stat341f11

Sampling - Sept 20, 2011

The meaning of sampling is to generate data points or numbers such that these data follow a certain distribution.
i.e. From [math]\displaystyle{ x \sim~ f(x) }[/math] sample [math]\displaystyle{ \,x_{1}, x_{2}, ..., x_{1000} }[/math]

In practice, it maybe difficult to find the joint distribution of random variables. Through simulating the random variables, we can inference from the data.

Sample from uniform distribution.

Computers can't generate random numbers as they are deterministic but they can produce pseudo random numbers.

Multiplicative Congruential

• involves three parameters: integers [math]\displaystyle{ \,a, b, m }[/math], and an initial value [math]\displaystyle{ \,x_0 }[/math] we call the seed
• a sequence of integers is defined as

[math]\displaystyle{ x_{k+1} \equiv (ax_{k} + b) \mod{m} }[/math]

Example: [math]\displaystyle{ \,a=13, b=0, m=31, x_0=1 }[/math] creates a uniform histogram.

MATLAB code for generating 1000 randoms using the multiplicative congruential method:

a=13;
b=0;
m=31;
x(1)=1;

for ii = 1:1000
    x(ii+1) = mod(a*x(ii)+b,m);
end

MATLAB code for displaying the values of x generated:

x;

MATLAB code for plotting the histogram of x:

hist(x);

Facts about this algorithm:

• In this example, the first 30 terms in the sequence are a permutation of integers from 1 to 30 then the sequence repeats itself.
• Values are between 0 and [math]\displaystyle{ m-1 }[/math].
• Dividing the numbers by [math]\displaystyle{ m-1 }[/math] yields numbers in the interval [math]\displaystyle{ [0,1) }[/math].
• MATLAB's rand function once used this algorithm with [math]\displaystyle{ a=7^5, b=0, m=2^{31}-1 }[/math], for reasons described in Park and Miller's 1988 paper "Random Number Generators: Good Ones are Hard to Find" (available online).

Inverse Transform Method

[math]\displaystyle{ P(a\lt x\lt b)=\int_a^{b} f(x) dx }[/math]

[math]\displaystyle{ cdf=F(x)=P(X\lt =x)=\int_{-\infty}^{x} f(x) dx }[/math]

Assume cdf & cdf ^-1 can be found

Theorem:

Take [math]\displaystyle{ U \sim~ \mathrm{Unif}[0, 1] }[/math] and let [math]\displaystyle{ x=F^{-1}(u) }[/math]. Then x has distribution function [math]\displaystyle{ F() }[/math], where [math]\displaystyle{ F(x)=P(X\lt =x) }[/math].

Let [math]\displaystyle{ F^{-1}() }[/math] denote the inverse of [math]\displaystyle{ F() }[/math]. Therefore [math]\displaystyle{ F(x)=u \implies x=F^{-1}(u) }[/math]

Take the the exponential distribution for example

[math]\displaystyle{ \,f(x)={\lambda}e^{-{\lambda}x} }[/math]

[math]\displaystyle{ \,F(x)=\int_0^x {\lambda}e^{-{\lambda}u} du }[/math]

[math]\displaystyle{ \,F(x)=1-e^{-{\lambda}x} }[/math]

Let: [math]\displaystyle{ \,F(x)=y }[/math]

[math]\displaystyle{ \,y=1-e^{-{\lambda}x} }[/math]

[math]\displaystyle{ \,ln(1-y)={-{\lambda}x} }[/math]

[math]\displaystyle{ \,x=\frac{ln(1-y)}{-\lambda} }[/math]

[math]\displaystyle{ \,F^{-1}(x)=\frac{-ln(1-x)}{\lambda} }[/math]

Therefore, to get a exponential distribution from a uniform distribution takes 2 steps.

• Step 1. Draw [math]\displaystyle{ U \sim~ \mathrm{Unif}[0, 1] }[/math]
• Step 2. [math]\displaystyle{ x=\frac{-ln(1-U)}{\lambda} }[/math]

Now we just have to show the generated points have a cdf of F(x)

[math]\displaystyle{ \,p(F^{-1}(u)\lt =x) }[/math]

[math]\displaystyle{ \,p(F(F^{-1}(u))\lt =F(x)) }[/math]

[math]\displaystyle{ \,p(u\lt =F(x)) }[/math]

[math]\displaystyle{ \,=F(x) }[/math]

QED

Discrete Case

This same technique can be applied to the discrete case. Generate a discrete random variable [math]\displaystyle{ \,x }[/math] that has probability mass function [math]\displaystyle{ \,p(x=x_i)=p_i }[/math] where [math]\displaystyle{ \,x_0\lt x_1\lt x_2... }[/math] and [math]\displaystyle{ \,\sum p_i=1 }[/math]

• Step 1. Draw [math]\displaystyle{ u \sim~ \mathrm{Unif}[0, 1] }[/math]
• Step 2. [math]\displaystyle{ \,x=x_i }[/math] if [math]\displaystyle{ \,F(x_{i-1})\lt u\lt =F(x_i) }[/math]

Example - Sept 22, 2011

Let x be a discrete random variable with the following probability mass function:

[math]\displaystyle{ \begin{align} P(X=0) = 0.3 \\ P(X=1) = 0.2 \\ P(X=2) = 0.5 \end{align} }[/math]

Given the pmf, we now need to find the cdf.

