Advanced Monte Carlo Methods:
General Principles of the Monte
Carlo Method
Prof. Dr. Michael Mascagni
E-mail: mascagni@math.ethz.ch, http://www.cs.fsu.edu/~mascagni
Seminar für Angewandte Mathematik
Eidgenössische Technische Hochschule, CH-8044 Zürich Switzerland
and
Department of Computer Science and School of Computational Science
Florida State University, Tallahassee, Florida, 32306-4120 USA
Numerical Integration: The
Canonical Monte Carlo Application
„
Numerical integration is a simple problem to
explain and thoroughly analyze
‰
‰
„
Deterministic methods
Monte Carlo (stochastic methods)
Integration is expectation: one can view all
Monte Carlo as integration in appropriate
setting
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 2 of 61
Numerical Integration (Cont.)
„
Methods for approximating
definite integrals
‰ Rectangle Rule
‰ Trapezoidal Rule
„
„
Divide the curve into N strips of
thickness h=(b-a)/N
Sum the area of each trip
‰
‰
Approximate to that of a
trapezium
Simpson’s Rule
„
Calculate piecewise quadratic
approximation instead
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 3 of 61
The Monte Carlo Method
„
General Principles
‰
„
Every Monte Carlo computation that leads to
quantitative results may be regarded as estimating
the value of a multiple integral in the appropriate
setting
Efficiency
‰
Definition
„
„
„
„
Suppose there are two Monte Carlo methods
Method 1: n1 units of computing time, σ12
Method 2: n2 units of computing time, σ22
n1σ 12
Methods comparison:
n2σ 22
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 4 of 61
Monte Carlo Integration
„
Consider a simple integral
1
θ = ∫ f ( x)dx
0
„
Definition of expectation of a function on random
variable η
1
E ( f (η )) = ∫ f ( x) p ( x)dx
0
„
1
If η is uniformly distributed, then E ( f (η )) = ∫ f ( x)dx = θ
0
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 5 of 61
Crude Monte Carlo
„
Crude Monte Carlo
‰
If ξ1, …, ξn are independent random numbers
„
‰
Uniformly distributed
then fi =f(ξi ) are random variates with expectation θ
1
f =
n
‰
‰
‰
n
∑
i =1
fi
is an unbiased estimator of θ
The variance is
1
1
2
E (( f − θ ) ) = ∫ ( f ( x) − θ ) 2 dx = σ 2 / n
n0
The standard error is
„
σ/n1/2
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 6 of 61
Hit-or-Miss Monte Carlo
„
„
Suppose 0 ≤ f(x) ≤ 1 when 0 ≤ x ≤ 1
Main idea
‰
‰
‰
draw a curve y=f(x) in the unit square 0 ≤ x, y≤ 1
is the proportion of the are of the square beneath
the curve
1
or we can write f ( x) = ∫ g ( x, y )dy
0
g ( x, y ) = 0 if f(x) < y
= 1 if f(x) ≥ y
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 7 of 61
Analysis of Hit-or-Miss Monte Carlo
„
θ can be estimated as the a double integral
1 1
θ = ∫ ∫ g ( x, y )dxdy
0 0
„
The estimator of hit-or-miss Monte Carlo
n*
1 n
g = ∑ g (ξ 2i −1 , ξ 2i ) =
n i =1
n
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 8 of 61
Hit-or-Miss Monte Carlo (Cont.)
„
Hit-or-Miss Monte Carlo
‰
‰
„
We take n points at random in the unit square, and count the
proportion of them which lie below the curve y=f(x)
The points are either in or out of the area below the curve
„ The probability that a point lies under the curve is θ
The Hit-or-Miss Monte Carlo is a Bernoulli trial
‰
the estimator of Hit-or-Miss Monte Carlo is binomial
distributed
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 9 of 61
Binomial Distribution Revisited
„
Binomial Distribution
‰
‰
Discrete probability distribution Pp(n|N) of obtaining exactly n
successes out of N Bernoulli trials
Each Bernoulli trial is true with probability p and false with
probability q=1-p
„
Mean: Np; Variance: N(1-p)p
=
=
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 10 of 61
Comparison of Hit-or-Miss Monte
Carlo and Crude Monte Carlo
„
Standard error of Hit-or-Miss Monte Carlo
θ (1 − θ )
n
„
Standard error of Crude Monte Carlo
1
∫ ( f −θ )
2
dx
0
n
„
Hit-or-Miss Monte Carlo is always worse than Crude Monte Carlo
‰
Why?
