + bX) has a normal distribution with a mean of <math>\displaystyle a</math> and a standard deviation of <math>\displaystyle b</math> (which is equivalent to a variance of <math>\displaystyle b^2</math>).  Using this information with the Box-Muller transform, we can generate values sampled from some random variable <math>\displaystyle Y\sim N(a,b^2) </math> for arbitrary values of <math>\displaystyle a,b</math>.
 +
 
 +
# Generate a sample u from Norm(0, 1) using the Box-Muller transform.
 +
# Set v = a + bu.
 +
 
 +
The values for v generated in this way will be equivalent to sample from a <math>\displaystyle N(a, b^2)</math>distribution.  We can modify the MatLab code used in the last section to demonstrate this.  We just need to add one line before we generate the histogram:
 +
 
 +
<pre style='font-size:16px'>
 +
v = a + b * x;
 +
</pre>
 +
 
 +
For instance, this is the histogram generated when b = 15, a = 125:
 +
 
 +
[[File:Hist normal.jpg|center|500]]
 +
 
 +
'''Example 2: Multi-variate Normal'''
 +
 
 +
The Box-Muller method can be extended to higher dimensions to generate multivariate normals. The objects generated will be nx1 vectors, and their variance will be described by nxn covariance matrices.
 +
 
 +
<math>\mathbf{z} = N(\mathbf{u}, \Sigma)</math> defines the n by 1 vector <math>\mathbf{z}</math> such that:
 +
 
 +
* <math>\displaystyle u_i</math> is the average of <math>\displaystyle z_i</math>
 +
* <math>\!\Sigma_{ii}</math> is the variance of <math>\displaystyle z_i</math>
 +
* <math>\!\Sigma_{ij}</math> is the co-variance of <math>\displaystyle z_i</math> and <math>\displaystyle z_j</math>
 +
 
 +
If <math>\displaystyle z_1, z_2, ..., z_d</math> are normal variables with mean 0 and variance 1, then the vector <math>\displaystyle (z_1, z_2,..., z_d) </math> has mean 0 and variance <math>\!I</math>, where 0 is the zero vector and <math>\!I</math> is the identity matrix. This fact suggests that the method for generating a multivariate normal is to generate each component individually as single normal variables.
 +
 
 +
The mean and the covariance matrix of a multivariate normal distribution can be adjusted in ways analogous to the single variable case. If <math>\mathbf{z} \sim N(0,I)</math>, then <math>\Sigma^{1/2}\mathbf{z}+\mu \sim N(\mu,\Sigma)</math>. Note here that the covariance matrix is symmetric and nonnegative, so its square root should always exist.
 +
 
 +
We can compute <math>\mathbf{z}</math> in the following way:
 +
 
 +
# Generate an n by 1 vector <math>\mathbf{x} = \begin{bmatrix}x_{1} & x_{2} & ... & x_{n}\end{bmatrix}</math> where <math>x_{i}</math> ~ Norm(0, 1) using the Box-Muller transform.
 +
# Calculate <math>\!\Sigma^{1/2}</math> using singular value decomposition.
 +
# Set <math>\mathbf{z} = \Sigma^{1/2} \mathbf{x} + \mathbf{u}</math>.
 +
 
 +
The following MatLab code provides an example, where a scatter plot of 10000 random points is generated.  In this case x and y have a co-variance of 0.9 - a very strong positive correlation.
 +
 
 +
<pre style='font-size:16px'>
 +
x = zeros(10000, 1);
 +
y = zeros(10000, 1);
 +
for ii = 1:10000
 +
    u1 = rand;
 +
    u2 = rand;
 +
    R2 = -2 * log(u1);
 +
    theta = 2 * pi * u2;
 +
    x(ii) = sqrt(R2) * cos(theta);
 +
    y(ii) = sqrt(R2) * sin(theta);
 +
end
 +
 
 +
E = [1, 0.9; 0.9, 1];
 +
[u s v] = svd(E);
 +
root_E = u * (s ^ (1 / 2)) * u';
 +
 
 +
z = (root_E * [x y]');
 +
z(1,:) = z(1,:) + 0;
 +
z(2,:) = z(2,:) + -3;
 +
 
 +
scatter(z(1,:), z(2,:))
 +
</pre>
 +
 
 +
Note: The svd command computes the matrix singular value decomposition.
 +
 
 +
[u,s,v] = svd(E) produces a diagonal matrix s of the same dimension as E, with nonnegative diagonal elements in decreasing order, and unitary matrices u and v so that E = u*s*v'.
 +
 
 +
This code generated the following scatter plot:
 +
 
 +
[[File:scatter covar.jpg|center|500px]]
 +
 
 +
In Matlab, we can also use the function "sqrtm()" or "chol()" (Cholesky Decomposition) to calculate square root of a matrix directly. Note that the resulting root matrices may be different but this does materially affect the simulation.
 +
Here is an example:
 +
 
 +
<pre style='font-size:16px'>
 +
E = [1, 0.9; 0.9, 1];
 +
r1 = sqrtm(E);
 +
r2 = chol(E);
 +
</pre>
 +
 
 +
R code for a multivariate normal distribution:
 +
 
 +
<pre style='font-size:16px'>
 +
n=10000;
 +
r2<--2*log(runif(n));
 +
theta<-2*pi*(runif(n));
 +
x<-sqrt(r2)*cos(theta);
 +
 
 +
y<-sqrt(r2)*sin(theta);
 +
a<-matrix(c(x,y),nrow=n,byrow=F);
 +
e<-matrix(c(1,.9,09,1),nrow=2,byrow=T);
 +
svde<-svd(e);
 +
root_e<-svde$u %*% diag(svde$d)^1/2;
 +
z<-t(root_e %*%t(a));
 +
z[,1]=z[,1]+5;
 +
z[,2]=z[,2]+ -8;
 +
par(pch=19);
 +
plot(z,col=rgb(1,0,0,alpha=0.06))
 +
</pre>
 +
 
 +
[[File:m_normal.png|center|500px]]
 +
 
 +
=== Bernoulli Distribution ===
 +
The Bernoulli distribution is a discrete probability distribution, which usually describes an event that only has two possible results, i.e. success or failure (x=0 or 1). If the event succeed, we usually take value 1 with success probability p, and take value 0 with failure probability q = 1 - p.
 +
 
 +
P ( x = 0) = q = 1 - p <br />
 +
P ( x = 1) = p  <br />
 +
P ( x = 0) + P (x = 1) = p + q = 1 <br />
 +
 
 +
If X~Ber(p), its pdf is of the form <math>f(x)= p^{x}(1-p)^{(1-x)}</math>, x=0,1
 +
<br> P is the success probability.
 +
 +
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>)
 +
we are interested in y=min(x1,x2)
 +
 
 +
<pre style="font-size:16px">
 +
 
 +
Procedure:
 +
 
 +
To simulate the event of flipping a coin, let P be the probability of flipping head and X = 1 and 0 represent
 +
flipping head and tail respectively:
 +
 
 +
1. Draw U ~ Uniform(0,1)
 +
 
 +
2. If U <= P
 +
 
 +
  X = 1
 +
 
 +
  Else
 +
 
 +
  X = 0
 +
 
 +
3. Repeat as necessary
 +
 
 +
</pre>
 +
 
 +
An intuitive way to think of this is in the coin flip example we discussed in a previous lecture. In this example we set p = 1/2 and this allows for 50% of points to be heads or tails.
 +
 
 +
* '''Code to Generate Bernoulli(p = 0.3)'''<br />
 +
<pre style="font-size:16px">
 +
i = 1;
 +
 
 +
while (i <=1000)
 +
    u =rand();
 +
    p = 0.3;
 +
    if (u <= p)
 +
        x(i) = 1;
 +
    else
 +
        x(i) = 0;
 +
    end
 +
    i = i + 1;
 +
end
 +
 
 +
hist(x)
 +
</pre>
 +
 
 +
However, we know that if <math>\begin{align} X_i \sim Bernoulli(p) \end{align}</math> where each <math>\begin{align} X_i \end{align}</math> is independent,<br />
 +
<math>U = \sum_{i=1}^{n} X_i \sim Binomial(n,p)</math><br />
 +
So we can sample from binomial distribution using this property.
 +
Note: We can consider Binomial distribution as the sum of n, ''independent'', Bernoulli distributions
 +
<div style="background:#CCFF33;border-radius:5px;box-shadow: 10px 10px 5px #888888;padding:30px;">
 +
* '''Code to Generate Binomial(n = 20,p = 0.7)'''<br />
 +
<pre style="font-size:16px">
 +
p = 0.7;
 +
n = 20;
 +
 
 +
for k=1:5000
 +
    i = 1;
 +
    for i=1:n
 +
        u=rand();
 +
        if (u <= p)
 +
            y(i) = 1;
 +
        else
 +
            y(i) = 0;
 +
        end
 +
    end
 +
 
 +
    x(k) = sum(y==1);
 +
end
 +
 
 +
hist(x)
 +
 
 +
</pre>
 +
 
 +
 
 +
 
 +
 
 +
</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.
 +
 
 +
Comments on Matlab:
 +
When doing operations on vectors, always put a dot before the operator if you want the operation to be done to every element in the vector.
 +
example: Let V be a vector with dimension 2*4 and you want each element multiply by 3.
 +
        The  Matlab code is 3.*V
 +
 
 +
some examples for using code to generate distribution.
 +
 
 +
== Class 7 - Tuesday, May 28 ==
 +
 
 +
 
 +
[[Note that the material in this lecture will not be on the exam; it was only to supplement what we have learned.]]
 +
===Universality of the Uniform Distribution/Inverse Method===
 +
 
 +
The inverse method is universal in the sense that we can potentially sample from any distribution where we can find the inverse of the cumulative distribution function.
 +
 
 +
Procedure:
 +
 
 +
1) Generate U~Unif (0, 1)<br>
 +
2) Set <math>x=F^{-1}(u)</math><br>
 +
3) X~f(x)<br>
 +
 
 +
'''Remark'''<br>
 +
1) The preceding can be written algorithmically for discrete random variables as <br>
 +
Generate a random number U ~ U(0,1] <br>
 +
If U < p<sub>0</sub> set X = x<sub>0</sub> and stop <br>
 +
If U < p<sub>0</sub> + p<sub>1</sub> set X = x<sub>1</sub> and stop <br>
 +
... <br>
 +
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>
 +
 
 +
Let <math>X</math><sub>1</sub>,<math>X</math><sub>2</sub> denote the lifetime of two independent particles:<br>
 +
<math>X</math><sub>1</sub>~exp(<math>\lambda</math><sub>1</sub>)<br>
 +
<math>X</math><sub>2</sub>~exp(<math>\lambda</math><sub>2</sub>)<br>
 +
 
 +
We are interested in <math>y=min(X</math><sub>1</sub><math>,X</math><sub>2</sub><math>)</math><br>
 +
Design an algorithm based on the Inverse-Transform Method to generate samples according to <math>f</math><sub>y</sub><math>(y)</math><br>
 +
 
 +
'''Solution:'''<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)=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>F_Y(y)=1-e^{\, -(\lambda_1 + \lambda_2) y}, y\geq 0</math><br>
 +
 
 +
<math>U=1-e^{\, -(\lambda_1 + \lambda_2) y}</math> => <math>y=\, {-\frac {1}{{\lambda_1 +\lambda_2}}} ln(1-u)</math><br>
 +
 
 +
'''Procedure:'''
 +
 
 +
Step1: Generate U~ U(0, 1)<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>
 +
 
 +
<math>X</math><sub>1</sub>~exp(<math>\lambda</math><sub>1</sub>)<br><math>X</math><sub>2</sub>~exp(<math>\lambda</math><sub>2</sub>)<br> ...<br> <math>X</math><sub>n</sub>~exp(<math>\lambda</math><sub>n</sub>)<br>.
 +
 
 +
And the algorithm using the inverse-transform method as follows:
 +
 
 +
step1: Generate U~U(0,1)
 +
 
 +
Step2: <math>y=\, {-\frac {1}{{ \sum\lambda_i}}} ln(1-u)</math><br>
 +
 
 +
 
 +
'''Example 2'''<br>
 +
Consider U~Unif[0,1)<br>
 +
<math>X=\, a (1-\sqrt{1-u})</math>,
 +
<br>where a>0 and a is a real number
 +
What is the distribution of X?<br>
 +
 
 +
'''Solution:'''<br>
 +
 
 +
We can find a form for the cumulative distribution function of X by isolating U as U~Unif[0,1) will take values from the range of F(X)uniformly. It then remains to differentiate the resulting form by X to obtain the probability density function.
 +
 
