Monte Carlo Methods: Direct Sampling and Integration, Slides of Stochastic Processes

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Lecture Two: Direct Sampling -

II

Monte Carlo Methods^ Monte Carlo Methods

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Direct Sampling : Part –II

b a

dx

x

g

x

f

I^

Suppose that

we have to find following integral:

Then let us first sample x using f(x) (say by direct method).

First we find its

normalization constant:

∞^ ∫

0

(^

A

dx x f

So, normalized pdf is given as

x a

dx

x

Af

x

F

Then the Cumulative distribution is given as

(^

x

Af

x

h

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Example 1: As an illustration of this we will try first to find the following integral:

1 0 4

1 0

3

∫

x

dx x

I

∫

1 0

2 dx

x

x

I

2

1

2

) (

1 0 2

1 0

1 0

=

⇒

=

=^

∫

∫^

A

x A

dx x A

dx x Af

Let us consider f(x) = x and g(x) = x

The exact answer is

Let us first find the normalization constant for f(x): Then comulative pdf is

ξ^

=^

∫

∫^

x

x

dx x

xdx

x

x

x

0 2

0

0

and the find the area as

. /) (

1 2

N x g

I^

i^

i

∑ ≈

So, we use this equationto generate x

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MATLAB

Program for the

example

% Program to find integral (f(x)g(x) type) % by Monte Carlo method % as a function of N N = 1000 ; rand('state', 0) % initialize area(1)= 0.25; xn(1)= N; exact_area(1)= 0.25; er(1) =0.0;^ for i = 2:

i sum = 0; for k = 1: N

exi = rand; x(k) = sqrt(exi);

end^ for kk = 1: N

sum = sum + x(kk)*x(kk);

end^ area(i) = sum/(2.*N); xn(i)= N;^ N = N + 1000;^ exact_area(i) = 0.25;^ error(i) = (area(i) - exact_area(i)); end plot(xn, area, xn, exact_area)

Example 1:

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Let us try to find the following integral:

π

π

=

=

∫^0

sin

xdx

x

I

∫

1 0

)

(sin

dx x

x

I^2

0 2

0

0

/ 2 1 2 ) ( π

π

π

π

=

⇒

=

=^

∫

∫^

A

x A

dx x A

dx x Af

Let us consider f(x) = x and g(x) = sinx.

The exact answer is

Let us first find the normalization constant for f(x): Then comulative pdf is

ξ π

ξ

=

⇒

=

=

=^

∫

∫

x

x A

dx x A

Axdx

x

x

x

0 2

0

0

2

So, we use the above equation to generate x and the find the area as

. /)

(

(^22)

N

x g

I^

i^

i

∑

≈

π

Example 2:

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The MATLAB program

for Example 2

% Program to find integral (f(x)g(x) type) % by Monte Carlo method % as a function of number of histories N = 1000 ; rand('state', 0) % initialize area(1)= pi; xn(1)= N; exact_area(1)= pi; er(1) =0.0;^ for i = 2:

i sum = 0; for k = 1: N

exi = rand; x(k) = pi*sqrt(exi);

end

for kk = 1: N

sum = sum + sin(x(kk)); end area(i) = pipisum/(2.*N); xn(i)= N; N = N + 1000; exact_area(i) = pi; error(i) = (area(i) - exact_area(i));

Example 2:^ end^ plot(xn, area, xn, exact_area)

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RESULTS

Example 2:^ Figure above shows the area versus the number of histories (N). Increasingthe N we see a decrease in the error. The error bars below are indicating thateffect.

Error in Area

Number of Trails in a sample

x 1000

Error in Area from MC method Zero Error line

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