#10MONTE CARLO SIMULATION

Systems 303 - Fall 2000

Instructor: Peter M. Hahn

hahn@seas.upenn.edu

MONTE CARLO SIMULATION

• Based upon the generation of random numbers• Used for solving stochastic (or deterministic)

problems where the passage of time plays no substantive role

• Widely used to solve certain problems in statistics that are not analytically tractable

• Specifically used for determining the critical values for the Kolmogorov-Smirnov test

• We will describe the concept, determine the number of random number samples required and give several examples

MODELING ALTERNATIVES

-Prob. Theory -Monte Carlo -Queuing Theory -Discrete Event-Diff. Eqns. Simulation -Markov Chains or Continuous etc. etc. Simulation

MONTE CARLO SIMULATION

• Consider the problem of evaluating a definite integral

where for simplicity r(x) ≥ 0 on [a,b], r(x) is bounded above by y = c and an analytical solution is not possible for r(x)

• The graph of r bounds a region R within the rectangle defined by x=a, x=b, y=0 and y=c

• We present a method for determining A

r(x)dxa

b

A

MONTE CARLO SIMULATION

y =r(x)

c

a b

y

x

R

METHOD FOR ESTIMATING A

• A is simply the area of region R

• Select point (x,y) at random in rectangle abc

• With probability p, (x,y) will satisfy y ≤ r(x)

• If we can determine p, then it is clear that

• The Monte Carlo method involves estimating p by generating statistically independent points (xi,yi) from the uniform distribution of points on the rectangle abc

pc(b a) A

METHOD FOR ESTIMATING A

• The pdf of points (xi,yi) is given by

• Actually, we can do the same by generating

fX,Y (x, y) 1

c(b a) a x b,0 y c

0 otherwise

fX (x) 1

(b a) a xb

0 otherwise

and fY (y) 1c

0 y c

0 otherwise

METHOD FOR ESTIMATING A• We have N samples (x1,y1), (x2,y2),…, (xN,yN)

• An estimate of p is the fraction of points in R

• How well approximates p depends on N

• The larger N the better the estimate

Let I(x,y) 1 if (x, y)R

0 otherwise

Then an estimate of p is

ˆ p 1N

I(x, y)i1

N

Number of sample points in R

N

ˆ p

BIASEDNESS OF ESTIMATOR OF p?

• If is an unbiased estimator, its variance is a measure of how good an estimate it is

• is indeed an unbiased estimator

ˆ p

E[ ˆ p ]1

NE I xi , yi

i1

N

1N

1c(b a)

dxdy

R

i1

N

1N

1c(b a)

N dxdy

R

Area of R

c(b a)p

ˆ p

HOW GOOD IS OUR ESTIMATOR?• The variance of is computed as followsˆ p 2 ˆ p E ˆ p 2 E ˆ p 2 E ˆ p 2 p2

E1

N 2 I xi , yi I x j , y j j1

N

i1

N

p2

E1

N 2 I x i , yi 2i1

N

1

N 2 I x i , yi I x j , y j j1

N

ij

N

p2

Np

N 2 N N 1 p2

N 2 p2 Np N 2 p2 Np2 N 2 p2

N 2

1N

p(1 p) pqN

CHEBYCHEV’S INEQUALITY (A BOUND)

X is an arbitrary RV with mean , mean -square X2

and variance X2 . The pdf of X is unknown.

X2 x 2 fX (x)dx

x 2 fX (x)dx

x 2 fX (x)dx

x

P X X2

2

Replacing X with X

P X X2

2

P X 1 X

2

2

CHEBYCHEV’S INEQUALITY (A BOUND)

-

A

B

P X A X

2

2 A *

But, B 1 A

Now B1 A *

P X 1 X2

2

SAMPLES REQUIRED FOR p ESTIMATE

P ˆ p p 1 2 ˆ p 2 1

pqN 2

Thus, if we wish

P ˆ p p 1 where , small, N has to be

N pq 2

It is necessary that p where < 1 so that

N 1 p

2 p

For 0.5, 0.01 and p 0.1, N 0.9

0.01 0.25 0.1 3600

SAMPLES REQUIRED FOR p ESTIMATE

• But, we really don’t know p at the start• What if p = 0.1 and we guess 0.3?