 +
<math>X=\, a (1-\sqrt{1-u})</math><br>
 +
=><math>1-\frac {x}{a}=\sqrt{1-u}</math><br>
 +
=><math>u=1-(1-\frac {x}{a})^2</math><br>
 +
=><math>u=\, {\frac {x}{a}} (2-\frac {x}{a})</math><br>
 +
<math>f(x)=\frac {dF(x)}{dx}=\frac {2}{a}-\frac {2x}{a^2}=\, \frac {2}{a} (1-\frac {x}{a})</math><br>
 +
[[File:Example_2_diagram.jpg]]
 +
 
 +
'''Example 3'''<br>
 +
 
 +
Suppose F<sub>X</sub>(x) = x<sup>n</sup>, 0 ≤ x ≤ 1, n ∈ N > 0. Generate values from X.<br>
 +
 
 +
'''Solution:'''<br>
 +
<br>
 +
1. Generate <math>U ~\sim~ Unif[0, 1)</math><br>
 +
2. Set <math>X = U^{1/n}</math><br>
 +
<br>
 +
For example, when <math>n = 20</math>,<br>
 +
<math>U = 0.6</math> => <math>X = U^{1/20} = 0.974</math><br>
 +
<math>U = 0.5 =></math> <math>X = U^{1/20} = 0.966</math><br>
 +
<math>U = 0.2</math> => <math>X = U^{1/20} = 0.923</math><br>
 +
<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
 +
If Y = max (X<sub>1</sub>, X<sub>2</sub>, ... , X<sub>n</sub>), where X<sub>1</sub>, X<sub>2</sub>, ... , X<sub>n</sub> are independent,  <br>
 +
F<sub>Y</sub>(y) = P(Y ≤ y) = P(max (X<sub>1</sub>, X<sub>2</sub>, ... , X<sub>n</sub>) ≤ y) = P(X<sub>1</sub> ≤ y, X<sub>2</sub> ≤ y, ... , X<sub>n</sub> ≤ y) = F<sub>x<sub>1</sub></sub>(y) F<sub>x<sub>2</sub></sub>(y) ... F<sub>x<sub>n</sub></sub>(y)<br>
 +
Similarly if <math> Y = min(X_1,\ldots,X_n)</math> then the cdf of <math>Y</math> is <math>F_Y = 1- </math><math>\prod</math><math>(1- F_{X_i})</math><br>
 +
<br>
 +
'''Method 1:''' Following the above result we can see that in this example, F<sub>X</sub> = x<sup>n</sup> is the cumulative distribution function of the max of n uniform random variables between 0 and 1 (since for U~Unif(0, 1), F<sub>U</sub>(x) = <br>
 +
'''Method 2:'''  generate X by having a sample of n independent U~Unif(0, 1) and take the max of the n samples to be x. However, the solution given above using inverse-transform method only requires generating one uniform random number instead of n of them, so it is a more efficient method.
 +
<br>
 +
 
 +
Generate the Y = max (X1, X2, ... , Xn), Y = min (X1, X2, ... , Xn), pdf and cdf, but (xi and xj are independent) i,j=1,2,3,4,5.....
 +
 
 +
'''Example 4 (New)'''<br>
 +
Now, we are having an similar example as example 1 just doing the maximum way.
 +
 
 +
Let X<sub>1</sub>,X<sub>2</sub> denote the lifetime of two independent particles:<br>
 +
<math>\, X_1, X_2 \sim exp(\lambda)</math><br>
 +
 
 +
We are interested in Z=max(X<sub>1</sub>,X<sub>2</sub>)<br>
 +
Design an algorithm based on the Inverse-Transform Method to generate samples according to f<sub>Z</sub>(z)<br>
 +
 
 +
<math>\, F_Z(z)=P[Z<=z] = F_{X_1}(z) \cdot F_{X_2}(z) = (1-e^{-\lambda z})^2</math><br>
 +
<math> \text{thus } F^{-1}(z) = -\frac{1}{\lambda}\log(1-\sqrt z)</math><br>
 +
 
 +
To sample Z: <br>
 +
<math>\, \text{Step 1: Generate } U \sim U[0,1)</math><br>
 +
<math>\, \text{Step 2: Let } Z = -\frac{1}{\lambda}\log(1-\sqrt U)</math>, therefore we can generate random variable of Z.<br><br>
 +
 
 +
'''Discrete Case:'''
 +
<font size="3">
 +
  u~unif(0,1)<br>
 +
  x <- 0, S <- P<sub>0</sub><br>
 +
  while u < S<br>
 +
        x <- x + 1<br>
 +
        S <- S + P<sub>0</sub><br>
 +
  Return x<br></font>
 +
 
 +
===Decomposition Method===
 +
The CDF, F, is a composition if <math>F_{X}(x)</math> can be written as:
 +
 
 +
<math>F_{X}(x) = \sum_{i=1}^{n}p_{i}F_{X_{i}}(x)</math> where
 +
 
 +
1)  p<sub>i</sub> > 0
 +
 
 +
2)  <math>\sum_{i=1}^{n}</math>p<sub>i</sub> = 1.
 +
 
 +
3)  <math>F_{X_{i}}(x)</math> is a CDF
 +
 
 +
The general algorithm to generate random variables from a composition CDF is:
 +
 
 +
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>
 +
 
 +
3)  Else if U < p<sub>1</sub> + p<sub>2</sub>, V = <math>F_{X_{2}}(x)</math><sup>-1</sup>
 +
 
 +
4)  Repeat from Step 1 (if N randomly generated variables needed, repeat N times)
 +
 
 +
<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.
 +
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 ===
 +
<b>Example 1</b> <br>
 +
<math>f(x) = \frac{5}{12}(1+(x-1)^4)  0\leq x\leq 2</math> <br>
 +
<math>f(x) = \frac{5}{12}+\frac{5}{12}(x-1)^4 = \frac{5}{6} (\frac{1}{2})+\frac {1}{6}(\frac{5}{2})(x-1))^4</math> <br>
 +
Let<math>f_{x_1}= \frac{1}{2}</math>  and <math>f_{x_2} = \frac {5}{2}(x-1)^4</math> <br>
 +
 
 +
Algorithm:
 +
Generate U~Unif(0,1) <br>
 +
If <math>0<u<\frac {5}{6}</math>, then we sample from f<sub>x1</sub> <br>
 +
Else if <math>\frac{5}{6}<u<1</math>, we sample from f<sub>x2</sub> <br>
 +
We can find the inverse CDF of f<sub>x2</sub> and utilize the Inverse Transform Method in order to sample from f<sub>x2</sub> <br>
 +
Sampling from f<sub>x1</sub> is more straightforward since it is uniform over the interval (0,2) <br>
 +
 
 +
divided f(x) to two pdf of x1 and x2, with uniform distribution, of two range of uniform.
 +
 
 +
<b>Example 2</b> <br>
 +
<math>f(x)=\frac{1}{4}e^{-x}+2x+\frac{1}{12}, \quad 0\leq x \leq 3 </math> <br>
 +
We can rewrite f(x) as <math>f(x)=(\frac{1}{4}) e^{-x}+(\frac{2}{4}) 4x+(\frac{1}{4}) \frac{1}{3}</math> <br>
 +
Let f<sub>x1</sub> = <math>e^{-x}</math>, f<sub>x2</sub> = 4x, and f<sub>x3</sub> = <math>\frac{1}{3}</math> <br>
 +
Generate U~Unif(0,1)<br>
 +
If <math>0<u<\frac{1}{4}</math>, we sample from f<sub>x1</sub> <br><br>
 +
If <math>\frac{1}{4}\leq u < \frac{3}{4}</math>, we sample from f<sub>x2</sub> <br><br>
 +
Else if <math>\frac{3}{4} \leq u < 1</math>, we sample from f<sub>x3</sub> <br>
 +
We can find the inverse CDFs of f<sub>x1</sub> and f<sub>x2</sub> and utilize the Inverse Transform Method in order to sample from f<sub>x1</sub> and f<sub>x2</sub> <br><br>
 +
We find F<sub>x1</sub> = <math> 1-e^{-x}</math> and F<sub>x2</sub> = <math>2x^{2}</math> <br>
 +
We find the inverses are <math> X = -ln(1-u)</math> for F<sub>x1</sub> and <math> X = \sqrt{\frac{U}{2}}</math> for F<sub>x2</sub> <br>
 +
Sampling from f<sub>x3</sub> is more straightforward since it is uniform over the interval (0,3) <br>
 +
 
 +
In general, to write an <b>efficient </b> algorithm for: <br>
 +
<math>F_{X}(x) = p_{1}F_{X_{1}}(x) + p_{2}F_{X_{2}}(x) + ... + p_{n}F_{X_{n}}(x)</math> <br>
 +
We would first calculate <math> {q_i} = \sum_{j=1}^i p_j, \forall i = 1,\dots, n</math>
 +
Then Generate <math> U \sim~ Unif(0,1) </math> <br>
 +
If <math> U < q_1 </math> sample from <math> f_1 </math> <br>
 +
else if <math> u<q_i </math> sample from <math> f_i </math> for <math> 1 < i < n </math><br>
 +
else sample from <math> f_n </math> <br>
 +
 
 +
when we divided the pdf of different range of f(x1) f(x2) and f(x3), and generate all of them and inverse, U~U(0,1)
 +
<div style="background:#66CCFF;padding:20px;border-radius:5px;">
 +
 
 +
=== Example of Decomposition Method ===
 +
 
 +
<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 <math>U =F_x(x) = \frac {1}{3} x+\frac {1}{3} x^2+\frac {1}{3} x^3</math>, solve for x.
 +
 
 +
<math>P_1=\frac{1}{3}, F_{x1} (x)= x, P_2=\frac{1}{3},F_{x2} (x)= x^2,
 +
P_3=\frac{1}{3},F_{x3} (x)= x^3</math>
 +
 
 +
'''Algorithm:'''
 +
 
 +
Generate <math>\,U \sim Unif [0,1)</math>
 +
 
 +
Generate <math>\,V \sim  Unif [0,1)</math>
 +
 
 +
if <math>0\leq u \leq \frac{1}{3}, x = v</math>
 +
 
 +
else if <math>u \leq \frac{2}{3},  x = v^{\frac{1}{2}}</math>
 +
 
 +
else <math>x=v^{\frac{1}{3}}</math> <br>
 +
 
 +
 
 +
'''Matlab Code:'''
 +
<pre style="font-size:16px">
 +
u=rand # U is
 +
v=rand
 +
if u<1/3
 +
x=v
 +
elseif u<2/3
 +
x=sqrt(v)
 +
else
 +
x=v^(1/3)
 +
end
 +
</pre>
 +
 
 +
 
 +
<span style="margin:0 auto;">=== Example of Decomposition Method(new) ===</span>
 +
 
 +
F<sub>x</sub>(x) = 1/2*x+1/2*x<sup>2</sup>, 0<= x<=1
 +
 
 +
let U =F<sub>x</sub>(x) = 1/2*x+1/2*x<sup>2</sup>, solve for x.
 +
 
 +
P<sub>1</sub>=1/2, F<sub>x1</sub>(x)= x, P<sub>2</sub>=1/2,F<sub>x2</sub>(x)= x<sup>2</sup>,
 +
 
 +
'''Algorithm:'''
 +
 
 +
Generate U ~ Unif [0,1)
 +
 
 +
Generate V~ Unif [0,1)
 +
 
 +
if 0<u<1/2, x = v
 +
 
 +
else x = v<sup>1/2</sup>
 +
 
 +
 
 +
'''Matlab Code:'''
 +
<pre style="font-size:16px">
 +
u=rand
 +
v=rand
 +
if u<1/2
 +
x=v
 +
else
 +
x=sqrt(v)
 +
end
 +
</pre>
 +
</div>
 +
 
 +
'''Extra Knowledge about Decomposition Method'''
 +
 
 +
There are different types and applications of Decomposition Method
 +
 
 +
1. Primal decomposition
 +
 
 +
2. Dual decomposition
 +
 
 +
3. Decomposition with constraints
 +
 
 +
4. More general decomposition structures
 +
 
 +
5. Rate control
 +
 
 +
6. Single commodity network flow
 +
 
 +
For More Details, please refer to http://www.stanford.edu/class/ee364b/notes/decomposition_notes.pdf
 +
 
 +
===Fundamental Theorem of Simulation===
 +
Consider two shapes, A and B, where B is a sub-shape (subset) of A.
 +
We want to sample uniformly from inside the shape B.
 +
Then we can sample uniformly inside of A, and throw away all samples outside of B, and this will leave us with a uniform sample from within B.
 +
(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.<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>
 +
 
