A New Method of Variance Reduction in Monte Carlo Integration

Fatin Sezgin

Present methods• Common random numbers• Antithetic variables• Latin hypercube sampling• Control variables• Importance sampling• Conditional Monte Carlo• Indirect estimation• Stratification

EXTENDED MONTE CARLO INTEGRATION

EMCI

Two ways of Monte-Carlo Integration

• The Crude Monte Carlo inserts the random number into the function and calculates the average of values obtained.

• Hit-or-Miss Monte Carlo casts n points into a space and finds the ratio of points falling within the integration regions.

Hit-or-Miss Monte Carlo

EXAMPLE The integration region for the 2X

Four curves

.)1(1)1(121)(11

)1()1(2)(1

2

2

2

2

xxfUpxxfUp

xxfDownxxfDown

−−=−−=−=−=

−=−===

4

ˆˆˆˆˆ 4321 IIIII +++

=

nxxx k =+++ ...21

1...21 =+++ kppp

kxk

xx

kk ppp

xxxnxxxP ...

!,...,!!!),...,,( 21

2121

21 =

jj npxE =)(

)1()( jjj pnpxVar −=

jiji pnpxxCov −=),(

∑=

=k

iiiXaY

1

)()(1

i

k

iiY XEaYE ∑

=

== µ

∑∑<=

+==ji

ijji

k

iiY aaaYVar

iσσσ 2)(

1

222

Comparison with single Hit-or-Miss curve

06.80276.02222.0

)16(4412.0)9(2

)4()()ˆ(

2

2

2

1 ===n

n

nTVarIVar

Sin(x) e-x and x3X

The coefficients for the sum of x variables in estimation of the total of sub-regions.

The variances of integral estimations to six different functions by using a single integral area and the

average of four sub-sections

Beta Function

The error of Sin(x) integration for sample sizes form10 thousand to 10 million

-0,008

-0,006

-0,004

-0,002

0,000

0,002

0,004

0,006

0,008

0,010

0,012

10000 100000 1000000 10000000

Down1

Down2

Up1

Up2

Mean

The factors affecting the magnitude of the variance reduction

• The amount of unused area will increase the variance of EMCI estimator.

• Multiple usage of sub-regions will decrease the efficiency of EMCI.

• If all sub-regions are included at least once in the summation formula, the least usage frequency can be subtracted from all cells.

Square root in unit Y axis

0,6500

0,6550

0,6600

0,6650

0,6700

0,6750

0,6800

0,6850

0,6900

0 200 400 600 800 1000 1200

d1d2u1u2mean

Square root in 1.41 Y axis

0,6400

0,6450

0,6500

0,6550

0,6600

0,6650

0,6700

0,6750

0,6800

0,6850

0,6900

0 200 400 600 800 1000 1200

d1d2u1u2mean

Speed comparisons

We tested the running times of single and Mean Estimator methods by a simulation comprising of 100000 runs each using n=1000 points. The concerned programs coded in Fortran Power Station 4.0 compiler are run on an Intel Pentium 4, CPU 3.06 GHz processor with 2.00 GB of RAM under Microsoft Windows XP professional version 2002 platform.

Timings

Comparison of variances

{ }

{ } ∫ −=−=

+−−=−

1

0

2

22

222

.))(1)((1)(1

)(1

dxxfxfn

xfEnn

InIxfE

nnI

nI

ch σσ

Variances of Crude Monte Carlo and EMCI Method

6(10 )−

6(10 )−

A distinct advantage There are cases very common in physics,

chemistry, medicine, genetics or biology where there is no explicit function defining the region or the volume to be estimated. A distinct advantage of our method is its applicability to these problems. In this application, instead of four different forms of function expression, different rotations of the figure can be used in Monte Carlo simulation.

Examples:

In multi-dimensional Nuclear Magnetic Resonance (NMR) experiments quantitative information can be obtained by peak volume integration. In this case the Hit-or-Miss technique is the most efficient way.

Another example is the usage of Monte-Carlo integration to find virial coefficients of some volumes in Molecular Physics.

Some references

• D.Q. Naiman and C.E. Priebe, Computing Scan Statistic p Values Using Importance Sampling, With Applications to Genetics and Medical Image Analysis, J. Comput. Graph. Statist., Vol. 10 (2), (2001) pp. 296–328.

• C.E. Priebe, D.Q. Naiman, and L.M. Cope, Importance sampling for spatial scan analysis: computing scan statistic p-values for marked point processes, Comput. Statist. Data Anal. Vol. 35, (2001) pp. 475- 485.

• R. Romano, D.B. Paris, F. Acernese, F. Barone, and A. Motta, Fractional volume integration in two-dimensional NMR spectra: CAKE, a Monte Carlo approach. Journal of Magnetic Resonance Vol. 192, (2008) pp. 294–301.

• Y. Takano and K.N. Liou, (1995) Radiative Transfer in cirrus clouds Part III. Light scattering by irregular crystals J. Atmospheric Sci., Vol. 52   No. 7 pp. 818-837.

• A.Y. Vlasov, X.M. You, and A.J. Masters, Monte-Carlo integration for virial coefficients re-visited: hard convex bodies, spheres with a square-well potential and mixtures of hard spheres, Molecular Phys., Vol. 100, No. 20, (2002) pp. 3313-3324.

SPM Analysis for PET Scan Brain Image Gaussian Random Field Analysis

In multi-dimensional Nuclear Magnetic Resonance (NMR) experiments quantitative information can be obtained by peak volume integration. In this case the Hit-or-Miss technique is the most efficient way.

Rotating and flipping an image

Thank you for your attention.