Extraction and Detection Problems and Reproducing Kernel Hilbert Spaces

Source abstract

Problems involving the extraction, detection and prediction of signals in the presence of noise are among the central problems of statistical communication theory. Over the past few years I have sought to develop an approach to such problems which (i) would simultaneously apply to time series which are stationary or non-stationary, discrete parameter or continuous parameter, univariate or multivariate, and (ii) would distinguish between the statistical and analytical aspects of these problems, and in particular would clarify the role played by various widely employed analytical techniques (such as the Wiener-Hopf equation and eigenfunction expansions). In developing this approach, two basic concepts are used : the notion of the probability density functional of a time series and the notion of a reproducing kernel Hilbert space. The aim of this paper is to sketch some of the main results which may be obtained by means of this approach. In sections 1 and 2, it is shown how one may define and obtain a formula for the probability density functional of a normal time series. This formula is used to study the structure of optimum estimators (section 3) and detectors (section 4) by expressing them in a coordinate free way in terms of inner products in a reproducing kernel Hilbert space. Various ways of evaluating such inner products are discussed in sections 5 and 6. In section 7, it is shown how reproducing kernel Hilbert spaces provide a solution to the problems of minimum mean square error linear and non-linear prediction.