Regularization Methods for the Stable Identification of Probabilistic Characteristics of Stochastic Structures

This chapter covers the method of experimental data analysis and processing in cyber-physical systems for medical monitoring, control of manufacturing processes and management of industrial facilities. The suggested methods are used to develop mathematical models of dynamic systems with stochastic properties for managing complex structural subsystems of cyber-physical systems. The most important computational stage of the simulation is thereat the identification of multimodal (in general) densities of the random variable distribution. A matrix conditioning analysis is herein suggested with minimizing relevant functionals of the identification problem. For the method of identifying multimodal densities of random variable distribution a matrix condition analysis is suggested with minimizing the relevant functionals of the problem. It is shown that under ill-conditioning of the equivalent system of equations an algorithm for regularization of solutions is needed. The regularization of the basic method for identifying distribution densities based on the ridge regression-algorithm (RRA) is proposed and substantiated. The classical RRA is improved and modified for local regularization showing the advantage of the high-order unstable SLAEs over the classical Tikhonov method. The suggested regularization algorithms and programs are universal, applicable to the study of random structures in natural science, biomedicine, and computational mathematics.