Approximation of integrals via Monte Carlo methods, with an application to calculating radar detection probabilities

Abstract

The performance analysis of a radar detection scheme requires estimation of probabilities of false alarm and detection, under various clutter scenarios. These probabilities, which often appear as definite integrals, are frequently analytically difficult to evaluate. Hence, numerical approximation schemes are employed. Monte Carlo estimators use statistical simulation to evaluate such integrals. As with any approximation scheme, there are limitations and drawbacks in its application. One of the major difficulties with Monte Carlo estimators is that very large sample sizes may be required, in order to achieve a reasonable estimate. This is especially true in the context of estimating probabilities of rare events, such as radar false alarms. The purpose of this report is to examine the Monte Carlo estimation of integrals in general. After formulating the scheme, applications to the evaluation of two special functions are considered. The success of an estimator will be decided on its performance in terms of providing a reasonable estimate for the smallest sample size possible. The major application in this report will be to obtain estimates for a detection probability in a binary integration context. Under the assumption of a Swerling target model, an expression for the binary integrated probability of detection is obtained in Shnidman's 1998 paper entitled Binary integration for Swerling target fluctuations (IEEE Transactions on Aerospace and Electronic Systems, Volume 34, pp. 1043-1053). We apply Monte Carlo simulations, together with some functional approximations, to estimate this probability for Swerling 1 and 3 target models. The approximation of definite integrals using Monte Carlo simulations is the focus of the work presented here. The methodology methodology of estimation by sampling is introduced, and is applied to the approximation of two special functions of mathematics: the Gamma and Beta functions. A significant application, in the context of radar detection theory, is based upon the work of [Shnidman 1998]. The latter considers problems associated with the optimal choice of binary integration parameters. We apply the techniques of Monte Carlo simulation to estimate binary integration detection probabilities. CDR MPG

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Australian Government
Department of Defence
Defence Science and
Technology Organisation
Approximation of Integrals via Monte Carlo
Methods, with an Application to Calculating
Radar Detection Probabilities
Graham V. Weinberg and Ross Kyprianou
Electronic Warfare and Radar Division
Systems Sciences Laboratory
DSTO-TR-1692
ABSTRACT
The approximation of definite integrals using Monte Carlo simulations is the
focus of the work presented here. The general methodology of estimation by
sampling is introduced, and is applied to the approximation of two special
functions of mathematics: the Gamma and Beta functions. A significant ap-
plication, in the context of radar detection theory, is based upon the work of
[Shnidman 1998]. The latter considers problems associated with the optimal
choice of binary integration parameters. We apply the techniques of Monte
Carlo simulation to estimate binary integration detection probabilities.
APPROVED FOR PUBLIC RELEASE

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DSTO-TR-1692
Published by
DSTO Systems Sciences Laboratory
PO Box 1500
Edinburgh, South Australia, Australia 5111
Telephone:
Facsimile:
(08) 8259 5555
(08) 8259 6567
@ Commonwealth of Australia 2005
AR No. AR-013-341
March, 2005
APPROVED FOR PUBLIC RELEASE

Page 3

DSTO-TR 1692
Approximation of Integrals via Monte Carlo Methods, with
an Application to Calculating Radar Detection Probabilities
EXECUTIVE SUMMARY
The performance analysis of a radar detection scheme requires estimation of probabilities
of false alarm and detection, under various clutter scenarios. These probabilities, which
often appear as definite integrals, are frequently analytically difficult to evaluate. Hence,
numerical approximation schemes are employed. Monte Carlo estimators use statistical
simulation to evaluate such integrals. As with any approximation scheme, there are limi-
tations and drawbacks in its application. One of the major difficulties with Monte Carlo
estimators is that very large sample sizes may be required, in order to achieve a reasonable
estimate. This is especially true in the context of estimating probabilities of rare events,
such as radar false alarms.
The purpose of this report is to examine the Monte Carlo estimation of integrals in general.
After formulating the scheme, applications to the evaluation of two special functions are
considered. The success of an estimator will be decided on its performance in terms of
providing a reasonable estimate for the smallest sample size possible.
The major application in this report will be to obtain estimates for a detection probability
in a binary integration context. Under the assumption of a Swerling target model, an
expression for the binary integrated probability of detection is obtained in Shnidman's
1998 paper entitled Binary integration for Swerling target fluctuations (IEEE Transactions
on Aerospace and Electronic Systems, Volume 34, pp. 1043-1053). We apply Monte Carlo
simulations, together with some functional approximations, to estimate this probability
for Swerling 1 and 3 target models.
iii

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DSTO-TR-1692
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DSTO-TR 1692
Authors
Graham V. Weinberg
Electronic Warfare and Radar Division
Graham V. Weinberg was educated at The University of Mel-
bourne, graduating in 1996 with a Bachelor of Science degree
with Honours in Mathematics and Statistics. Following this he
was conferred in 2001 with a Doctor of Philosophy degree in
Applied Probability, also at The University of Melbourne. His
doctoral thesis examined probability approximations for ran-
dom variables and point processes, using a technique known
as the Stein-Chen Method. The latter enables a probabilis-
tic assessment of the errors in distributional approximations of
stochastic processes.
His post-doctoral work began with a 6 month contractual po-
sition with a finance group known as Webmind, investigating
artificial intelligence applied to stock market prediction. This
was based in the Department of Psychology at The University
of Melbourne. Following this was employment as a Research
Fellow in the Teletraffic Research Centre, at the University of
Adelaide. The latter involved working as a telecommunications
consultant to business and industry. Since early 2002 he has
been employed as a Research Scientist with DSTO, examining
radar performance analysis techniques for the P3 Orion mar-
itime patrol aircraft. From mid 2004 he has changed directions
in terms of research, now working on problems related to radar
resource management and waveform analysis.