Monte_Carlo与积分计算

x
f ( x ) dx
x0

x0 h
f ( x ) dx
1 3
x0
h[ f ( x 0 ) 4 f ( x1 ) f ( x 2 )]
Monte Carlo method
E N
1 2
x1 x 0 x 2 x1 h
purely statistical, not rely on the dimension !
ˆ I z/2 s I ˆ
By the way…
Sample Mean Method
No one ever said that you have to use the uniform distribution Example:
b
I : x
a
1 /2
e
2x
dx

1

20
x
1 /2
2e
grid points add up quickly in high dimensions
bad choices of grid may misrepresent g(x)
第九章 Monte Carlo积分
Monte Carlo method can be used to compute
E ( M ) Np
2 ( M ) Np (1 P )
ˆ 2 ( Iˆ ) 2 p h ( b a ) 2 h (b a )
2 2
M ˆ ˆ E ( I ) E p h (b a ) E h (b a ) N h (b a ) N E ( M ) h (b a ) p I
Start Set N : large integer
s1 = 0, s2 = 0
Loop N times
xn = (b-a) un + a yn = (xn)
s1 = s1 + yn , s2 = s2 + yn2 Estimate mean m’=s1/N Estimate variance V’ = s2/N – m’2 End
Sample Mean Method
Simulation Results:
Simulation
1
true = 19.08554, n=100,000
ˆ I
19.10724
2
3 4 5
19.08260
18.97227 19.06814 19.13261
Don’t ever give an estimate without a confidence interval!
p : I h(b - a)
Hit-or-Miss Method
So, if we can estimate p, we can estimate I:
ˆ ˆ I ph(b - a)
ˆ where p is our estimate of p
Hit-or-Miss Method
We can easily estimate p:
ˆ Let I SM be the sample mean estimator of I
g(X1), g(X2), …, g(Xn) iid
Let Yi=g(Xi) for i=1,2,…,n Then
ˆ I (b - a) Y
and we can once again invoke the CLT.
For n “large enough” (n>30),
Sample Mean Method
Hit-or-Miss Method Sample Mean Method Variance Reduction Technique Variance Reduction using Rejection Technique
Importance Sampling Method
Sample Mean Method
2x
I[a, b](x) dx

1 2
EX

1 /2
I[a, b](X)

where X~exp(rate=2).
Sample Mean Method
Comparison of Hit-and-Miss and Sample Mean Monte Carlo
ˆ Let I HM be the hit-and-miss estimator of I
an approximation
ˆ I g(x i ) b a 2 (b - a) i 1 2 sI ˆ n n 1
n
Sample Mean Method
ˆ I (b - a)
g(X ) n
i i 1
1
n
X1, X2, …, Xn iid ->
This estimator is “unbiased”:
Sample Mean Method
1 n ˆ] E (b - a) E[I g(X i ) n i 1
(b - a)
b
E[g(X n
i 1
1
n
i
)]
b
(b - a)
g(x) b - a dx n
SM error
I
( b a ) 0 . 6745
V' N
Sample Mean Method
b
I g(x) dx
a
Write this as:
b b
I g(x) dx (b - a) g(x)
a a
1 b-a
dx
(b - a) Eg(X)
where X~unif(a,b)
I = (b-a) h (M/N)
End
Hit-or-Miss Method
Error Analysis of the Hit-or-Miss Method It is important to know how accurate the result of simulations are note that M is binomial(M,p)
Sample Mean Method


(b - a)
2
n 2 (b - a)
n
Var[g(X1 )]
(indent)
2

(b - a) n
a 2 b
g(x) E[g(X)]
2
b
1 b-a
dx
I 1 g(x) b - a b - a dx a
Sample Mean Method
Importance Sampling Method
Hit-or-Miss Method
• Evaluation of a definite integral
I

b
( x )dx
h
X X X X X X O
a
h ( x ) for any x
•Probability that a random point reside inside the area
第九章 Monte Carlo积分
Goal: Evaluate an integral:
b
I g(x) dx
a
Why use random methods? Computation by “deterministic quadrature” can become expensive and inaccurate.
M=0
h
X X X X X X O O O O O O O
Loop N times
Choose a point x in [a,b]
x ( b a ) u1 a y hu 2
a
Choose a point y in [0,h]
b
if [x,y] reside inside then M = M+1
2 ˆ I N(I, I ) ˆ
So, a confidence interval for I is roughly given by
ˆ I z/2 I ˆ
but since we don’t know I , we’ll have to be ˆ content with the further approximation:
Sample Mean Method
Example:
3
Ie answer is e3-1
19.08554)
write this as
3
I 3 e
0
x
1 3
dx 3E[e ]
where X~unif(0,3)
X
Sample Mean Method
Sample Mean Method
I (b - a) Eg(X)
where X~unif(a,b)
So, we will estimate I by estimating E[g(X)] with
ˆ E[g(X)]
g(X ) n
i i 1
1
n
where X1, X2, …, Xn is a random sample from the uniform(a,b) distribution.
Monte Carlo method WINS, when d >> 3
第九章 Monte Carlo积分
Hit-or-Miss Method Sample Mean Method Variance Reduction Technique Variance Reduction using Rejection Technique