Direct Sampling-Stochastic Process-Lecture Slides, Slides of Stochastic Processes

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Lecture Eight:

Direct Sampling

Monte Carlo Methods^ Monte Carlo Methods

Sampling Terminology & Concepts ‰

We

Collect

data

that

will

provide

information

on

the

phenomena

of

interest. ‰

Using data we draw conclusions. ‰

Then researcher generalizes from that experiment to the class of all suchexperiments. ‰

However, we are not absolutely certain about our conclusions. ‰

Using statistical techniques, we can measure and manage the degree ofuncertainty in our results. Target Population is defined as the entire collection of objects or individualsabout which we need some information. Sample Population is only a part of the entire collection of objects orindividuals. Random Sample Population is a part of the entire collection of objects orindividuals that are randomly selected from Target Population.

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Sampling Random Variables Monte

Carlo

calculation

is

a

numerical

stochastic

process.

It

is

usually required that random variables be drawn from distributionfunctions that derive the process.

For example, the integral is given.

∫^

dx

x

g

x

f^

Its Evaluation requires that values of x must be drawn using

f(x)

and

the

value

of

average

of

g(x)

over

such

set

of

x

values

be

calculated. This is the process of “sampling x from f(x)” and it is an essential techniquefor any Monte Carlo simulation.^ We may also say that the sampling process is an algorithm thatcan produce a sequence of values of x (random variables) suchthat for any W that belongs to a universal set:

. 1

) (

}

{^

≤

= Ω ∈

∫^

dx x f

x P

k

Techniques for Direct Sampling It is about the method of selection of a sequence of values of independentvariables, x, such that, in the limit of a large number of such selections, thedistribution follow the pdf,

f(x). This is “sampling the function f(x)”.

The cumulative distribution function (cdf) is defined as

dx

x

f

x

F

x ∫ a

=^

(^

Where the range of x is in [a, b].

This function plays a central role in the Monte Carlo work.

The normalization

The F(x) represents the probability that a random variable from f(x) liesbetween a and x.condition also requires that

(^

=^

∫^

dx

x

f

b

F

b a

Example 2:

Direct Sampling

Let us perform the sampling of function f(x).

Suppose that the given

pdf for y is an

exponential probability distribution.

∞ < <

−

=^

y

y

y f^

0 )

exp(

) (

λ

λ

(^

y F Then the cumulative distribution is The

F(y)

represents the probability that a random variable from

f(y)

lies between 0 and y.

λ^

=^ ∫

exp( 1 )

exp(

) (

0

y

y

y F

y

log(

log(

exp(

ξ

λ

ξ

λ

ξ

λ

y

y

MATLAB Program on

Exponential

Distribution Sampling

%^ MC Sampling for f(y) = lambdaexp(-lambday) % Exponential Distribution sampling program in MATLAB**^ N = 10000 ;

% No of histories to be generated

max_bins = 500;

% Max. no of bins

xlambda = 2.0; rand('state', 0)

% initialize to zero

for j=1:max_bins

% initialize

ibin(j) = 0; xmid(j) = 0; ntheory(j)=0; end for i= 1:N

% Start Monte Carlo loop

yy = rand;

% choose random F

ff(i) = -log(yy)/xlambda; % calculate function end max_ff = 5;

% find max value of func estimated

size_bin = max_ff/max_bins;

% find size of bins

for kk=1: N

% put all in bins

l_limit=0; u_limit=0; for ii=1: max_bins l_limit = (ii-1)size_bin; u_limit = iisize_bin; if((ff(kk)>l_limit)&(ff(kk)<=u_limit)) jj=ii; end end ibin(jj) = ibin(jj) + 1; % one more for the bin end for k = 1:max_bins

xmid(k) = (k - 0.5)size_bin;*

% compute mid-point

% compute the expected pdf

fx = xlambdaexp(-xlambdaxmid(k)); ntheory(k) = Nfxsize_bin;**

% no by theory

end^ plot(xmid, ibin,'r+',xmid, ntheory,'b.')

0

1

2

3

4

5

5 0^0 2 5 0 2 0 0 1 5 0 1 0 0

Example 2: N = 10000, No. of Bins = 500, Lambda = 2.

Direct Sampling

ExponentialDistributionSampling

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(^

)^

.

0

, 1

;

1

) (^

∞ ≤ ≤

>

=^

x

n

bx a

x w^

n

.

) 1

(

1

) ( 0

1

∞ ∫

−

−

=^

n ba

n

dx x w

First we find its normalization constant: So, normalized pdf is given as

∫^

−

−

=

x

n a bx

dx x f

x F

0

1 ) /

(^1) (

1 1 ) ( ) (

( )[^

]^

. 1

0

; 1

) (

) (^

) 1 /( 1

1

< <

−

=

⇒

=^

− −

−

ξ

ξ

ξ

ξ^

n

a b

x

F x Then the Cumulative distribution is given as

(^

)^

.

0

, 1

;

) 1

( ) (

1

∞ ≤ ≤

>

− +

=

−

x

n

bx a

ba

n

x f^

n n

Example 3:

Direct Sampling

Consider another example of a pdf with the following form:

0

1

2

3

4

5

0 1000900800700600500400300200100 f(x)

Monte Carlo MethodTheory x

Parameters Used: a =10,

b = 1,

n = 2

N = 10000, No. of bins

=

200

Range for x

== (0, 5) (^

)

.

1

−

x n

bx a

ba n

x f^

n n

Example 3:

Direct Sampling

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Example 4: Sampling a Gaussian Distribution The Gaussian or a Normal distribution function is not easily sampled in directmanner. The pdf is given as following: And cumulative distribution function is given as Thus to sample the area (probability), we must take a slice of the area:

.

0

);

exp( 2 ) (^

2

∞ ≤ ≤

−

=^

x

x

x f

∫

∫^

−

=

=

x

x^ dx e

dx x f

x F

0

2 2 ) ( ) (

π

This is error function is not simply invertible. However, one can use the trickof changing from rectangular to polar coordinates:

θ

π π

d drr

e

dy dx e e

dy dx y f x f

r

y x^2

2 2 (^44)

) ( ) (

−

− −

=

0

1

2

3

4

5

0 400 300 200 100

Example 4: Sampling a Gaussian Distribution Using a MATLAB program, we have following:^ Parameters Used:^ N = 10000,^ No. of bins = 200^ Range for x = (0, 5)

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Consider a pdf

with the following form:

. 1

0

2 ) (^

≤ ≤

=^

x

x

x f ∫^

1 0

(^

dx x f It is correctly normalized since, Cumulative distribution is given as

∫^

x

x

dx x f

x F

0

2

(^

x

F

F

x

So, choosing a number of values of

ξ

from uniform distribution

and taking square root will give such values of x that will have f(x) as

pdf.

Example 5: Sampling a pdf

0 2500 2000 1500 1000 500

No of histories = 50000,No. of Bins

= 40

pdf =

f(x)

= 2x, [0, 1]

f(x)

x-value

Monte Carlo resultsTheoretical pdf

We developed a MATLABprogram for sampling from apdf

with the following form:

. 1

0

2 ) (^

≤ ≤

=

x

x

x Example 5: Sampling a pdf f^ For about 50 thousandhistories the resultsfrom Monte Carlosimulations startedmatching very closelyto the actual pdf.

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