Kirichenko L. Models and methods for estimating the parameters of self-similar and multifractal stochastic processes

Object of study are self-similar and multifractal stochastic processes in technical, economic and biological systems. The purpose of research is development of a comprehensive approach to the estimation of parameters of self-similar and multifractal stochastic processes of the short time series, based on the using a combination of effective inter-related methods of analysis of related fractal and the correlation structures of process. Research methods - methods оf fractal, and statistical analysis, methods of probability theory and stochastic processes, methods of nonlinear dynamics and methods of wavelet analysis, simulation modeling. Equipment - a personal computer. Theoretical and practical studies - proposed research methods and mathematical models of fractal processes are useful for modeling and analysis of the technical, economic and other systems that have fractal properties; the window methods of fractal analysis are useful for monitoring, diagnosis and prognosis of critical phenomena for time series of different nature. Scientific innovation - first the comprehensive approach to the estimation of parameters of self-similar and multifractal random processes from a short time series is proposed, which provides an unbiased interval estimation of a parameter of self-similarity; first the method for determining the of monofractal and multifractal properties using a sample function of the generalized Hurst exponent is proposed; first the model of the stochastic process with the given parameter of self-similarity and characteristic of the distribution tail is proposed, first the method of wavelet estimation of the Hurst exponent for the shot series with significant trend and cyclical components is proposed, which is based on a preliminary analysis of the wavelet energy spectrum and the using of wavelet packet transform. The developed models and analysis of fractal telecommunication traffic and congestion prediction were introduced in the State enterprise "Plant behalf Malyshev". The methods of window analysis of multifractal structure were introduced in company "Market-report." The results of studies of multifractal properties of biomedical signals were introduced in Kharkiv Regional Clinical Oncology Center. The methods of the study of random and chaotic processes were introduce in educational process at the Department of Applied Mathematics of Kharkov National University of Radio Electronics and Physical-Technical Institute of the National Technical University of Ukraine "Kyiv Polytechnic Institute". All results are confirmed by the relevant implementing acts. Scientific theoretical and practical results of the thesis should be used to solve practical problems of the analysis of technical, informational, biological, and other systems that have fractal properties, and for monitoring, diagnosis and prognosis of critical phenomena for the time series of different nature; for modeling complex systems related to data traffics, which have long-term dependence in order to study the best performance.