{"title":"Developing the Theory of Stochastic Canonic Expansions and Its Applications","authors":"I. N. Sinitsyn","doi":"10.1134/s1054661823040429","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>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.</p>","PeriodicalId":35400,"journal":{"name":"PATTERN RECOGNITION AND IMAGE ANALYSIS","volume":"37 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"PATTERN RECOGNITION AND IMAGE ANALYSIS","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1054661823040429","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
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.
摘要 卡农展开（CE）理论的创立与洛夫（Loeve）、科尔莫戈罗夫（Kolmogorov）、卡尔胡宁（Karhunen）和普加乔夫（Pugachev）等人的名字有关，可追溯到 1940-1950 年代。在应用于随机系统（SS）和技术的分析、建模和综合问题时，考虑了 CE 和小波 CE 理论的发展。在随机函数相关理论（SFs）的框架内，有关 CE 的直接和逆普加乔夫定理被扩展到随机线性函数的情况。推导了线性和准线性 SF 的 CE。特别关注了用 CE 对强非线性变换进行等效回归线性化的问题。提出了基于 CE 的非线性回归算法。以 Haar 小波为基础，发展了参数变化指定域内的小波 CE 理论。对于随机元素（SE），提出了直接和逆普加乔夫定理，并结合线性变换理论发展了两个 SE 的联合 CE 关联理论。给出了在有基础的线性空间中通过 SE 的 CE 求解线性算子方程的方法。特别关注的是带基巴拿赫空间中 SE 的 CE。发展了 SF 和 SE 的 CE 分布一般理论的一些要素。特别关注基于独立分量 CE 的方法。提出了一些计算拉顿-尼科迪姆导数的新方法。所考虑的 CEs 和小波 CEs 在分析、建模和合成问题中的应用如下：SSs 和技术、建模、识别和识别过滤、计量和生物识别技术和系统，以及协同组织技术经济系统 (OTES)。结论包含进一步研究的推论和建议。参考文献清单包含 43 项内容。
期刊介绍：
The purpose of the journal is to publish high-quality peer-reviewed scientific and technical materials that present the results of fundamental and applied scientific research in the field of image processing, recognition, analysis and understanding, pattern recognition, artificial intelligence, and related fields of theoretical and applied computer science and applied mathematics. The policy of the journal provides for the rapid publication of original scientific articles, analytical reviews, articles of the world''s leading scientists and specialists on the subject of the journal solicited by the editorial board, special thematic issues, proceedings of the world''s leading scientific conferences and seminars, as well as short reports containing new results of fundamental and applied research in the field of mathematical theory and methodology of image analysis, mathematical theory and methodology of image recognition, and mathematical foundations and methodology of artificial intelligence. The journal also publishes articles on the use of the apparatus and methods of the mathematical theory of image analysis and the mathematical theory of image recognition for the development of new information technologies and their supporting software and algorithmic complexes and systems for solving complex and particularly important applied problems. The main scientific areas are the mathematical theory of image analysis and the mathematical theory of pattern recognition. The journal also embraces the problems of analyzing and evaluating poorly formalized, poorly structured, incomplete, contradictory and noisy information, including artificial intelligence, bioinformatics, medical informatics, data mining, big data analysis, machine vision, data representation and modeling, data and knowledge extraction from images, machine learning, forecasting, machine graphics, databases, knowledge bases, medical and technical diagnostics, neural networks, specialized software, specialized computational architectures for information analysis and evaluation, linguistic, psychological, psychophysical, and physiological aspects of image analysis and pattern recognition, applied problems, and related problems. Articles can be submitted either in English or Russian. The English language is preferable. Pattern Recognition and Image Analysis is a hybrid journal that publishes mostly subscription articles that are free of charge for the authors, but also accepts Open Access articles with article processing charges. The journal is one of the top 10 global periodicals on image analysis and pattern recognition and is the only publication on this topic in the Russian Federation, Central and Eastern Europe.
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