 +
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===
 +
 
 +
Use Acceptance and Rejection Method to sample from <math>f_X(x)=b*x^n*(1-x)^n</math> , <math>n>0</math>, <math>0<x<1</math>
 +
 
 +
Solution:
 +
This is a beta distribution,  Beta ~<math>\int _{0}^{1}b*x^{n}*(1-x)^{n}dx = 1</math>
 +
 
 +
U<sub>1~Unif[0,1)
 +
 
 +
 
 +
U<sub>2~Unif[0,1)
 +
 
 +
fx=<math> bx^{1/2}(1-x)^{1/2} <= bx^{-1/2}\sqrt2  ,0<=x<=1/2 </math>
 +
 
 +
 
 +
 
 +
The beta distribution maximized at 0.5 with value <math>(1/4)^n</math>.
 +
So, <math>c=b*(1/4)^n</math><br />
 +
Algorithm: <br />
 +
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><br />
 +
  then X=U_1
 +
  Else return to step 1.
 +
 
 +
Discrete Case:
 +
Most discrete random variables do not have a closed form inverse CDF. Also, its CDF <math>F:X \rightarrow [0,1]</math> is not necessarily onto. This means that not every point in the interval <math> [0,1] </math> has a preimage in the support set of X through the CDF function.<br />
 +
 
 +
Let <math>X</math> be a discrete random variable where <math>a \leq X \leq b</math> and <math>a,b \in \mathbb{Z}</math> . <br>
 +
To sample from <math>X</math>, we use the partition method below: <br>
 +
 
 +
<math>\, \text{Step 1: Generate u from } U \sim Unif[0,1]</math><br>
 +
<math>\, \text{Step 2: Set } x=a, s=P(X=a)</math><br />
 +
<math>\, \text{Step 3: While } u>s, x=x+1, s=s+P(X=x)</math> <br />
 +
<math>\, \text{Step 4: Return } x</math><br />
 +
 
 +
==Class 8 - Thursday, May 30, 2013==
 +
 
 +
In this lecture, we will discuss algorithms to generate 3 well-known distributions: Binomial, Geometric and Poisson. For each of these distributions, we will first state its general understanding, probability mass function, expectation and variance. Then, we will derive one or more algorithms to sample from each of these distributions, and implement the algorithms on Matlab. <br \>
 +
 
 +
===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 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>
 +
P(X=1)= p
 +
P(X=0)= q = 1-p
 +
Therefore, P(X=0) + P(X=1) = p + q = 1
 +
 
 +
'''Algorithm: '''
 +
 
 +
1) Generate <math>u\sim~Unif(0,1)</math> <br>
 +
2) If <math>u \leq p</math>, then <math>x = 1 </math><br>
 +
else <math>x = 0</math> <br>
 +
The answer is: <br>
 +
when <math> U \leq p, x=1</math> <br>
 +
when <math>U \geq p, x=0</math><br>
 +
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===
 +
 
 +
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 />
 +
Or f(x) = <math>(n!/x!(n-x)!)</math> p<sup>x</sup>(1-p)<sup>(n-x)</sup>, x=0,1,...n <br />
 +
 
 +
Mean (x) = E(x) = <math> np </math>
 +
Variance = <math> np(1-p) </math><br/>
 +
 
 +
Generate n uniform random number <math>U_1,...,U_n</math> and let X be the number of <math>U_i</math> that are less than or equal to p.
 +
The logic behind this algorithm is that the Binomial Distribution is simply a Bernoulli Trial, with a probability of success of p, repeated n times. Thus, we can sample from the distribution by sampling from n Bernoulli. The sum of these n bernoulli trials will represent one binomial sampling. Thus, in the below example, we are sampling 1000 realizations from 20 Bernoulli random variables. By summing up the rows of the 20 by 1000 matrix that is produced, we are summing up the 20 bernoulli outcomes to produce one binomial sampling. We have 1000 rows, which means we have realizations from 1000 binomial random variables when this sum is done (the output of the sum is a 1 by 1000 sized vector).<br />
 +
To continue with the previous example, let X be the number of heads in a series of ''n'' independent coin tosses - where for each toss, the probability of coming up with a head is ''p'' - then ''X~Bin(n, p)''. <br />
 +
MATLAB tips: to get a pdf f(x), we can use code binornd(N,P). N means number of trials and p is the probability of success. a=[2 3 4],if set a<3, will produce a=[1 0 0]. If you set "a == 3", it will produce [0 1 0]. If a=[2 6 9 10], if set a<4, will produce a=[1 0 0 0], because only the first element (2) is less than 4, meanwhile the rest are greater. So we can use this to get the number which is less than p.<br />
 +
 
 +
Algorithm for Bernoulli is given as above
 +
 
 +
'''Code'''<br>
 +
<pre style="font-size:16px">
 +
>>a=[3 5 8];
 +
>>a<5
 +
ans= 1 0 0
 +
 
 +
>>rand(20,1000)
 +
>>rand(20,1000)<0.4
 +
>>A = sum(rand(20,1000)<0.4)  #sum of raws ~ Bin(20 , 0.3)
 +
>>hist(A)
 +
>>mean(A)
 +
Note: `1` in the above code means sum the matrix by column
 +
 
 +
>>sum(sum(rand(20,1000)<0.4)>8)/1000
 +
This is an estimate of Pr[A>8].
 +
 
 +
</pre>
 +
 
 +
[[File:Binomial_example.jpg|300px]]
 +
 
 +
remark: a=[2 3 4],if set a<3, will produce a=[1 0 0]. If you set "a == 3", it will produce [0 1 0].
 +
using code to find some value what i want to get from the matrix. It`s useful to define some matrixs.
 +
 
 +
Relation between Bernoulli Distribution and Binomial Distribution:
 +
For instance, we want to find numbers ≤0.3. Uniform collects which is ≤0.3, and Binomial calculates how many numbers are there ≤0.3.
 +
 
 +
===The Geometric Distribution===
 +
Geometric distribution is a discrete distribution. There are two types geometric distributions, the first one is the probability distribution of the number of X Bernoulli fail trials, with probability 1-p, needed until the first success situation happened, X come from the set { 1, 2, 3, ...}; the other one is the probability distribution of the number Y = X − 1 of failures, with probability 1-p, before the first success, Y comes from the set { 0, 1, 2, 3, ... }.
 +
 
 +
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 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>2 </sup>. etc.<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 />
 +
x    Pr<br />
 +
1    P<br />
 +
2    P(1-P)<br />
 +
3    P(1-P)<sup>2</sup><br />
 +
.    .<br />
 +
.    .<br />
 +
.    .<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.
 +
 
 +
Generally speaking, if X~G(p) then its pdf is of the form f(x)=(1-p)<sup>(x-1)</sup>*p, x=1,2,...<br />
 +
The random variable X is the number of trials required until the first success in a series of independent''' Bernoulli trials'''.<br />
 +
 
 +
 
 +
 
 +
Other properties
 +
 
 +
 
 +
Probability mass function : P(X=k) = p(1-p)<sup>(k-1)</sup>
 +
 
 +
Tail probability : P(X>n) = <math>(1-p)^n</math>
 +
 
 +
The CDF : P(X<n) = 1 - <math>(1-p)^n</math>
 +
 
 +
 
 +
<span style="background:#F5F5DC">
 +
 
 +
Mean of x = 1/p
 +
Var(x) = (1-p)/p^2
 +
 
 +
There are two ways to look at a geometric distribution.
 +
 
 +
<b>1st Method</b>
 +
 
 +
We look at the number of trials before the first success. This includes the last trial in which you succeeded. This will be used in our course.
 +
 
 +
pdf is of form f(x)=>(1-p)<sup>(x-1)</sup>*(p), x = 1, 2, 3, ...
 +
 
 +
<b>2nd Method</b>
 +
 
 +
This involves modeling the failure before the first success. This does not include the last trial in which we succeeded.
 +
 
 +
pdf is of form f(x)=> ((1-p)^x)*p , x = 0, 1, 2, ....
 +
 
 +
</span>
 +
 
 +
 
 +
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 />
 +
 
 +
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/>
 +
 
 +
<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>
 +
 
 +
P(X>x) = e<sup>(-<math>\lambda</math> * x)</sup> = e<sup>log(1-p)*x</sup> = (1-p)<sup>x</sup> <br/>
 +
 
 +
Note that floor(Y)>X  ->  Y >= X+1 (X is an integer) <br/>
 +
 
 +
proof how to use EXP distribution to find P(X>x)=(1-p)^x
 +
 
 +
<br>
 +
Suppose X has the exponential distribution with rate parameter <math> \lambda > 0 </math> <br>
 +
the <math>\left \lfloor X \right \rfloor </math> and <math>\left \lceil X \right \rceil </math> have geometric distribution on <math> \mathcal{N} </math> and <math> \mathcal{N}_{+} </math> respectively each with success probability <math> 1-e^ {- \lambda} </math> <br>
 +
 
 +
Proof: <br>
 +
<math>\text{For } n \in \mathcal{N} </math><br//>
 +
 
 +
<math>\begin{align}
 +
P(\left \lfloor X \right \rfloor = n)&{}= P( n \leq X < n+1) \\
 +
                                    &{}= F( n+1) - F(n) \\
 +
\text{By algebra and simplification:} \\
 +
P(\left \lfloor X \right \rfloor = n)&{}= (e^ {-\lambda})^n \cdot (1 - e^ {-\lambda}) \\
 +
                                    &{}= Geo (1 - e^ {-\lambda}) \\
 +
 
 +
\text{Proof of ceiling part follows immediately.} \\
 +
\end{align}</math> <br//>
 +
 
 +
 
 +
 
 +
 
 +
 
 +
'''Algorithm:''' <br />
 +
1) Let  <math>\lambda = -\log (1-p) </math><br />
 +
2) Generate a <math>Y \sim Exp(\lambda )</math> <br />
 +
3) We can then let <math>X = \left \lfloor Y \right \rfloor + 1, where X\sim Geo(p)</math> <br />
 +
note: <math>\left \lfloor Y \right \rfloor >2        ->    Y>=3</math><br />
 +
        <math> \left \lfloor Y \right \rfloor >5        ->    Y>=6</math><br />
 +
<br />
 +
 
 +
<math>\left \lfloor Y \right \rfloor>x </math>    ->  Y>= X+1 <br />
 +
 
 +
<math>P(Y>=X)</math><br />
 +
Y ~ Exp (<math>\lambda</math>)<br />
 +
pdf of Y :  <math>\lambda e^{-\lambda}</math><br />
 +
cdf of Y :  <math>1- e^{-\lambda}</math><br />
 +
cdf <math>P(Y<x)=1-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>P(Y>=x)=e^{-\lambda x}=e^{log(1-p)x}=(1-p)^x</math><br />
 +
<math>E[x]=1/P </math><br />
 +
<math>Var= (1-P)/(P^2)</math><br />
 +
P(X>x)<br />
 +
=P(floor(y)+1>x)<br />
 +
=P(floor(y)>x-1)<br />
 +
=P(y>=x)
 +
 
 +
use <math>e^{-\lambda}=1-p</math> to figure out the mean and variance.
 +
'''Code'''<br>
 +
<pre style="font-size:16px">
 +
>>p=0.4;
 +
>>l=-log(1-p);
 +
>>u=rand(1,1000);
 +
>>y=(-1/l)*log(u);
 +
>>x=floor(y)+1;
 +
>>hist(x)
 +
 
 +
===Note:===
 +
mean(x)~E[X]=> 1/p
 +
Var(x)~V[X]=> (1-p)/p^2
 +
 
 +
A specific Example:
 +
Consider x=5
 +
>> sum(x==5)/1000 -> chance that will succeed at fifth trial;
 +
>> ans =
 +
        0.0780
 +
>> sum(x>10)/1000 -> chance that will succeed after 10 trials
 +
>> ans =
 +
        0.0320
 +
 
 +
</pre>
 +
 
 +
Note that the above mean is the average amount of times you should try until you get a successful case.<br/>
 +
 
 +
[[File:Geometric_example.jpg|300px]]
 +
 
 +
<span style="background:#F5F5DC">
 +
EXAMPLE for geometric distribution: Consider the case of rolling a die: </span>
 +
 
 +
X=the number of rolls that it takes for the number 5 to appear.
 +
 
 +
We have X ~Geo(1/6), <math>f(x)=(1/6)*(5/6)^{x-1}</math>, x=1,2,3....
 +
 
 +
Now, let <math>\left \lfloor Y \right \rfloor=e^{\lambda}</math> => x=floor(Y) +1
 +
 
