第九章 monte carlo 积分. monte carlo...
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第九章 第九章 Monte CarloMonte Carlo 积分积分Monte Carlo 法的重要应用领域之一:计算积分和多重积分适用于求解:
1. 被积函数、积分边界复杂,难以用解析方法或一般的数值方法求解;
2. 被积函数的具体形式未知,只知道由模拟返回的函数值。
本章内容:
用 Monte Carlo 法求定积分的几种方法: 均匀投点法、期望值估计法、重要抽样法、半解析法、…
第九章 第九章 Monte CarloMonte Carlo 积分积分Goal: Evaluate an integral:
b
a
I dx g(x)
Why use random methods?
Computation by “deterministic quadrature” can become expensive and inaccurate.
grid points add up quickly in high dimensions
bad choices of grid may misrepresent g(x)
第九章 第九章 Monte CarloMonte Carlo 积分积分Monte Carlo method can be used to compute
integral of any dimension d (d-fold integrals)
Error comparison of d-fold integrals
Simpson’s rule,… dNE /1
21 NE purely statistical,
not rely on the dimension !
Monte Carlo method WINS, when d >> 3
Monte Carlo method
approximating the integral of a function f using quadratic polynomials
)]()(4)([3
1)()( 210
0
00
xfxfxfhdxxfdxxfhx
x
x
x
hxxxx 1201
第九章 第九章 Monte CarloMonte Carlo 积分积分
Hit-or-Miss Method
Sample Mean Method
Variance Reduction Technique
Variance Reduction using Rejection Technique
Importance Sampling Method
Hit-or-Miss Method•Evaluation of a definite integral
dxxIb
a )(
xxh any for )(
a b
hX
X
X
XX
X
O
O
OO
O
OO
•Probability that a random point reside inside the area
N
M
hab
Ip
)(
N
MhabI )(
N : Total number of points
M : points that reside inside the region
Hit-or-Miss MethodSample uniformly from the rectangular region [a,b]x[0,h]
a)-h(bI
: p
The probability that we are below the curve is
So, if we can estimate p, we can estimate I:
a)-h(bpI ˆˆ
where is our estimate of pp
Hit-or-Miss MethodWe can easily estimate p:
throw N “uniform darts” at the rectangle
NM
pˆ let
let M be the number of times you end up under the curve y=g(x)
Hit-or-Miss Method
a b
hX
X
X
XX
X
O
O
OO
O
OO
Start
Set N : large integer
M = 0
Choose a point x in [a,b]
Choose a point y in [0,h]
if [x,y] reside inside then M = M+1
I = (b-a) h (M/N)
End
LoopN times
auabx 1)(
2huy
Hit-or-Miss MethodError 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)
)1()()( 2 PNpMNpME
IpabhMEN
abh
abhN
MEabhpEIE
)()()(
)()(ˆ)ˆ(
N
ppabhM
N
abh
abhN
MabhpI
)1()()(
)(
)()(ˆ)ˆ(
222
2
22
222
N
Mpp ˆ )1(
)()ˆ(
N
MM
N
abhI
21)1(
)()ˆ(
NN
ppabhI
第九章 第九章 Monte CarloMonte Carlo 积分积分
Hit-or-Miss Method
Sample Mean Method
Variance Reduction Technique
Variance Reduction using Rejection Technique
Importance Sampling Method
Sample Mean MethodStart
Set N : large integer
s1 = 0, s2 = 0
xn = (b-a) un + a
yn = (xn)
s1 = s1 + yn , s2 = s2 + yn2
Estimate mean ’=s1/NEstimate variance V’ = s2/N – ’2
End
LoopN times
N
VabI SM
error
'6745.0)(
Sample Mean Method
dx g(x) Ib
a
Write this as:
dxa-b
1 g(x)a)-(b dx g(x) I
b
a
b
a
g(X)E a)-(b where X~unif(a,b)
Sample Mean Method
g(X)E a)-(b I where X~unif(a,b)
So, we will estimate I by estimating E[g(X)] with
n
1i
) ˆig(X
n1
[g(X)]E
where X1, X2, …, Xn is a random sample from the uniform(a,b) distribution.
Sample Mean Method
dx e I3
0
x
Example:
(we know that the answer is e3-1 19.08554)
write this as
]3E[e dx 31
e3 I X3
0
x
where X~unif(0,3)
Sample Mean Method write this as
]3E[e dx 31
e3 I X3
0
x
where X~unif(0,3)
estimate this with
n
1i
xX ien1
3][eE3 I ˆˆ
where X1, X2, …, Xn are n independent unif(0,3)’s.
Sample Mean MethodSimulation Results: true = 19.08554,
n=100,000
1 19.10724
I
2 19.08260
3 18.97227
4 19.06814
5 19.13261
Simulation
Sample Mean MethodDon’t ever give an estimate without a confidence interval!
This estimator is “unbiased”:
n
1ii )g(X
n1
a)-(bE ]IE[
n
1ii )]E[g(X
n1
a)-(b
b
a
dx a-b
1 g(x)
n1
a)-(b b
a
dx g(x) I
Sample Mean Method]IVar[ : σ2
Iˆ
ˆ
n
1ii )g(X
n1
a)-(bVar
n
1ii2
2
)g(XVarn
a)-(b
(indep) )]Var[g(Xn
a)-(b
n
1ii2
2
(indent) )]Var[g(X na)-(b
1
2
b
a
22
dxa-b
1 E[g(X)]g(x)
na)-(b
b
a
22
dx a-b
1a-b
Ig(x)
na)-(b
Sample Mean Method
n
1ii )g(X
n1
a)-(b I
X1, X2, …, Xn iid -> g(X1), g(X2), …, g(Xn) iid
Let Yi=g(Xi) for i=1,2,…,n
Then
Y a)-(b I ˆ
and we can once again invoke the CLT.
