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Electromagnetic waves propagation and coupling:
The statistical approach to information retrieval
W. Tabbara 1 , Ph. De Doncker 2 , M. Hélier 1 , D. Lecointe 1
1 Supélec/DRE, Plateau de Moulon, 91192 Gif-sur-Yvette, France
2 Université Libre de Bruxelles, CP 165/51, Av. Roosevelt 50, 1050 Brussels, Belgium
tabbara@lss.supelec.fr
Key words: Electromagnetic compatibility, statistics, kriging.
Abstract
The aim of this paper is to provide examples of applications of probabilistic and statistical methods (ProStat) to
problems dealing, on one hand with the coupling of an electromagnetic wave to cavities, cables and circuits, and others
related to indoor propagation. The kriging approach, developed in the field of geostatistics, will be emphasized. It
allows the prediction of the values of an observable, within some given set, from a reduced number of values taken in
the same set, providing also an estimation of the accuracy of the predicted values.
The statistical approach
The coupling of an electromagnetic wave to a structure depends on a large number of factors such as the dimensions of
the structure, the electromagnetic parameters of the media, the polarization of the incident wave... The parametric
analysis of an observable (field, current, reflection coefficient...) resulting from the interaction is conducted, by means
of numerical modeling or experimentation. If an exact solution is sought, a numerical code can be executed repeatedly
leading to a high cost in terms of computing time. Similar conclusions as to the cost of repeated experimentations can
be drawn when measurements provide the desired results. Thus, it is important, in particular when complex systems are
investigated, to consider the possible means of reducing the cost of the analysis, and to estimate the accuracy of the
results.
The cost-reduction problem can be looked at in a somewhat unusual manner. Each combination of values of the factors,
define one of the many possible configurations of the system. The latter are then considered as the outcome of a random
experiment, and the following analysis is conducted:
1- Problems of practical interest involve a large number of factors each having more or less influence on the
observable. In order to reduce the computational cost we only need to keep the most important factors.
Moreover, the factors may not be independent parameters, thus there is a need to point out the correlation
between them. To achieve these two goals, one may resort to the analysis of variance (ANOVA) and to
regression techniques [1, 2]. The ANOVA, that allows one to investigate the influence of one factor at a time, is
a fast and simple method but it does not provide information on the correlations between factors. Regression is
a more elaborate approach that may be used in place of ANOVA and is able to give correlation information. It
may also be used to interpolate the values of the observable in some data base. The interpolating function is
then a linear combination of functions of the factors, called regressors (these are usually polynomials in many
variables) and the coefficients of the linear combination are obtained by applying a mean square minimization
to some cost function.
2- A configuration of the system is defined by considering a combination of values of the factors. The latter and
the observable are considered as random variables. For some simple systems, it is possible to determine
numerically the probability law of the observable from those of the factors. In that case, one usually employs
the Monte-Carlo [1] technique where a large number of combinations of the values of the factors are chosen at
random and the value of the observable is computed, or measured, for each of these combinations. A histogram
of the observable is then drawn and fitted with a function known as the probability density function (pdf) of the
observable. The knowledge of the pdf permits the computation of the probability of occurrence of any value of
the observable, it also alows to compute the quantile which is a convenient observable in EMC problems.
3- Build a small sample of configurations of the system. Then, for each configuration in the sample, the value of
the observable is computed (measured) by standard techniques and this set of values provides the data base.
4- When one considers a configuration that does not belong to the data base, it can be defined by a combination of
values of the factors. The value of the observable, for this configuration, is predicted by interpolating the values
in the data base by mean of Kriging [3, 4], a multi-factor parametric approach to system modeling which leads
to adequate prediction, together with an estimation of the accuracy of the predicted values of an observable.
With this technique, we study the evolution of an observable with respect to all the factors simultaneously, and
not one at a time as in a classical parametric analysis.