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 11 of 61
Comparison of Hit-or-Miss Monte
Carlo and Crude Monte Carlo (Cont.)
„
„
Note: all of our integrands are in L2 as they have finite
variance
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 12 of 61
Why Hit-or-Miss Monte Carlo is
worse?
„
Fact
‰
‰
The hit-or-miss to crude sampling is equivalent to
replacing g(x, ξ) by its expectation f(x)
The y variable in g(x,y) is a random variable
„
„
„
Leads to uncertainty
Places extra uncertainty in the final results
Can be replace by exact value
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 13 of 61
General Principle of Monte Carlo
„
„
If, at any point of a Monte Carlo calculation,
we can replace an estimate by an exact value,
we shall replace an estimate by an exact value,
we shall reduce the sampling error in the final
result
Mark Kac: “You use Monte Carlo until you
understand the problem”
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 14 of 61
Curse of Dimensionality
„
Curse of Dimensionality
‰
„
If one needs n points to achieve certain accuracy for an 1-D
integral, to achieve the same accuracy one needs ns points for sdimensional integral
Complexity of high-dimensional integration (tensor
product rules
‰
‰
‰
‰
Rectangle Rule
Trapezoidal Rule
Simpson’s Rule
Convergence Rate
„
O(n-α/s)
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 15 of 61
High-Dimensional Monte Carlo
Integration
„
Consider the following integral
ρ ρ
θ = ∫ ...∫ f ( x )dx
1
„
1
Definition of expectation of a function on random variable η
that is uniformly distributed
1
1
0
0
ρ
ρ ρ
E ( f (η )) = ∫ ...∫ f ( x )dx = θ
„
Standard error
‰
‰
0
0
Independent of Dimension
σ/n1/2
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 16 of 61
Confidence Interval
„
Estimator for the standard error
1 n
2
s =
(
f
−
f
)
∑
i
n − 1 i =1
2
„
Confidence Intervals
‰
66%: [ f − s, f + s ]
‰
95%: [ f − 2s, f + 2s ]
‰
99%: [ f − 3s, f + 3s]
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 17 of 61
Variance Reduction Methods
„
Variance Reduction Techniques
‰
Employs an alternative estimator
„
„
„
„
Unbiased
More deterministic
Yields a smaller variance
Methods
‰
‰
‰
‰
‰
‰
Stratified Sampling
Importance Sampling
Control Variates
Antithetic Variates
Regression Methods
Orthonormal Functions
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 18 of 61
Stratified Sampling
„
Idea
‰
‰
„
Analysis of Stratified Sampling
‰
‰
„
Break the range of integration into several pieces
Apply crude Monte Carlo sampling to each piece separately
Estimator
Variance
Conclusion
‰
If the stratification is well carried out, the variance of
stratified sampling will be smaller than crude Monte Carlo
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 19 of 61
Stratified Sampling
„
„
First we divide the integration interval into k
subintervals:
The estimator is then:
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 20 of 61
Stratified Sampling (Cont.)
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 21 of 61
Stratified Sampling (Cont.)
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 22 of 61
Stratified Sampling (Cont.)
„
This variance may be less than that from
crude Monte Carlo with good stratification
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 23 of 61
Stratified Sampling (Cont.)
