Developing the Theory of Stochastic Canonic Expansions and Its Applications

• Published in: Pattern recognition and image analysis Vol. 33; no. 4; pp. 862 - 887
• Main Author: Sinitsyn, I. N.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-12-2023; Springer Nature B.V
• Subjects: Algorithms; Banach spaces; Canonical forms; Computer Science; Image Processing and Computer Vision; Linear operators; Linear transformations; Mathematical analysis; Moscow; Operators (mathematics); Pattern Recognition; Regression models; Scientific Schools of the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences; Stochastic models; Stochastic systems; Synthesis; the Russian Federation; Theorems; Vector spaces; Wavelet analysis; stochastic system; biometric technology; metrological technology; stochastic function; stochastic element; optimal; wavelet canonic expansion; and conditionally optimal filtering; suboptimal; analysis and modeling; synergic organizational technoeconomic system; stochastic technology
• ISSN: 1054-6618; 1555-6212
• DOI: 10.1134/S1054661823040429

The creation of the theory of canonic expansions (CEs) is related with the names Loeve, Kolmogorov, Karhunen, and Pugachev and dates back to the 1940–1950s. The development of the theory of CEs and wavelet CEs is considered in application to the problems of the analysis, modeling, and synthesis of stochastic systems (SSs) and technologies. The direct and inverse Pugachev theorems about CEs are extended to the case of stochastic linear functionals within the framework of the correlational theory of stochastic functions (SFs). The CEs of linear and quasi-linear SFs are derived. Particular attention is paid to the problems of the equivalent regression linearization of strongly nonlinear transformations by CEs. The nonlinear regression algorithms on the basis of CEs are proposed. The theory of wavelet CEs within the specified domain of the change of the argument on the basis of Haar wavelets is developed. For stochastic elements (SEs), the direct and inverse Pugachev theorems are formulated and the correlational theory of joint CEs for two SEs is developed together with the theory of linear transformations. The solution of linear operator equations by the CEs of SEs in linear spaces with a basis is given. Special attention is focused on the CEs of SEs in Banach spaces with a basis. Some elements of the general theory of distributions for the CEs of SFs and SEs are developed. Particular attention is paid to the method based on CEs with independent components. Some new methods for the calculation of Radon–Nikodym derivatives are proposed. The considered applications of CEs and wavelet CEs to analysis, modeling, and synthesis problems are as follows: SSs and technologies, modeling, identification and recognition filtering, metrological and biometric technologies and systems, and synergic organizational technoeconomic systems (OTESs). The conclusion contains inferences and propositions for further studies. The list of references contains 43 items.