 +
Let <math>e^{-\lambda}=5/6</math>
 +
 
 +
<math>P(X>x) = P(Y>=x)</math> (from the class notes)
 +
 
 +
We have <math>e^{-\lambda *x} = (5/6)^x</math>
 +
 
 +
Algorithm: let <math>\lambda = -\log(5/6)</math>
 +
 
 +
1) Let Y be <math>e^{\lambda}</math>, exponentially distributed
 +
 
 +
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>
 +
 
 +
 
 +
<span style="background:#F5F5DC">GENERATING NEGATIVE BINOMIAL RV USING GEOMETRIC RV'S</span>
 +
 
 +
Property of negative binomial Random Variable: <br/>
 +
 
 +
The negative binomial random variable is a sum of r independent geometric random variables.<br/>
 +
 
 +
Using this property we can formulate the following algorithm:<br/>
 +
 
 +
Step 1: Generate r geometric rv's each with probability p using the procedure presented above.<br/>
 +
Step 2: Take the sum of these r geometric rv's. This RV follows NB(r,p)<br/>
 +
 
 +
remark the step 1 and step 2. Looking for the floor Y, and e^(-mu)=1-p=5/6, and then generate x.
 +
 
 +
===Poisson Distribution===
 +
If <math>\displaystyle X \sim \text{Poi}(\lambda)</math>, its pdf is of the form <math>\displaystyle \, f(x) = \frac{e^{-\lambda}\lambda^x}{x!}</math> , where <math>\displaystyle \lambda </math> is the rate parameter.<br />
 +
 
 +
definition:In probability theory and statistics, the Poisson distribution (pronounced [pwasɔ̃]) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
 +
For instance, suppose someone typically gets 4 pieces of mail per day on average. There will be, however, a certain spread: sometimes a little more, sometimes a little less, once in a while nothing at all.[2] Given only the average rate, for a certain period of observation (pieces of mail per day, phonecalls per hour, etc.), and assuming that the process, or mix of processes, that produces the event flow is essentially random, the Poisson distribution specifies how likely it is that the count will be 3, or 5, or 10, or any other number, during one period of observation. That is, it predicts the degree of spread around a known average rate of occurrence.
 +
The Derivation of the Poisson distribution section shows the relation with a formal definition.(from Wikipedia)
 +
 
 +
Understanding of Poisson distribution:
 +
 
 +
If customers '''independently''' come to bank over time, all following exponential distributions with rate <math>\lambda</math> per unit of time, then
 +
X(t) = # of customer in [0,t] ~ Poi<math>(\lambda t)</math>
 +
 
 +
Its mean and variance are<br />
 +
<math>\displaystyle E[X]=\lambda</math><br />
 +
<math>\displaystyle Var[X]=\lambda</math><br />
 +
An useful property: If <math>X_i \sim \mathrm{Pois}(\lambda_i)\, i=1,\dots,n</math> are independent and <math>\lambda=\sum_{i=1}^n \lambda_i</math>, then <math>Y = \left( \sum_{i=1}^n X_i \right) \sim \mathrm{Pois}(\lambda)</math>
 +
 
 +
A Poisson random variable X can be interpreted as the maximal number of i.i.d. (Independent and Identically Distributed) exponential variables(with parameter <math>\lambda</math>) whose sum does not exceed 1.<br />
 +
The traditional understanding of the Poisson distribution as the total number of events in a specific interval can be understood here since the above definition simply describes the Poisson as the sum of waiting times for n events in an interval of length 1.
 +
<br />
 +
<br />
 +
<math>\displaystyle\text{Let } Y_j \sim \text{Exp}(\lambda), U_j \sim \text{Unif}(0,1)</math><br>
 +
<math>Y_j = -\frac{1}{\lambda}\log(U_j) \text{ from Inverse Transform Method}</math><br><br>
 +
 
 +
<math>\begin{align}
 +
X &= \max \{ n: \sum_{j=1}^{n} Y_j \leq 1 \} \\
 +
  &= \max \{ n: \sum_{j=1}^{n} - \frac{1}{\lambda}\log(U_j) \leq 1 \} \\
 +
  &= \max \{ n: \sum_{j=1}^{n} \log(U_j) >= -\lambda \} \\
 +
  &= \max \{ n: \log(\prod_{j=1}^{n} U_j) >= -\lambda \} \\
 +
  &= \max \{ n: \prod_{j=1}^{n} U_j >= e^{-\lambda} \} \\
 +
  &= \min \{ n: \prod_{j=1}^{n} U_j >= e^{-\lambda} \} - 1 \\
 +
\end{align}</math><br><br />
 +
 
 +
Note: From above, we can use Logarithm Rules <math>\log(a)+\log(b)=\log(ab)</math> to generate the result.<br><br />
 +
'''Algorithm:''' <br />
 +
1) Set n=1, a=1 <br />
 +
2) Generate <math>U_n \sim U(0,1),  a=aU_n </math> <br />
 +
3) If <math>a >= e^{-\lambda}</math> , then n=n+1, and go to Step 2. Else, x=n-1 <br />
 +
 
 +
using inverse-method to proof mean and variance of poisson distribution.
 +
 
 +
===MATLAB Code for generating Poisson Distribution===
 +
<pre style='font-size:16px'>
 +
>>l=2; N=1000
 +
>>for ii=1:N
 +
      n=1;
 +
      a=1;
 +
      u=rand;
 +
      a=a*u;
 +
      while a>exp(-l)
 +
            n=n+1;
 +
            u=rand;
 +
            a=a*u;
 +
      end
 +
      x(ii)=n-1;
 +
  end
 +
>>hist(x)
 +
>>Sum(x==1)/N      # Probability of x=1
 +
>>Sum(x>3)/N        # Probability of x > 3
 +
</pre>
 +
 
 +
[[File:Poisson_example.jpg|300px]]
 +
 
 +
=== Another way to generate random variable from poisson distribution ===
 +
<br/>
 +
Note: <math>P(X=x)=\frac {e^{-\lambda}\lambda^x}{x!}, \forall x \in \N</math><br/>
 +
Let <math>\displaystyle p(x) = P(X=x)</math> denote the pmf of <math>\displaystyle X</math>.<br/>
 +
Then ratio is <math>\frac{p(x+1)}{p(x)}=\frac{\lambda}{x+1}, \forall x \in \N</math> <br/>
 +
Therefore, <math>p(x+1)=\frac{\lambda}{x+1}p(x)</math> <br/>
 +
Algorithm: <br/>
 +
1. Set <math>\displaystyle x=0</math><br/>
 +
2. Set <math>\displaystyle F=p=e^{-\lambda}</math> <br/>
 +
3. Generate <math>\displaystyle U \sim~ \text{Unif}(0,1)</math> <br/>
 +
4. If <math>\displaystyle U<F</math>, output <math>\displaystyle x</math><br/>
 +
Else<br/>
 +
<math>\displaystyle p=\frac{\lambda}{x+1} p</math><br/>
 +
<math>\displaystyle F=F+p</math><br/>
 +
<math>\displaystyle x = x+1</math><br/>
 +
Go to 4.<br/>
 +
 
 +
This is indeed the inverse-transform method, with a clever way to calculate the CDF on the fly.
 +
 
 +
u=rand(0.1000)
 +
hist(x)
 +
 
 +
== Class 9 - Tuesday, June 4, 2013 ==
 +
 
 +
=== Beta Distribution ===
 +
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
 +
 
 +
:<math>\displaystyle \text{ } f(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1} </math> where <math>0 \leq x \leq 1</math> and <math>\alpha</math>>0, <math>\beta</math>>0<br>
 +
and
 +
<math>f(x;\alpha,\beta)= 0 </math> otherwise
 +
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 variance of the beta distribution decreases monotonically if <math> \alpha = \beta </math> and as <math> \alpha = \beta </math> increases, the variance decreases.
 +
 
 +
The formula for the cumulative distribution function of the beta distribution is also called the incomplete beta function ratio (commonly denoted by Ix) and is defined as F(x) = I(x)(p,q)
 +
 
 +
To generate random variables of a Beta distribution, there are multiple cases depending on the value of <math>\alpha </math> and <math> \beta </math>:
 +
 
 +
'''Case 1:''' If <math>\alpha=1</math> and <math>\beta=1</math>
 +
 
 +
:<math>\displaystyle \text{Beta}(1,1) = \frac{\Gamma(1+1)}{\Gamma(1)\Gamma(1)}x^{1-1}(1-x)^{1-1}</math><br>
 +
:<math> = \frac{1!}{0!0!}x^{0}(1-x)^{0}</math><br>
 +
:<math> = 1 </math><br>
 +
 
 +
Note: 0! = 1. <br>
 +
Hence, the distribution is:<br>
 +
:<math>\displaystyle \text{Beta}(1,1) = U (0, 1) </math><br>
 +
If the Question asks for sampling Beta Distribution, we can sample from Uniform Distribution which we already know how to sample from<br>
 +
Algorithm:<br>
 +
Generate U~Unif(0,1)<br>
 +
 
 +
'''Case 2:''' Either <math>\alpha=1</math> or <math>\beta=1</math>
 +
 
 +
 
 +
e.g. <math>\alpha=1</math>
 +
We don't make any assumption about <math>\beta</math> except that it is a positive integer. <br\>
 +
 
 +
:<math>\displaystyle \text{f}(x) = \frac{\Gamma(1+\beta)}{\Gamma(1)\Gamma(\beta)}x^{1-1}(1-x)^{\beta-1}=\beta(1-x)^{\beta-1}</math><br>
 +
 
 +
:<math>\beta=1 </math>
 +
 
 +
:<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>
 +
 
 +
By integrating <math>f(x)</math>, we find the CDF of X is <math>F(x) = x^{\alpha}</math>.
 +
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].
 +
 
 +
'''Algorithm'''
 +
 
 +
:1. Generate U~Unif(0,1)<br\>
 +
:2. Assign <math>x = u^\frac {1}{\alpha}</math>
 +
 
 +
After we have simplified this example, we can use other distribution methods to solve the problem.
 +
 
 +
'''MATLAB Code to generate random n variables using the above algorithm'''
 +
<pre style='font-size:16px'>
 +
 
 +
x = rand(1,n).^(1/alpha)       
 +
</pre>
 +
 
 +
'''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\>
 +
:  <math>Y_2</math> follows gamma <math>(\beta,1)</math><br\>
 +
 
 +
'''Note:
 +
1. <math>\alpha</math> and <math>\beta</math> are shape parameters here and 1 is the scale parameter.<br\>'''
 +
:then <math>Y=\frac {Y_1}{Y_1+Y_2}</math> follows Beta <math>(\alpha,\beta)</math><br\>
 +
2.Exponential: <math>-\frac{1}{\lambda} \log(u)</math> <br\>
 +
3.Gamma: <math>-\frac{1}{\lambda} \log(u_1 * \cdots * u_t)</math><br\>
 +
 
 +
'''Algorithm'''<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\>
 +
*3. Set
 +
:<math> Y = \frac{Y_1}{Y_1+Y_2}</math><br>
 +
 
 +
Please see the following example for Matlab code.  <br> 
 +
 
 +
 
 +
'''Case 4:'''<br\> Use The Acceptance-Rejection Method <br\>
 +
The beta density is<br />
 +
<math>\displaystyle \text{Beta}(\alpha,\beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1} </math> where <math>0 \leq x \leq 1</math><br>
 +
Assume <math>\alpha,\beta \geq 1</math>. Then <math>\displaystyle f(x)</math> has the maximum at <math>\frac{\alpha-1}{\alpha+\beta-2}</math>.<br />
 +
(Please note that we could find the  maximum by taking the derivative of f(x), let f'(x)=0 and then use maximum likelihood estimate to find what the maximum value is)<br>
 +
Define<br />
 +
<math> c=f(\frac{\alpha-1}{\alpha+\beta-2}</math>) and choose <math>\displaystyle g(x)=1</math>.<br />
 +
The A-R method becomes<br />
 +
1.Generate independent <math>\displaystyle U_1</math> and <math>\displaystyle U_2</math> from <math>\displaystyle UNIF[0,1]</math> until <math>\displaystyle cU_2 \leq f(U_1)</math>;<br />
 +
2.Return <math>\displaystyle U_1</math>.<br />
 +
'''MATLAB Code for generating Beta Distribution'''
 +
<pre style='font-size:16px'>
 +
>>Y1 = sum(-log(rand(10,1000)))            #Gamma(10,1), sum 10 exponentials for each of the 1000 samples
 +
 