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 24 of 61
„
Idea
‰
„
Importance Sampling
Concentrate the distribution of the sample points in the parts of the
interval that are of most importance instead of spreading them out
evenly
Importance Sampling
1
θ =∫
0
‰
1
1
f ( x)
f ( x)
f ( x)dx = ∫
g ( x)dx = ∫
dG ( x)
g ( x)
g ( x)
0
0
where g and G satisfy
x
G ( x) =
∫ g ( y ) dy
1
G (1) =
0
‰
∫ g ( y ) dy = 1
0
G(x) is a distribution function
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 25 of 61
Importance Sampling
„
Variance
1
σ 2 f / g = ∫ ( f ( x ) / g ( x ) − θ ) 2 dG ( x )
0
„
How to select a good sampling function?
‰
‰
How about g=cf?
g must be simple enough for us to know its integral
theoretically.
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 26 of 61
Control Variates
„
Control Variates
1
1
0
0
θ = ∫ φ ( x)dx + ∫ [ f ( x) − φ ( x)]dx
‰
‰
‰
φ(x) is the control variate with known integral
Estimator
1
„ t-t’+θ’ is the unbiased estimator
t=
„ θ’ is the first (known) integral
n
Variance
„ var(t-t’+θ’)=var(t)+var(t’)-2cov(t,t’)
„
∑
i
1
f (ξ i ), t ' = ∑ φ (ξ i )
n i
if 2cov(t,t’)<var(t’), then the variance is smaller than crude Monte
Carlo
‰
t and t’ should have strong positive correlation
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 27 of 61
Antithetic Variates
„
Main idea
‰
‰
„
Estimator
‰
‰
„
Select a second estimate that has a strong negative
correlation with the original estimator
t’’ has the same expectation of t
[t+t’’]/2 is an unbiased estimator of θ
var([t+t’’]/2)=var(t)/4+var(t’’)/4+cov(t,t’’)/2
Commonly used antithetic variate
‰
‰
(t+t’’)/2=f(ξ)/2+ f(1-ξ)/2
If f is a monotone function, f(ξ) and f(1-ξ) are negatively
correlated
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 28 of 61
Antithetic Variates (Cont.)
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 29 of 61
Antithetic Variates (Cont.)
„
„
„
General definition of “Antithetic Variates”: Any
method that introduces a set of estimators that
mutually compensate for the others’ variance
Theorem states any unbiased combination of
variables can be rearranged into a linear combination
that achieves the minimal possible variance
Such combinations invariably are antithetic
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 30 of 61
Antithetic Variates (Cont.)
„
„
„
General definition of “Antithetic Variates”: Any
method that introduces a set of estimators that
mutually compensate for the others’ variance
Theorem states any unbiased combination of
variables can be rearranged into a linear combination
that achieves the minimal possible variance
Some examples
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 31 of 61
Antithetic Variates: Examples
„
Consider examples from stratification (I)
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 32 of 61
Antithetic Variates: Examples
„
Consider examples from stratification (II)
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 33 of 61
Antithetic Variates: Examples (Cont.)
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 34 of 61
Antithetic Variates: Examples (Cont.)
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 35 of 61
Antithetic Variates: Examples (Cont.)
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 36 of 61
Antithetic Variates: Examples (Cont.)
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 37 of 61
Orthonormal Functions
„
General method of Monte Carlo integration
based on orthonormal functions (Ermakov &
Zolotukhin)
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 38 of 61
Orthonormal Functions (Cont.)
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 39 of 61
Orthonormal Functions (Cont.)
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 40 of 61
Orthonormal Functions (Cont.)
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 41 of 61
Orthonormal Functions (Cont.)
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 42 of 61
Orthonormal Functions (Cont.)
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 43 of 61
Orthonormal Functions (Cont.)
„
An simple, 1-D example, n=0
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 44 of 61
Orthonormal Functions (Cont.)
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 45 of 61
Orthonormal Functions (Cont.)
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 46 of 61
Orthonormal Functions (Cont.)
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 47 of 61
Orthonormal Functions (Cont.)
„
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 48 of 61
Orthonormal Functions (Cont.)
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 49 of 61
Orthonormal Functions (Cont.)
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 50 of 61
Orthonormal Functions (Cont.)
