The Principle of the Information-Divergence Minimum in the Problem of Spectral Analysis of the Random Time Series Under the Condition of Small Observation Samples

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UDC 537.86+519.2 We consider the issue of small observation samples in the problem of spectral analysis of the random time series. It is proposed to solve the considered problem using the information-theoretic approach and a new algorithm based on the principle of minimum divergence of the cognominal spectral estimates yielded by the results of several independent observations in the Kullback– Leibler information metric. An example of a practical realization of the algorithm is considered and its asymptotic properties are studied. Spectral analysis of the random time series belongs to the most rapidly developing directions in the field of statistical data processing. The problem of small samples has traditionally been a topical issue in this field [1]. This problem has been studied in many works, which stimulated progress in the field of spectral analysis. The nonlinear recurrent methods [2], which are characterized by high resolution under the finite-sample conditions, are in particular referred to the most important recent results. With the advent of them, the problem of choosing the best mathematical tool, i.e., the mathematical model and the method of spectral analysis, with respect to each particular problem has become the most topical in modern studies. In this case, we actually mean choosing the best estimate of the power spectral density within the limits of the finite set {Gj(f )} of the alternatives obtained from the samples xj with the finite sizes nj < ∞ [3]. The above-mentioned problem becomes more urgent if this size decreases. The present work is devoted to the study of this problem in the proposed formulation. In this work, the problem of ill-posed analysis is solved on the basis of the Kullback–Leibler information metric [4] and a new mathematical tool (the principle of minimum information divergence [15] and the whitening-filter method [6]), i.e., it is required to perform mathematical description of the observed process with maximum (in the information-theoretical sense) accuracy using limited ap rioridata. 2. FORMULATION OF THE PROBLEM