Sample Mean Method
)N(I, I 2I
ˆ
For n “large enough” (n>30),
So, a confidence interval for I is roughly given by
I/2 z I ˆˆ
but since we don’t know , we’ll have to be content with the further approximation:
I
I/2 s z I ˆˆ
Sample Mean Method
b
a
2x1/2 dx e x: I
By the way…
No one ever said that you have to use the uniform distribution
Example:
0
b][a,2x1/2 dx (x)I 2e x
21
(X)I XE21
b][a,1/2
where X~exp(rate=2).
Sample Mean MethodComparison of Hit-and-Miss and Sample Mean Monte Carlo
Let be the hit-and-miss estimator of IHMI
)IVar( )IVar( SMHMˆˆ
Then
Let be the sample mean estimator of ISMI
Sample Mean MethodComparison of Hit-and-Miss and Sample Mean Monte Carlo
Sample mean Monte Carlo is generally preferred over Hit-and-Miss Monte Carlo because:
the estimator from SMMC has lower variance
SMMC does not require a non-negative integrand (or adjustments)
H&M MC requires that you be able to put g(x) in a “box”, so you need to figure out the max value of g(x) over [a,b] and you need to be integrating over a finite integral.
第九章 第九章 Monte CarloMonte Carlo 积分积分
Hit-or-Miss Method
Sample Mean Method
Variance Reduction Technique
Variance Reduction using Rejection Technique
Importance Sampling Method
Variance Reduction Technique
IntroductionMonte Carlo Method and Sampling Distribution
Monte Carlo Method : Take values from random sample From central limit theorem,
3 rule
Most probable error
Important characteristics
N/ 22
9973.0)33( XP
NError
6745.0
NError /1 Error
Variance Reduction Technique
IntroductionReducing error
*100 samples reduces the error order of 10 Reducing variance Variance Reduction Technique
The value of variance is closely related to how samples are taken
Unbiased samplingBiased sampling
More points are taken in important parts of the population
Variance Reduction TechniqueMotivation
If we are using sample-mean Monte Carlo MethodVariance depends very much on the behavior of (x)(x) varies little variance is small(x) = const variance=0
Evaluation of a integral
Near minimum points contribute less to the summationNear maximum points contribute more to the summation
More points are sampled near the peak ”importance sampling strategy”
N
nnY
xN
ababI
1
)()('
第九章 第九章 Monte CarloMonte Carlo 积分积分
1.2 Hit-or-Miss Method
1.3 Sample Mean Method
2.1 Variance Reduction Technique
2.3 Variance Reduction using Rejection Technique
2.4 Importance Sampling Method
Variance Reduction Technique
Variance Reduction for Hit-or-Miss method
•In the domain [a,b] choose a comparison function
dxxwA
dxxwxW
xxw
b
a
x
)(
)()(
)()(
)(xwAu )(1 AuWx
a b
w(x)
X
X
XX
X
O
O
OO
O
OO
(x)
Points are generated on the area under w(x) function
Random variable that follows distribution w(x)
Variance Reduction Technique
Points lying above (x) is rejected
N
NAI
'
nnn uxwy )(
)( if 0
)( if 1
nn
nnn xy
xyq
q 1 0P(q) r 1-r
AIr /
a b
w(x)
X
X
XX
X
O
O
OO
O
OO
(x)
Variance Reduction Technique
Error Analysis)1()( ,)( rrQVrQE
)(QAEArI
N
IAII RJ
error
)(67.0
Hit or Miss method
Error reduction
habA )(
0Error then IA
Variance Reduction Technique
Start
Set N : large integer
N’ = 0
Generate u1, x= W-1(Au1)
Generate u2, y=u2 w(x)
If y<= f(x) accept value N’ = N’+1Else : reject value
I = (b-a) h (N’/N)
End
LoopN times
N
IAII RJ
error
)'('67.0
a b
w(x)
X
X
XX
X
O
O
OO
O
OO
(x)
第九章 第九章 Monte CarloMonte Carlo 积分积分
1.2 Hit-or-Miss Method
1.3 Sample Mean Method
2.1 Variance Reduction Technique
2.3 Variance Reduction using Rejection Technique
2.4 Importance Sampling Method
Importance Sampling Method
Basic ideaPut more points near maximumPut less points near minimum
F(x) : transformation function (or weight function_
x
dxxfxF )()(
)(
)(1 yFx
xFy
Importance Sampling Method
)(/)(/ xfdydxxfdxdy
b
a
b
adxxf
xf
xdy
xf
xI )(
)(
)(
)(
)(
b
af dxxfx )()(
)(
)()(
xf
xx
if we choose f(x) = c(x) ,then variance will be smallThe magnitude of error depends on the choice of f(x)
f
b
adxxfxI )()(
Importance Sampling Method
Estimate of error
N
nnf x
NI
1
)(1
N
VI f
error
)(67.0
22 )()( fffV
N
II fIS
error
22
67.0
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