We have:

[math]\displaystyle{ F(x) = \begin{cases} 0 & x \lt 0 \\ 0.3 & x \lt 1 \\ 0.5 & x \lt 2 \\ 1 & x \lt 3 \end{cases} }[/math]

We can now apply the inverse transform method to obtain our random numbers from this distribution.

Pseudo Code for generating the random numbers:

Draw U ~ Unif[0,1] 
     if U < 0.3 
        x = 0 
     else if  
        0.3 < U < 0.5 
        x = 1 
     else 
        0.5 <= U < 1 
        x = 2

Matlab code for generating 1000 random numbers in the discrete case:

for ii = 1:1000
    u = rand;
    
    if u < 0.3
       x(ii) = 0;
       else if u < 0.5
           x(ii) = 1;
           else
             x(ii) = 2;
       end
    end

Pseudo code for the Discrete Case:

1. Draw U ~ Unif [0,1]

2. If [math]\displaystyle{ U \leq P_0 }[/math], deliver [math]\displaystyle{ x = x_0 }[/math]

3. Else if [math]\displaystyle{ U \leq P_0 + P_1 }[/math], deliver [math]\displaystyle{ x = x_1 }[/math]

4. Else If [math]\displaystyle{ U \leq P_0 +....+ P_k }[/math], deliver [math]\displaystyle{ x = x_k }[/math]

Although this method is useful, it isn't practical in many cases since we can't always obtain [math]\displaystyle{ F }[/math] or [math]\displaystyle{ F^{-1} }[/math]

Acceptance/Rejection Sampling

Sampling from [math]\displaystyle{ f(x) }[/math] can be difficult. Another general method for generating random numbers is acceptance/rejection sampling.

Suppose we assume the following:

1. There exists another distribution [math]\displaystyle{ g(x) }[/math] that is easier to work with and that you know how to sample from, and

2. There exists a constant c such that [math]\displaystyle{ f(x) \leq c \cdot g(x) }[/math] for all x

Under these assumptions, we can sample from [math]\displaystyle{ f(x) }[/math] by sampling from [math]\displaystyle{ g(x) }[/math]

General Idea

Looking at the image below we have graphed [math]\displaystyle{ c \cdot g(x) }[/math] and [math]\displaystyle{ f(x) }[/math].

Using the acceptance/rejection method we will accept some of the points from [math]\displaystyle{ g(x) }[/math] and reject some of the points from [math]\displaystyle{ g(x) }[/math]. The points that will be accepted from [math]\displaystyle{ g(x) }[/math] will have a distribution similar to [math]\displaystyle{ f(x) }[/math]. We can see from the image that the values around [math]\displaystyle{ x_1 }[/math] will be sampled more often under [math]\displaystyle{ c \cdot g(x) }[/math] than under [math]\displaystyle{ f(x) }[/math], so we will have to reject more samples taken at x₁. Around [math]\displaystyle{ x_2 }[/math] the number of samples that are drawn and the number of samples we need are much closer, so we accept more samples that we get at [math]\displaystyle{ x_2 }[/math]

Procedure

1. Draw y ~ g

2. Draw U ~ Unif [0,1]

       If U [math]\displaystyle{ \begin{align}\leq \frac{f(x)}{c \cdot g(x)} \end{align} }[/math]
          Then [math]\displaystyle{  x=y }[/math]

       Else

          return to 1

Example

Sample from Beta(2,1)

In general:

Beta([math]\displaystyle{ \alpha, \beta) = \frac{\gamma (\alpha + \beta)}{\gamma(\alpha)\gamma(\beta)} }[/math] [math]\displaystyle{ x^{\alpha-1} }[/math] [math]\displaystyle{ (1-x)^{\beta-1} }[/math]

Note: [math]\displaystyle{ \gamma(n) = (n-1)! }[/math] if n is a positive integer

[math]\displaystyle{ \begin{align} f(x) = Beta(2,1) = \frac{\gamma(3)}{\gamma(2)\gamma(1)} x^1(1-x)^0 \\ = \frac{2!}{1! 0!}\cdot (1) \\ = 2 \end{align} }[/math]

We want to choose [math]\displaystyle{ g(x) }[/math] that is easy to sample from. So we choose [math]\displaystyle{ g(x) }[/math] to be uniform distribution.

We now want a constant c such that [math]\displaystyle{ f(x) \leq c \cdot g(x) }[/math]

So,[math]\displaystyle{ c \geq \frac{f(x)}{g(x)} }[/math] [math]\displaystyle{ c = max \frac {f(x)}{g(x)} }[/math] [math]\displaystyle{ = max \frac {2x}{1} }[/math] [math]\displaystyle{ = 2 }[/math]

1. Draw y ~ g(x)

2. Draw y ~ Unif [0.1]

Select U ~ Unif [0,1]

if [math]\displaystyle{ u \leq \frac{2y}{2 \cdot 1} }[/math]

   then x=y 

else

   return to 2

Matlab code

while ii < 1000
    y = rand;
    u = rand;
    
    if u <= y
        x(ii)=y;
        ii=ii+1;
    end
end
hist(x)