 +
>>Y2 = sum(-log(rand(5,1000)))              #Gamma(5,1), sum 5 exponentials for each of the 1000 samples
 +
 
 +
%NOTE: here, lamda is 1, since the scale parameter for Y1 & Y2 are both 1
 +
 
 +
>>Y=Y1./(Y1+Y2)                              #Don't forget to divide elements using "." Where Y follows Beta(10,5)
 +
 
 +
>>figure
 +
 +
>>hist(Y1)                                  #Gamma curve
 +
 
 +
>>figure
 +
 
 +
>>hist(Y2)                                  #Gamma curve
 +
 
 +
>>figure
 +
 
 +
>>hist(Y)                                    #Do this to check that the shape fits beta. ~Beta(10,5).
 +
 
 +
>>disttool                                  #Check the beta plot.
 +
 
 +
</pre>
 +
This is the histogram of Y, precisely simulated version of Beta (10,5)
 +
 
 +
 
 +
[[File:Beta(10,5)_Simulated.jpg|300px]]
 +
 
 +
This is the pdf of various beta distributions
 +
 
 +
[[File:325px-Beta_distribution_pdf.png|300px]]
 +
 
 +
[[File:untitled.jpg|300px]]<br />
 +
MATLAB tips: rand(10,1000) produces one 10*1000 matrix and sum(rand(10,1000)) produces a 10*1000 matrix
 +
and each element in the matrix follows CDF of uniform distribution.
 +
 
 +
Example for the code to explain the beta distribution.
 +
 
 +
 
 +
'''Another MATLAB Code for generating Beta Distribution using AR method'''
 +
<pre style='font-size:16px'>
 +
>>alpha = 3
 +
>>beta = 2
 +
>> a = sum (-log(rand(alpha,1000)))
 +
>> b = sum (-log(rand(beta,1000)))
 +
>> aandb=sum(-log(rand(alpha+beta,1000)))
 +
>> t = (alpha - 1)/(alpha + beta -2)
 +
>> c = (andb/(a*b))*t^(alpha-1)*(1-t)^(beta-1)
 +
>> u1 = rand
 +
>> u2 = rand
 +
>> x = (andb/(a*b))*u1^(alpha-1)*(1-u1)^(beta-1)
 +
>> while c*u2>x
 +
>> u1 = rand
 +
>> u2 = rand
 +
>> x = (andb/(a*b))*u1^(alpha-1)*(1-u1)^(beta-1)
 +
>> end
 +
>> u1
 +
</pre>
 +
 
 +
=== 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 need to take into account the following two cases:
 +
 
 +
====Case 1====
 +
if the <math>x_1, x_2 \cdots, x_d</math>'s are independent, then<br/>
 +
<math>f(x) = f(x_1,\cdots, x_d) = f(x_1)\cdots f(x_d)</math><br/>
 +
we can sample from each component <math>x_1, x_2,\cdots, x_d</math> individually, and then form a vector.<br/>
 +
 
 +
based on the property of independence, we can derive the pdf or pmf of <math>x=x_1,x_2,x_3,x_4,x_5,\cdots</math>
 +
 
 +
====Case 2====
 +
If <math>X_1, X_2, \cdots , X_d</math> are not independent<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),\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.
 +
In each case, we have to consider the previous cases.
 +
<math>f(x_1)</math> is one-dimensional, some as <math>f(x_2|x_1)</math> and all others.
 +
In general, one could consider the covariance matrix <math> C </math> of random variables <math> X_1</math>,…,<math>X_d </math>. <br>
 +
Suppose we now have the Cholesky factor <math> G</math> of <math> C </math> (i.e. <math> C = GG^T </math>).  In matlab, we use Chol(C) <br>
 +
For any d-tuple <math> X := (X_1 ,\ldots , X_d) </math> (i.e random variable generated by <math> X_1,\ldots , X_d </math> respectively)
 +
<math> GX </math> would yield the desired distribution. <br/>
 +
 
 +
'''Note''' (Product Rule)<br/>
 +
1.) All cases can use this (independent or dependent): <math>f(x) = f(x_1, x_2)= f(x_1) f(x_2|x_1)</math> <br/>
 +
2.) If we determine that <math>x_1</math> and <math> x_2</math> are ''independent'', then we can use <math>f(x) = f(x_1, x_2)= f(x_1)f(x_2)</math> <br/>
 +
*ie. If late for class=<math>x_1</math> and sick=<math>x_2</math>, then these are dependent variables so can only use equation 1 (<math>f(x) = f(x_1, x_2)= f(x_1) f(x_2|x_1)</math>)<br/>
 +
*ie. If late for class=<math>x_1</math> and milk is white=<math>x_2</math>, then these are independent variables so can use both equations 1 and 2. <br/>
 +
 
 +
the case show the formula of the X = (X1,X2,…,Xd), a d-dimensional vector, when they are not independent of each x. we use conditional function to define the probability function of x with d-dimensional.
 +
 
 +
====Example====
 +
Generate uniform random vectors
 +
 
 +
1)  x = (x<sub>1</sub>, …, x<sub>d</sub>) from the d-dimensional rectangle <br/>
 +
2)  D = { (x<sub>1</sub>, …, x<sub>d</sub>) : a<sub>i</sub> <= x<sub>i</sub> <= b<sub>i</sub>  ,  i = 1, …, d} <br/>
 +
 +
Algorithm: <br/>
 +
1)  For i = 1 to d <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/>
 +
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/>
 +
 
 +
An example of the 2-D case is given below:
 +
 
 +
<pre style='font-size:14px'>
 +
>>a=[1 2];
 +
>>b=[4 6];
 +
>>for i=1:2
 +
      u(i) = rand();
 +
      x(i) = a(i) + (b(i) - a(i))*u(i);
 +
  end
 +
 
 +
>>hold on          => this is to retain current graph when adding new graphs
 +
>>rectangle('Position',[1 2 3 4])  => draw the boundary of the rectangle
 +
>>axis([0 10 0 10])    => change the size of axes
 +
>>plot(x(1),x(2),'.')
 +
 
 +
</pre>
 +
[[File:2d_ex.jpg|300px]]
 +
 
 +
==== Matlab Code: ====
 +
 
 +
<pre style='font-size:14px'>
 +
function x = urectangle (d,n,a,b)
 +
for ii = 1:d;
 +
    u(ii,:) = rand(1,n);
 +
    x(ii,:) = a+ u(ii,:)*(b-a);
 +
    %keyboard                      #makes the function stop at this step so you can evaluate the variables
 +
end
 +
 
 +
>>x=urectangle(2, 100, 2, 5);
 +
>>scatter(x(1,:),x(2,:))
 +
 
 +
>>x=urectangle(2, 10000, 2, 5);        #generate 10000 numbers (instead of 100)
 +
>>x=urectangle(3, 10000, 2, 5);        #changed to 3-dimensional
 +
>>scatter3(x(1,:), x(2,:), x(3,:))
 +
>>axis square
 +
</pre>
 +
[[File:Urectangle_2d.jpg|300px]][[File:Urectangle_3d.jpg|300px]]
 +
 
 +
=== Vector Acceptance-Rejection Method ===
 +
 
 +
The acceptance-rejection method can be extended to n-dimensional cases, with the same concept:
 +
 
 +
If a random vector is to be generated uniformly from G, an irregular shape in the nth dimension, and W is a regular shape arbitrarily close to G in the nth dimension, then acceptance-rejection method can be applied as follows:
 +
 
 +
1. Sample from the regular shape W
 +
 
 +
2. Accept sample points if they are inside G
 +
 
 +
<br>'''Example:''' <br>
 +
Generate a random vector Z that is uniformly distributed over region G
 +
 
 +
G: d-dimensional unit ball, <math>G = \big\{{x: \sum_{i}{x_i}^2 \leq 1}\big\}</math>
 +
 
 +
w: d-dimensional hypercube, <math>W = \big\{{-1 \leq x_i \leq 1}\big\}_{i=1}^d</math>
 +
 
 +
'''Procedure:'''<br />
 +
Step 1: <math>U_1 \sim~ U(0,1),\cdots, U_d \sim~ U(0,1)</math><br />
 +
Step 2: <math>X_1 = 1 - 2U_1, \cdots, X_d = 1 - 2U_d, R = \sum_i X_i^2</math><br />
 +
Step 3: If <math>R \leq 1,  Z=(X_1, ..... , X_d)</math><br />
 +
Else go to step 1
 +
 
 +
it is an example of the vector A/R, regular shape is W likes the proposal distribution g(x), G is the target distribution g(x) <br\>
 +
 
 +
Suppose we sampled from the target area W uniformly, let Aw, Ag indicate the area of W and G, g(x)=1/Aw and f(x)=1/Ag
 +
 
 +
 
 +
The following is a picture relating to the example
 +
 
 +
[[File:Untitled.jpg]]
 +
 
 +
Matlab code:
 +
<pre style='font-size:16px'>
 +
u = rand(d,n);
 +
z = 1- 2 *u;
 +
R = sum(z.^2);
 +
jj=1;
 +
 
 +
  for ii=1:n
 +
 
 +
      if R(ii)<=1
 +
 
 +
        x(:,jj)=z(:,ii);
 +
        jj=jj+1;
 +
 
 +
      end
 +
 
 +
  end
 +
 
 +
  output = x;
 +
 
 +
end
 +
</pre>
 +
 
 +
==Class 10 - Thursday June 6th 2013 ==
 +
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
 +
 
 +
<pre style='font-size:16px'>
 +
1)  U1~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: ====
 +
 
 +
<pre style='font-size:16px'>
 +
function output = Unitball(d,n)
 +
 
 +
u = rand(d,n);
 +
z = 1- 2 *u;
 +
R = sum(z.^2);
 +
jj=1;
 +
 
 +
  for ii=1:n
 +
 
 +
      if R(ii)<=1
 +
 
 +
        x(:,jj)=z(:,ii);
 +
        jj=jj+1;
 +
 
 +
      end
 +
 
 +
  end
 +
 
 +
  output = x;
 +
 
 +
end
 +
 
 +
>> data = Unitball(d, n)
 +
>> scatter(data(1,:), data(2,:))    %plot 2d graph
 +
 
 +
R(ii) computes the sum of the square of each element of a vector, so if it is less than 1,
 +
then the vector is in the unit ball.
 +
 
 +
x(:,jj) means all the numbers in the jj column.
 +
 
 +
z(:,ii) means all the numbers in the ii column starting from 1st column until the nth
 +
column, which is the last one.
 +
 
 +
higher dimension, less efficient and we need more data points
 +
 
 +
Save it with the name of the pattern.
 +
 
 +
</pre>
 +
 
 +
 
 +
[[File:Unitball.png|400px]]
 +
 
 +
<pre style='font-size:16px'>
 +
Execution:
 +
 
 +
>>[x]=Unitball(2,10000);
 +
>>scatter(x(1,:),x(2,:));    %plot 2D circle
 +
>>axis square;                %make the x-y axis has same size                 
 +
>>size(x)
 +
 
 +
ans =
 +
 
 +
          2        7839
 +
 
 +
>>scatter(x(1,:),x(2,:))
 +
</pre>
 +
 
 +
scatter(x(1,:),x(2,:)) the (x(1,:) means all the numbers in the first row are parameter.
 +
 
 +
'''Calculate the efficiency:'''
 +
<pre style='font-size:16px'>
 +
 
 +
>>c=7839/10000                %Efficiency = points accepted / total points
 +
 
 +
c =
 +
 
 +
    0.7839
 +
 
 +
</pre>
 +
 
 +
We can use the above program to calculate how many points in the circle condition are in the square.
 +
 
 +
'''Estimate <math>\displaystyle \pi</math>'''
 +
:*We know the radius is 1
 +
:*Then the area of the square is <math>(1-(-1))^2=4</math><br\>
 +
:*Then the area of the circle is <math>\pi</math><br\>
 +
:*<math>\pi</math> is approximated to be <math>4\times c=4 \times 0.7839=3.1356</math> in the above example <br\>
 +
 
 +
<pre style='font-size:16px'>
 +
>> 4*size(x,2)/10000
 +
 
 +
ans =
 +
 
 +
    3.1356
 +
 
 +
>> [x]=Unitball(3,10000);
 +
>> scatter3(x(1,:),x(2,:),x(3,:)) %plot 3d ball
 +
>> axis square
 +
>> size(x,2)/10000  %returns the size of the dimension of X specified by scalar 2
 +
 