• A third try will give us the required N=3600

We calculate N 0.7

(0.01)(0.25)(0.3)934

Since ( ˆ p ) pq

N

0.1 0.9 934

0.01

The value of ˆ p we get should be within 0.03 of 0.1

So if ˆ p 0.13 a new N 0.87

(0.01)(0.25)(0.13)2677

MONTE CARLO SIMULATION

• Monte Carlo is commonly used today to determine outcomes of complex processes

• The process is represented by steps or equations in a general purpose computer

• The computer model is then exercised in a fashion similar to the real process to estimate the desired probabilities

• Often Monte Carlo is used for determining probability of rare but important events

• The rarer the event, the longer the simulation

AMMUNITION DEPOT EXAMPLE

• Bombers attempting to destroy a depot of odd shape fly over it in the E-W direction

• The pdf of bomb impact is two-dimensional Gaussian centered around the aim point with E-W = 600m and N-S = 300m

• The problem is to determine the percentage of bombs that hit the depot

• This example is hand simulated in Section 2.3 of B,C,N&N

AMMUNITION DEPOT EXAMPLE

AMMUNITION DEPOT EXAMPLE

AMMUNITION DEPOT EXAMPLE• The aiming point is considered the (0,0)

point of the following distribution

• X and Y values can be computer-generated with the ‘direct transformation technique’

• Assuming p = 3/7, = 0.01 and = 0.5

fX,Y (x, y) fX (x)fY (y) 1

2X Y

e 1

2x 2

X2 y2

Y2

where X 600m and Y 300m

N 1 p 2 p

0.57

0.01 0.25 0.429 532.4

DIRECT TRANSFORMATION

MONTE CARLO SIMULATION

• Suppose we can express the desired system measure as the RV X=g(Y1,…,Yk,Z1,…,Zm) where the Y’s represent random inputs and the Z’s represent control variables.

• Using Monte Carlo we can estimate E[h(X)] or P(x1<h(X)<x2). h(.) a real fn or constant.

• We generate Y1,…,Yk using their statistical distributions and a random number generator

• We read in Z1,…,Zm from a data file or key these in manually

MONTE CARLO ALGORITHM

N=number of simulation iterations

i=1, H = 0, P =0 (Initialize)

Read in Z1,…,Zm

Do while (i ≤ N) {Generate Y1i,…,Yki

Xi=g(Y1i,…,Yki,Z1,…,Zm)

H = H + h(Xi) where h(Xi) can be equal to Xi

If (x1<h(Xi)<x2 ) P = P + 1}

E[h(X)] = H/N

P(x1<h(X)<x2) = P/N

MONTE CARLO SIMULATION• Good practice - collect cumulative variance for

averages and Chebychev bounds for probabilities in order to revise number of samples N, if needed

• For either we need the cumulative sample mean:

After the n - th sample the sample mean is

X (n)1

nXi

i1

n

We add a sample Xn1

X (n 1) 1

n 1nX (n) Xn1

CUMULATIVE SAMPLE VARIANCE

After the n - th sample the cumulative sample variance is ˜ S 2 (n)

We add a sample Xn1

˜ S 2 (n 1) 1n

n 1 ˜ S 2 (n) Xn1 X (n 1) 2 Thus,

˜ S 2 (1) 0

˜ S 2 (2) 0 X2 X (2) 2

˜ S 2 (3)12

˜ S 2 (2) X3 X (3) 2 ˜ S 2 (4)

1

32 ˜ S 2 (3) X4 X (4) 2 etc.,

COMPUTER PRODUCTION EXAMPLE

• How many computers P to produce? WhereD = Market demand ~ 200 + exp(0.02)

L = Total labor cost 100+P

M= Total material cost 10+3P

I = Income per computer ~ N(5,1)

• We wish to maximize profit g(P) = I·min{D,P} - L - M

• For HW#10 conduct a Monte Carlo simulation to estimate E[g(250)]. Assume N=10,000 provides an accurate estimate