„
The formulae (quart) and (quint) exactly treat
all quartic and quintic polynomials exactly
„
„
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 51 of 61
Regression
„
Variation in Raw Experimental Data
‰
Two parts
„
The first part consists of an entirely random variation
‰
„
The second part arises because the observations are influenced by
certain concomitant conditions of the experiment
‰
‰
‰
We may do little about it
We may record these condition
Determine how they influence the raw observations
Regression
„
„
„
Calculate (estimate) the second part
Subtract it out from the reckoning
Leave only those variations in the observations which are not due to
the concomitant conditions
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 52 of 61
Regression Model
„
Model
‰
The random observations ηi (i=1, 2, … n) are associated with a set
of concomitant numbers xij (j=1,2, .., p)
„
„
„
‰
xij describe the experimental condition under which the
observations ηi was taken
ηi is the sum of a purely random component δi and a linear
combination ∑j β j xij of the concomitant numbers
βi are called regression coefficient
The minimum-variance unbiased linear estimator of βi is
„ b=(X’V-1X)-1X’V-1 η
„
„
X: nxp matrix xij
V: nxn variance covariance matrix of δi
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 53 of 61
Regression Methods
„
Regression Method
‰
‰
Suppose we have several unknown estimates θ1, θ2, …, θp
A set of estimators t1, t2, …, tn
„ Eti=xi1 θ1+xi2 θ2 + … + xip θp (i=1, 2, …, n)
„ Et=Xθ
„
‰
xij are a set of known constants
Minimum-variance unbiased linear estimator of θ={θ1, θ2, …,
θp }
„
„
t*=(X’V-1X)-1X’V-1 t
V is unknown
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 54 of 61
Regression Methods (Cont.)
„
Consider an alternative estimator which uses
an arbitrary V0
‰
‰
‰
„
t0*=(X’V0-1X)-1X’V0-1 t
Et0*= θ
However, t0* is not a minimum-variance estimator
If V0 is close to V, then t0* will have a very
nearly minimum variance
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 55 of 61
Regression Method in Practice
„
Regression Method in Practice
‰
‰
‰
Calculate N independent sets of estimates t1, t2, …, tn
Each result is denoted by t1k, t2k, …, tnk (k=1,2,…N)
vij can be estimated by
N
vij 0 = ∑ (tik − ti )(t jk − t j ) /( N − 1)
k =1
N
ti = ∑ tik / N
k =1
‰
Then, the estimator of θ is t
„
t0*=(X’V0-1X)-1X’V0-1
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 56 of 61
Example of Regression
Methods
t1 =
1
2
f (ξ ) + 12 f (1 − ξ )
t2 =
1
4
f ( 12 ξ ) + 14 f ( 12 − 12 ξ ) + 14 f ( 12 + 12 ξ ) + 14 f (1 − 12 ξ )
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 57 of 61
Buffon Needle Problem Revisited
„
If we through a needle of length L onto a
square grid with Δx =Δy = 1 then Mantel gives
a quadratic estimator for the Buffon Needle
‰
‰
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 58 of 61
Comparison of the Variance
Reduction Methods
„
„
Definitions
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 59 of 61
Comparison of the Variance
Reduction Methods (Cont.)
Method Used
Var. Ratio Labor Ratio Efficiency
Hit-or-miss
0.34
1/1
0.34
Stratified, 4 strata
13
1/1.3
10
Importance, g(x)=x
29.9
1/3
10
Control Var., ø(x)=x
60.4
1/2
30
Antithetic Variate
62
1/2
31
985
1/2
490
Antithetic (II)*
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 60 of 61
Comparison of the Variance
Reduction Methods (Cont.)
Method
Var. Ratio Labor Ratio Efficiency
Antithetic (II)* 2-way
15600 1/4
3900
Antithetic (II)* 4-way
249000 1/8
31000
Antithetic (II)* 8-way 3980000 1/16
250000
Antithetic (II)* special 2950000 1/6
460000
Orthonormal
240000
720000 1/3
Prof. Dr. Michael Mascagni: Advanced Monte Carlo Methods
General Principles of Monte Carlo
Slide 61 of 61