 +
ans =
 +
 
 +
    0.5231
 +
 
 +
>> [x]=Unitball(5,10000);
 +
>> size(x,2)/10000
 +
 
 +
ans =
 +
 
 +
    0.1648
 +
 
 +
</pre>
 +
3d unit ball
 +
 
 +
[[File:3-dimensional unitball.jpg|400px]]
 +
 
 +
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.
 +
 
 +
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 ===
 +
 
 +
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 \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.
 +
 
 +
As the dimension '''increase''', the efficiency of the algorithm will '''decrease''' exponentially.
 +
 
 +
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.
 +
 
 +
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>
 +
<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 ====
 +
The basic idea of Stochastic Process (also called random process) is a collection of some random variables,
 +
<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)
 +
 
 +
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>
 +
Also, Different Stochastic Process has different properties.
 +
 
 +
In the course, we study two Stochastic Process models.
 +
 
 +
The two stochastic Process models we will study are:
 +
 
 +
1. Poisson Process-This is continuous time counting process that satisfies a couple of properties that are listed in the next section. The Poisson process is understood to be a good model for events such as incoming phone calls, number of traffic accidents, and goals during a game of hockey or soccer. It is also an example of a birth-death process.<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>
 +
 +
 
 +
=====Example=====
 +
The state space is the set of English words, and <math>x_t</math> are words that are said. Another example involves the stock market: the set of all non-negative numbers is the state space, and <math>x_t</math> are stock prices.
 +
 
 +
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.
 +
 
 +
Births in a hospital occur randomly at an average rate
 +
 
 +
The number of cases of a disease in different towns
 +
 
 +
==== Poisson Process ====
 +
[[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.
 +
 
 +
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 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\>
 +
-By definition, <math>N(a,b]=N_b-N_a</math><br\>
 +
-The two random variables <math>N(a,b]</math> and <math>N(c,d]</math> are independent if <math>(a,b]</math> and <math>(c,d]</math> do not intersect.<br\>
 +
 
 +
 
 +
'''Definition:''' An arrival counting process <math>N=(N_t)</math> is called (Homogeneous) Poisson Process (PP) with rate <math>\lambda > 0</math> if
 +
 
 +
A. The numbers of points in non-overlapping intervals are independent. <br \>
 +
B. The number of points in interval <math>I(a,b]</math> has a poisson distribution with mean <math>\lambda (b-a)</math> ,where (b-a) represents the length of I.
 +
 
 +
In particular, observe that if <math>N=(N_t)</math> is a Poisson process of rate <math>\lambda>0</math>, then the moments are
 +
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 />
 +
'''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.)
 +
 
 +
<pre style='font-size:16px'>
 +
z=randn(length(mu),n); %Here length(mu)=2 since s is a 2*2 matrix;
 +
s=[1 0.7;0.7 1]
 +
A=chol(s)% This only works for a positive definite A, otherwise we need to consider some other form of decomposition
 +
x=mu'*ones(1,n) + A*z; %ones(1,n) is a function expand the size of mu from 1*1
 +
                      %matrix to 1*n matrix;
 +
</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>
 +
 
 +
 
 +
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'>
 +
d = length(mu);
 +
R = chol(Sigma);
 +
 
 +
U1=rand(n,d)
 +
U2=rand(d,d)
 +
D=-2*log(U1)
 +
tet=2*pi*U2
 +
Z=(D.^0.5)*cos(tet)' %since in lecture 6, we set up x and y are independent normal
 +
                      %distribution, so in here we can choose either cos and sin
 +
 
 +
X = Z*R + ones(n,1)*mu';
 +
</pre>
 +
 
 +
==== '''Central Limit Theorem''' ====
 +
 
 +
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>
 +
 
 +
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.)
 +
 
 +
We illustrate with an example using 1000 observations each of 20 independent exponential random variables.
 +
 
 +
<pre style='font-size:16px'>
 +
>> X = exprnd (20,20,1000); % 1000 instances of 20 exponential random numbers with mean 20
 +
>> hist(X(1,:))
 +
>> hist(X(1:2,:))
 +
...
 +
>>hist(X(1:20,:)) -> approaches to normal
 +
</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==
 +
=== Announcement ===
 +
Midterm covers up to the middle of last lecture, which means stochastic process will not be on midterm. There won't be any Matlab syntax questions. And Students can contribute to any previous classes. We might however be asked to write down algorithms.
 +
 
 +
===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
 +
:<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>
 +
(a) '''Independence:''' The numbers of arrivals in non-overlapping intervals are independent  <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>
 +
 
 +
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.
 +
 
 +
For a small interval (t,t+h], where h is small<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>
 +
 
 +
<math>\lambda h</math> can be thought of as the probability of observing an arrival in the interval t to t+h.<br>
 +
 
 +
Similarly, the probability of not observing an arrival in this interval is 1-<math>\lambda </math> h.<br>
 +
 
 +
 
 +
'''Generate a Poisson Process'''<br />
 +
 
 +
1. set <math>T_{0}=0</math> and n=1<br/>
 +
 
 +
2. <math>U_{n} \sim~ U(0,1)</math><br />
 +
 
 +
3. <math>T_{n} = T_{n-1}-\frac {1}{\lambda} log (U_{n})  </math> (declare an arrival)<br />
 +
 
 +
4. if <math>T_{n} \gneq T</math> stop<br />
 +
&nbsp;&nbsp;&nbsp;&nbsp;else<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>
 +
*Note : Recall that exponential random variable is the waiting time  until one event of interested occurs.
 +
 
 +
'''Review of Poisson - Example'''
 +
 
 +
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=4)=\frac {e^{-1.8}(1.8)^4}{4!} </math>
 +
 
 +
<math>P(X\geq1) = 1 - P(x=0) = 1- e^{-1.8}</math>
 +
 
 +
<span style="background:#F5F5DC">
 +
<math>P(N_3> 3 | N_2)=P(N_1 > 2)</math>
 +
</span>
 +
 
 +
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 ===
 +
Suppose we want to generate the first n events of a Poisson process with rate <math>{\lambda}</math>. We showed that the times between successive events are independent exponential random variables, each with rate <math>{\lambda}</math>. Therefore, we can first generate n random numbers U<sub>1</sub>, U<sub>2</sub>,...,U<sub>n</sub> and use the inverse transform method to find the corresponding exponential variable X<sub>1</sub>, X<sub>2</sub>,...,X<sub>n</sub>. Then X<sub>i</sub> can be interpreted as the time between the (<i>i</i>-1)st and the <i>i</i>th event of the process. <br>
 +
The actual time of the <i>j</i>th event is the sum of the first j interarrival times. Therefore, the generated values of the first n event times are <math>\sum_{i=1}^{j} X_i</math>, j = 1...n.
 +
<br>
 +
 
 +
Homogeneous poisson process refers to the rate of occurrences remaining constant for all periods of time. <br>
 +
<br>U<sub>n</sub>~U(0,1)<br>
 +
<math>T_n-T_{n-1}=-\frac {1}{\lambda}  log(U_n)</math> <br>
 +
 
 +
The waiting time between each occurrence follows the exponential distribution with parameter <math>\lambda </math>.
 +
<math>T_n</math> represents the time elapsed at the n<sup>th</sup> occurence.
 +
 
 +
1) Set T<sub>0</sub> = 0 ,and n = 1 <br>
 +
2) U<sub>n</sub> follow U(0,1) <br>
 +
3) T<sub>n</sub> - T<sub>n-1</sub> =<math> -\frac {1}{\lambda} </math>  log (U<sub>n</sub>)    (Declare an arrival)<br>
 +
4) if T<sub>n</sub> >T stop;
 +
else n = n + 1, go to step 2 <br>
 +
 
 +
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.
 +
 
 +
<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.]]'''
 +
 
 +
==== MatLab Code ====
 +
<pre style='font-size:16px'>
 +
 
 +
T(1)=0; % Matlab does not allow 0 as an index, so we start with T(1).
 +
ii=1;
 +
l=2;
 +
TT=5;
 +
 
 +
while T(ii)<=TT
 +
  u=rand;
 +
  ii=ii+1;
 +
  T(ii)=T(ii-1) - (1/l)*log(u);
 +
end
 +
 
 +
plot(T, '.')
 +
 
 +
</pre>
 +
<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 maximum value of the points should be TT. <br>
 +
[[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 distribution.
 +
 
 +
===Markov chain===
 +
"A Markov Chain is a stochastic process where: <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/>
 +
 
 +
Source: "http://math.scu.edu/~cirving/m6_chapter8_notes.pdf" <br/>
 +
 
 +
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")
 +
 
 +
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 />
 +
<math>f(x_1,x_2,...,x_n)=f(x_1)f(x_2\mid x_1)f(x_3\mid x_2,x_1)...f(x_n\mid x_{n-1},x_{n-2},....)</math>
 +
 
 +
In Markov Chain<br />
 +
<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. 
 +
 
 +
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. 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>
 +
 
 +
Formal Definition:
 +
The process <math> \{x_n: n \in T\} </math> is a markov chain if:<br />
 +
<math> Pr(x_n|x_{n-1},...,x_1) = Pr(x_n|x_{n-1}) \ \ \forall n\in T </math> and <math> \forall x\in X</math>
 +
 
 +
<span style="background:#F5F5DC">CONTINUOUS TIME MARKOV PROCESS</span>
 +
 
 +
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. 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 />
 +
 
 +
 
 +
====Transition Matrix====
 +
 
 +
Transition Probability: <math> P_{ij} = P(X_{t+1} =j | X_t =i) </math> is the one-step transition probability from state i to state j.
 +
 
 +
Transition Probability: <math> P_{ij}(k) = P(X_{t+1}(k) =j | X_t(k) =i) </math> is the k-step transition probability from state i to state j.
 +
 
 +
The matrix P whose elements are transition Probabilities <math> P_{ij} </math> is a one-step transition matrix.
 +
 
 +
Example:<br />
 +
 
 +
[[File:Transition_Map.png]]
 +
 
 +
<math>\begin{align}
 +
P_{ab} &=P(X_{t+1} &=b &\mid X_{t} &=a) &= 0.3 \\
 +
P_{aa} &=P(X_{t+1} &=a &\mid X_{t} &=a) &= 0.7 \\
 +
P_{ba} &=P(X_{t+1} &=a &\mid X_{t} &=b) &= 0.2 \\
 +
P_{bb} &=P(X_{t+1} &=b &\mid X_{t} &=b) &= 0.8 \\
 +
\end{align}</math><br />
 +
 
 +
<math> P= \left [ \begin{matrix}
 +
0.7 & 0.3 \\
 +
0.2 & 0.8
 +
\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
 +
 
 +
Properties of Transition Matrix: <br />
 +
 
 +
1. <math> 1 \geq P_{ij} \geq 0 </math><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>
 +
 
 +
<math> \forall x \in \Omega </math>, the probability of the next state is given according to the distribution <math> P(x,\cdot) </math> <br>
 +
 
 +
This means our model can be simulated as a sequence of random variables <math> (X_0, X_1, X_2, \ldots ) </math> with state space <math> \Omega </math> and transition matrix <math> P = [P_{ij}] </math> where <math> \forall t \in \N, 0 \leq s \leq t+1, x_s \in \Omega, </math> <br/>
 +
 
 +
we have to following property (Markov property): <br/>
 +
<math> P(X_{t+1}= x_{t+1} \vert \cap^{t}_{s=0} X_s = x_s) = P(X_{t+1} =x_{t+1} \vert X_t =x_t) = P(x_t,x_{t+1}) </math> <br>
 +
 
 +
And  <math> \forall x \in \Omega, \sum_{y\in\Omega} P(x,y) =1; \; \forall x,y\in\Omega, P_{xy}  = P(x,y) \geq 0 </math><br>
 +
 
 +
Moreover if <math> \forall x,y \in \Omega, \exists k, P^k (x,y) > 0 </math> <br>
 +
<math> |\Omega| < \infty </math> (i.e Any two states can be translated somehow) <br>
 +
 
 +
Then one might consider the periodicity of the chain and derive a notion of cyclic behavior. <br>
 +
 
 +
=== Examples of Transition Matrix ===
 +
 
 +
[[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)
 +
 
 +
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(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?
  
  
=== Sampling (Generating random numbers)===  
+
<math>\begin{align}
Some people believe that activities such as rolling a dice and flipping a coin are not truly random but are '''deterministic''' – that is, the result can be reliably calculated using things such as physics and math.
+
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 \\
A computer cannot generate truly random numbers since 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.
+
\end{align}</math><br />
 
  
 +
== Class 12 - Thursday,June 13, 2013 ==
 +
<b>Time</b>
 +
Jun 17, 2013 2:30 PM - 3:30 PM
 +
===Midterm Review===
  
 
=== Multiplicative Congruential Algorithm ===
 
=== Multiplicative Congruential Algorithm ===
This is an algorithm used to generate uniform, pseudo random numbers<br />
+
<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.
 +
 
 +
'''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>
 +
 
 +
<math>\begin{align}X_{2} = 2 * 8  + 1\mod 13 = 4\end{align}</math> ... and so on<br>
 +
 
 +
 
 +
2. <math>\begin{align}X_{0} = 44 ,a = 13 , c = 17 , m = 211\end{align}</math><br>
 +
     
 +
<math>\begin{align}X_{1} = 13 * 44 + 17\mod 211 = 167\end{align}</math><br>
 +
 
 +
<math>\begin{align}X_{2} = 13 * 167  + 17\mod 211 = 78\end{align}</math><br>
 +
 
 +
<math>\begin{align}X_{3} = 13 * 78  + 17\mod 211 = 187\end{align}</math> ... and so on<br>
 +
 
 +
=== Inverse Transformation Method ===
 +
'''For continuous cases:''' <br/ >
 +
1. U~U(0,1)<br/ >
 +
2. X=F<sup>-1</sup>(u)<br/ >
 +
 
 +
'''Note:''' <br/>
 +
In Uniform Distribution <math>P(X<=a)=a </math><br/ >
 +
proof:<math>P(X<=x) = P(F^{-1}(u)<=x)=P(u<=F(x)) = F(x)</math><br/ >
 +
 
 +
'''For discrete cases:''' <br/ >
 +
1. U~U(0,1)<br/ >
 +
2. x=x<sub>i</sub> if F(x<sub>i-1</sub>)<math><</math>u<math>\leq</math>F(x<sub>i</sub>)<br/ >
 +
 
 +
===Acceptance-Rejection Method===
 +
cg(x)>=f(x)
 +
<math>c=max\frac{f(x)}{g(x)}</math>
 +
<br><math>\frac{1}{c}</math> is the efficiency of the method/probability of acceptance
 +
 
 +
1. Y~g(x)<br />
 +
2. U~U(0,1)<br />
 +
3. If <math>U<=\frac{f(y)}{c*g(y)}</math> then X=Y
 +
else go to step 1
 +
 
 +
Proof that this method generates the desired distribution:
 +
<math>P(Y|accepted)=\frac{P(accepted|Y)P(Y)}{P(accepted)}=\frac{\frac{f(y)}{cg(y)} g(y)}{\int_{y}^{ } \frac{f(y)}{cg(y)} g(y) dy}=\frac{\frac{f(y)}{c}}{\frac{1}{c}\cdot 1}=f(y)</math>
 +
 
 +
===Multivariate===
 +
 
 +
f(x<sub>1</sub>,....,x<sub>n</sub>)=f(x<sub>1</sub>)f(x<sub>2</sub>|x<sub>1</sub>)...f(x<sub>n</sub>|x<sub>n-1</sub>,...x<sub>1</sub>)
 +
in general we need knowledge of conditional distribution if x<sub>1</sub>....x<sub>n</sub> are independent
 +
i.e. f(x<sub>1</sub>...x<sub>n</sub>)=f(x<sub>1</sub>)f(x<sub>2</sub>)...f(x<sub>n</sub>)
 +
 
 +
=== Vector A/R Method===
 +
 
 +
This method is not efficient for high dimensional vectors<br/>
 +
 
 +
- Sample uniformly from a space W that contains the sample space G of interest<br/>
 +
- Accept if the point is inside G <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/>
 +
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.
 +
 
 +
 
 +
-<math>\frac{A_G}{A_W}</math> = <math>\frac{1}{c}</math> is the efficiency of the algorithm. That is the amount of points accepted over the amount of points rejected. This acceptance rate drops greatly for each dimension added.
 +
 
 +
Note that <math>\frac{A_G}{A_W}</math> decreases exponentially as the dimensional goes up, thus its not efficient for for high dimensional.
 +
 
 +
=== Common distribution ===
 +
 
 +
===Exponential===
 +
 
 +
Models the waiting time until the first success.<br>
 +
<math>X\sim~Exp(\lambda)</math> <br />
 +
 
 +
<math>f(x) = \lambda e^{-\lambda x} \, , x>0 </math><br/>
 +
 
 +
<math>1.\, U\sim~U(0,1)</math>
 +
<br />
 +
<math>2.\, x = \frac{-1}{\lambda} log(U)</math>
 +
 
 +
===Normal===
 +
Box-Muller method
 +
 
 +
1.<math>U_{1},U_{2}\sim~ U(0,1)</math> <br/>
 +
2.<math>R^{2}=-2log(U_{1}), R^{2}\sim~ Exp(1/2)</math>
 +
 
 +
<math>\theta = 2\pi U_{2},\theta\sim~ U(0,2\pi) </math><br/>
 +
3.<math>X=Rcos(\theta), Y=Rsin(\theta), X,Y\sim~ N(0,1)</math><br/ >
 +
<br/ >
 +
 
 +
To obtain any normal distribution X once a set of standard normal samples Z is generated, apply the following transformations: <br/ >
 +
<math>Z\sim N(0,1)\rightarrow  X \sim N(\mu,\sigma ^{2})</math><br/ >
 +
<math>X=\mu+\sigma~Z</math><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{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(t,λ) <br>
 +
t: The number of exponentials and the shape parameter<br>
 +
λ: The mean of the exponentials and the scale parameter<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>=\frac {-1}{\lambda}[\log(u_1)+\log(u_2)+.....+\log(u_t)]</math>
 +
 
 +
<math>=\frac {-1}{\lambda}\log(\prod_{j=1}^{t} U_j)</math>
 +
 
 +
This is a special property of gamma distribution.
 +
 
 +
=== Bernoulli ===
 +
 
 +
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>
 +
 
 +
To generate a Bernoulli random variable we use the following procedure:
 +
 
 +
<math> 1. U\sim~U(0,1)</math><br>
 +
<math> 2. if\, u <= p, then\, x=1\,</math><br />
 +
<math> else\, x=0</math><br/>
 +
where 1 stands for success and 0 stands for failure.<br>
 +
 
 +
===Binomial===
 +
 
 +
The sum of n independent Bernoulli trials
 +
<br\>
 +
<math> X\sim~ Bin(n,p)</math><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/>
 +
Return to 1<br/>
 +
 
 +
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.
 +
 
 +
The theory behind the algorithm is the fact that the sum of n independent and identically distributed Bernoulli trials, Ber(p), follows a binomial Bin(n,p) distribution.
 +
 
 +
Example:
 +
 
 +
Suppose rolling a die, success= lands on 5, fail ow
 +
 
 +
p=1/6, 1-p=5/6, rolling for 10 times, n=10
 +
 
 +
simulate this binomial distribution.
 +
 
 +
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>
 +
3)Return to 1)
 +
 
 +
=== Beta Distribution ===
 +
<math>\displaystyle \text{Beta}(\alpha,\beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1} </math> where <math>0 \leq x \leq 1</math> and <math>\alpha</math>>0, <math>\beta</math>>0<br>
 +
and
 +
<math>f(x;\alpha,\beta)= 0 </math> otherwise
 +
 
 +
:<math>\displaystyle \text{Beta}(1,1) = U (0, 1) </math><br>
 +
 
 +
 
 +
:<math>\displaystyle \text{Beta}(\alpha,1)={f}(x) = \frac{\Gamma(\alpha+1)}{\Gamma(\alpha)\Gamma(1)}x^{\alpha-1}(1-x)^{1-1}=\alpha x^{\alpha-1}</math><br>
 +
 
 +
 
 +
'''Gamma Distribution'''
 +
 
 +
'''Algorithm'''<br\>
 +
*1. Sample from          Y1 ~ Gamma (<math>\alpha</math>,1)  where <math>\alpha</math> is the shape, and 1 is the scale.<br\>
 +
*2. Sample from          Y2 ~ Gamma (<math>\beta</math>,1)  where <math>\alpha</math> is the shape, and 1 is the scale.<br\>
 +
*3. Set <math> Y = \frac{Y_1}{Y_1+Y_2},  </math> Then Y ~ <math>\beta ( \alpha , \beta )</math>, where we suppose <math>\alpha , \beta</math> are integers.
 +
 
 +
=== Geometric ===
 +
 
 +
This distribution models the number of failures before the first success.
 +
 
 +
X~Geo(p)
 +
 
 +
<math>f(x)=p*(1-p)^{x-1}, x=1, 2, ........</math><br>
 +
If Y~Exp<math>(\lambda)</math> where <math>\lambda=-log(1-p)</math><br>
 +
then <math>X=floor[Y]+1</math> is Geo(p)<br>
 +
 
 +
'''Proof:'''
 +
    <math>F(x)=P(X<=x)</math><br>
 +
    <math>    =1-P(X>x)</math><br>
 +
    <math>    =1-P(floor [Y]+1>x)</math> as <math> floor [Y]+1>x</math> implies <math> Y\leq x </math> for all real-valued Y<br>
 +
    <math>    =1-P(Y>=x)</math><br>
 +
    <math>    =1-(1-P(Y<x))</math><br>
 +
    <math>    =1-e^{-\lambda*x}</math><br>
 +
    <math>    =1-(1-p)^x</math>, which is the CDF of Geo(p)<br>
 +
 
 +
The above method can also be viewed with the inverse method since it is not rejecting any points. The following uses inverse method to generate Geo(p).<br>
 +
<math>F(x)=P(X \leq x)</math><br>
 +
<math>F(x)=1- P(X>x)</math><br>
 +
<math>P(X \leq x)=1-(1-p)^x</math> since <math>P(X>x)=(1-p)^x</math><br>
 +
<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=(e^{-\lambda})^x=e^{-\lambda*x}</math> since <math>1-p=e^{-\lambda}</math><br>
 +
<math>log(1-y)=-\lambda*x</math><br>
 +
<math>x=-1/(\lambda)*log(1-y)</math><br>
 +
<math>F^-1(x)=-1/(\lambda)*log(1-x)</math><br>
 +
<br>
 +
'''Algorithm:'''<br />
 +
1. Generate U ~ U(0,1)<br>
 +
2. Generate <math>Y=-1/(\lambda)*log(1-u)</math> where <math>Y ~ \sim Exp(\lambda) </math><br>
 +
3. Set <math>X=[Y]+1</math><br>
 +
 
 +
Note: <br />
 +
If X~Unif (0,1), Y= floor(3X) = [3X]-> Y ~ DU[0,2] (DU means discrete uniform)<br />
 +
 
 +
If X~Unif (0,1), Y= floor(5U)-2 = [5U]-2 -> Y~ DU[-2,2]
 +
<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===
 +
 
 +
This distribution models the number of times and event occurs in a given time period
 +
 
 +
X~Poi<math>(\lambda)</math> <br>
 +
X is the maximum number of iid Exp(<math>\lambda</math>) whose sum is less than or equal to 1.<br>
 +
<math>X = \max\{n: \sum\limits_{i=1}^n Y_i \leq 1, Y_i \sim Exp(\lambda)\}</math><br>
 +
<math>  = \max\{n: \sum\limits_{i=1}^n \frac{-1}{\lambda} log(U_i)<=1 , U_i \sim U[0,1]\}</math><br>
 +
<math>  = \max\{n: \prod\limits_{i=1}^n U_i >= e^{-\lambda}, U_i \sim U[0,1]\}</math><br>
 +
 
 +
'''Algorithm'''<br\>
 +
*1. Set n=1, a=1<br\>
 +
*2. <math>U_n</math> ~ <math>U[0,1]</math> and set <math>a=aU_n</math><br\>
 +
*3. If <math>a \geq e^{-\lambda}</math> then: n=n+1 and go to Step 2. Else set X=n-1.
 +
 
 +
An alternate way to write an algorithm for Poisson is as followings:
 +
 
 +
1)  x = 0, F = P(X=0) = e^-λ = p
 +
 
 +
 
 +
2)  Generate U ~ U(0,1)
 +
 
 +
 
 +
3)  If U < F, then output x
 +
 
 +
 
 +
4)  Else p = (λ/(x+1)) * p
 +
 
 +
    F = F + p
 +
 
 +
    x = x+1
 +
 
 +
5) else go to step 2
 +
 
 +
Acknowledgments: from Spring 2012 stat 340 coursenotes
 +
 
 +
== Class 13 - Tuesday June 18th 2013 ==
 +
'''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/>
 +
 
 +
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/>
 +
 
 +
<math> P= \left [ \begin{matrix}
 +
0.7 & 0.3 \\
 +
0.2 & 0.8
 +
\end{matrix} \right] </math>
 +
 
 +
The two step transition probability matrix is:
 +
 
 +
<math> P P= \left [ \begin{matrix}
 +
0.7 & 0.3 \\
 +
0.2 & 0.8
 +
\end{matrix} \right] \left [ \begin{matrix}
 +
0.7 & 0.3 \\
 +
0.2 & 0.8
 +
\end{matrix} \right] </math>=<math>\left [ \begin{matrix}
 +
0.7(0.7)+0.3(0.2) & 0.7(0.3)+0.3(0.8)              \\
 +
0.2(0.7)+0.8(0.2) & 0.2(0.3)+0.8(0.8)
 +
\end{matrix} \right] </math>=<math>\left [ \begin{matrix}
 +
0.55 &  0.45                  \\
 +
0.3  & 0.7
 +
\end{matrix} \right] </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_2 = P_1 P_1 </math><br\>
 +
 
 +
<math>P_3 = P_1 P_2 </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 />
 +
The equation above is a special case of the Chapman-Kolmogorov equations.<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 />
 +
only depends on your current state, not your previous states. By intuition, we can multiply the 1-step transition <br />
 +
matrix n-times to get a n-step transition matrix.<br />
 +
 +
Example: We can see how <math>P_n = P^n</math> from the following:
 +
<br/>
 +
<math>\vec{\mu_1}=\vec{\mu_0}\cdot P</math> <br/>
 +
<math>\vec{\mu_2}=\vec{\mu_1}\cdot P</math> <br/>
 +
<math>\vec{\mu_3}=\vec{\mu_2}\cdot P</math> <br/>
 +
Therefore,
 +
<br/>
 +
<math>\vec{\mu_3}=\vec{\mu_0}\cdot P^3
 +
</math> <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 />
 +
 
 +
Example with Markov Chain:
 +
Consider a two-state Markov chain {<math>X_t; t = 0, 1, 2,...</math>} with states {1,2} and transition probability matrix
 +
 
 +
<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:
 +
 
 +
a)<math> P(X_1=1 | X_0=1) = P(1,1) = 1/2 </math>
 +
 
 +
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>
 +
 
 +
c)<math> P(X_2=1|X_0=1)= P_2(1,1) = 5/12 </math>
 +
 
 +
d)<math> P^2=P*P= \left [ \begin{matrix}
 +
5/12 & 7/12 \\
 +
7/18 & 11/18
 +
\end{matrix} \right] </math>
 +
 
 +
=== Marginal Distribution of Markov Chain ===
 +
We represent the probability of all states at time t with a vector <math>\underline{\mu_t}</math><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>\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>\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/>
 +
The vector <math>\underline{\mu_0}</math> is called the initial distribution. <br/>
 +
 
 +
<math> P^2~=P\cdot P </math> (as verified above)
 +
 
 +
In general,
 +
<math> P^n~= \Pi_{i=1}^{n} P</math> (P multiplied n times)<br/>
 +
<math>\mu_n~=\mu_0 P^n</math><br/>
 +
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/>
 +
0  a  a  b  a<br/>
 +
1  a  b  a  a<br/>
 +
2  b  a  a  b<br/>
 +
3  b  a  b  b<br/>
 +
4  a  a  a  b<br/>
 +
5  a  b  b  a<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 />
 +
 
 +
 
 +
 
 +
 
 +
Marginal Distribution
 +
 
 +
<math>\mu_1~ = \mu_0P</math> <br>
 +
<math>\mu_2~ = \mu_1P = \mu_0PP = \mu_0P^2</math> <br>
 +
 
 +
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 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/>
 +
 
 +
 
 +
'''Comments:'''<br/>
 +
As n gets bigger and bigger, <math>\mu_n</math> will possibly stop changing, so the quantity <math>\pi</math> <sub>i</sub> can also be interpreted as the limiting probability that the chain is in the state <math>j</math>
 +
 
 +
Comments: <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 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 ====
 +
<pre style='font-size:14px'>
 +
 
 +
In Matlab, you can find the stationary distribution by:
 +
 
 +
>> p=[.7 .3;.2 .8]              % Input the matrix P
 +
 
 +
p =
 +
 
 +
    0.7000    0.3000
 +
    0.2000    0.8000
 +
 
 +
>> p^2                          % one state to another state by 2 steps transition
 +
 
 +
ans =
 +
 
 +
    0.5500    0.4500
 +
    0.3000    0.7000
 +
 
 +
>> mu=[.9 .1]                                 
 +
 
 +
mu =
 +
 
 +
    0.9000    0.1000
 +
 
 +
>> mu*p                        %  enter mu=mu*P, repeat multiple times until the value of the vector mu remains unchanged
 +
 
 +
ans =
 +
 
 +
    0.6500    0.3500
 +
 
 +
>> mu=mu*p
 +
 
 +
mu =
 +
 
 +
    0.4002    0.5998
 +
 
 +
>> mu=mu*p                    %The vector mu will be your stationary distribution
 +
 
 +
mu =
 +
 
 +
    0.4000    0.6000
 +
 
 +
 
 +
>> p^100                      % it is limiting distribution of chain which finally gives a stable matrix
 +
                                                       
 +
ans =
 +
 
 +
    0.4000    0.6000
 +
    0.4000    0.6000
 +
 
 +
 
 +
</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. 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}
 +
1/3 & 1/3 & 1/3 \\
 +
1/4 & 3/4 & 0 \\
 +
1/2 & 0 & 1/2 \end{array} } \right]</math>
 +
 
 +
Solution:
 +
<math>\pi=\left[ {\begin{array}{ccc}
 +
\pi_0 & \pi_1 & \pi_2 \end{array} } \right]</math>
 +
 
 +
Using the stationary distribution property <math>\pi=\pi~P</math> we get, <br>
 +
<math>\pi_0=\frac{1}{3}\pi_0+\frac{1}{4}\pi_1+\frac{1}{2}\pi_2 </math><br>
 +
<math>\pi_1=\frac{1}{3}\pi_0+\frac{3}{4}\pi_1+0\pi_2 </math><br>
 +
<math>\pi_2=\frac{1}{3}\pi_0+0\pi_1+\frac{1}{2}\pi_2 </math><br>
 +
 
 +
And since <math>\pi</math> is a probability vector, <br>
 +
<math> \pi_{0}~ + \pi_{1} + \pi_{2} = 1 </math>
 +
 
 +
Solving the 4 equations for the 3 unknowns gets, <br>
 +
<math>\pi_{0}~=1/3</math>, <math>\pi_{1}~=4/9</math>, and <math>\pi_{2}~=2/9</math> <br>
 +
Therefore <math>\pi=\left[ {\begin{array}{ccc}
 +
1/3 & 4/9 & 2/9 \end{array} } \right]</math>
 +
 
 +
Example 2: Find the stationary distribution of P= <math>\left[ {\begin{array}{ccc}
 +
1/3 & 1/3 & 1/3 \\
 +
1/4 & 1/2 & 1/4 \\
 +
1/6 & 1/3 & 1/2 \end{array} } \right]</math>
 +
 
 +
Solution:
 +
<math>\pi=\left[ {\begin{array}{ccc}
 +
\pi_0 & \pi_1 & \pi_2 \end{array} } \right]</math>
 +
 
 +
Using the stationary distribution property <math>\pi=\pi~P</math> we get, <br>
 +
<math>\pi_0=\frac{1}{3}\pi_0+\frac{1}{4}\pi_1+\frac{1}{6}\pi_2 </math><br>
 +
<math>\pi_1=\frac{1}{3}\pi_0+\frac{1}{2}\pi_1+\frac{1}{3}\pi_2 </math><br>
 +
<math>\pi_2=\frac{1}{3}\pi_0+\frac{1}{4}\pi_1+\frac{1}{2}\pi_2 </math><br>
 +
 
 +
And since <math>\pi</math> is a probability vector, <br>
 +
<math> \pi_{0}~ + \pi_{1} + \pi_{2} = 1 </math>
 +
 
 +
Solving the 4 equations for the 3 unknowns gets, <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}
 +
\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">
 +
Generating Random Initial distribution<br>
 +
<math>\mu~=rand(1,n)</math><br>
 +
<math>\mu~=\frac{\mu}{\Sigma(\mu)}</math></span>
 +
 
 +
<span style="background:#F5F5DC">
 +
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>
 +
<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>
 +
<math> (\frac{1}{n}, \ldots , \frac{1}{n}) </math>
 +
 
 +
=== Properties of Markov Chain ===
 +
 
 +
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.
 +
 
 +
1. Reducibility <br>
 +
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>
 +
<math>P(X_{n_{ij}} =j \vert X_0 =i) > 0</math><br>
 +
 
 +
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 />
 +
 
 +
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 />
 +
 
 +
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 />
 +
 
 +
(The properties are from
 +
http://www2.math.uu.se/~takis/L/McRw/mcrw.pdf)
 +
 
 +
CHAPMAN-KOLMOGOROV EQUATION
 +
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>
 +
 
 +
== Class 14 - Thursday June 20th 2013 ==
 +
 
 +
Example: Find the stationary distribution of <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>
 +
 
 +
<math>\displaystyle \pi=\pi  p</math>
 +
 
 +
Solve the system of linear equations to find a stationary distribution
 +
 
 +
<math>\displaystyle \pi=(\frac{1}{3},\frac{4}{9}, \frac{2}{9})</math>
 +
 
 +
Note that <math>\displaystyle \pi=\pi  p</math> looks similar to eigenvectors/values <math>\displaystyle \lambda vec{u}=A vec{u}</math>
 +
 
 +
<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.
 +
 
 +
<math>=> \pi</math><sup>T</sup>= P<sup>T</sup><math>\pi</math><sup>T</sup><br/>
 +
Then <math>\pi</math><sup>T</sup> is an eigenvector of P<sup>T</sup> with eigenvalue = 1. <br />
 +
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 = [1/3 1/3 1/3; 1/4 3/4 0; 1/2 0 1/2]
 +
 
 +
pii = [1/3 4/9 2/9]
 +
 
 +
[vec val] = eig(P')            %% P' is the transpose of matrix P
 +
 +
vec(:,1) = [-0.5571 -0.7428 -0.3714]      %% this is in column form
 +
 
 +
a = -vec(:,1)
 +
 
 +
>> a =
 +
[0.5571 0.7428 0.3714]   
 +
 
 +
%% a is in column form
 +
 
 +
%% Since we want this vector a to sum to 1, we have to scale it
 +
 
 +
b = a/sum(a)
 +
 
 +
>> b =
 +
[0.3333 0.4444 0.2222] 
 +
 
 +
%% b is also in column form
 +
 
 +
%% Observe that b' = pii
 +
 
 +
</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>
 +
 
 +
That is <math>\pi_j=\lim[P^n]_{ij}</math> exists and is independent of i.<br/>
 +
 
 +
A Markov Chain is convergent if and only if its limiting distribution exists. <br/>
 +
 
 +
If the limiting distribution <math>\pi</math> exists, it must be equal to the stationary distribution.<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/>
 +
 
 +
'''Example:'''
 +
For a transition matrix <math> P= \left [ \begin{matrix}
 +
0 & 1 & 0 \\[6pt]
 +
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 ===
 +
<pre style='font-size:14px'>
 +
MATLAB
 +
>> P=[0, 1, 0;0, 0, 1; 1, 0, 0]
 +
 
 +
P =
 +
 
 +
    0    1    0
 +
    0    0    1
 +
    1    0    0
 +
 
 +
>> pii=[1/3, 1/3, 1/3]
 +
 
 +
pii =
 +
 
 +
    0.3333    0.3333    0.3333
 +
 
 +
>> pii*P
 +
 
 +
ans =
 +
 
 +
    0.3333    0.3333    0.3333
 +
 
 +
>> P^1000
 +
 
 +
ans =
 +
 
 +
    0    1    0
 +
    0    0    1
 +
    1    0    0
 +
 
 +
>> P^10000
 +
 
 +
ans =
 +
 
 +
    0    1    0
 +
    0    0    1
 +
    1    0    0
 +
 
 +
>> P^10002
 +
 
 +
ans =
 +
 
 +
    1    0    0
 +
    0    1    0
 +
    0    0    1
 +
 
 +
>> P^10003
 +
 
 +
ans =
 +
 
 +
    0    1    0
 +
    0    0    1
 +
    1    0    0
 +
 
 +
>> %P^10000 = P^10003
 +
>> % This chain does not have limiting distribution, it has a stationary distribution. 
 +
 
 +
This chain does not converge, it has a cycle.
 +
</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>
 +
 
 +
<ma