Methodological support of functional Bayes synthesis of stochastic systems based on wavelet canonical expansions

Article is devoted to methodological support and software tools for scalar nonlinear stochastic systems with high availability (StSHA) described by Pugachev nonlinear equations for parameters and additive noises and in case of functional Bayes criteria (FBC). Survey of wavelet analysis and synthesis technologies for scalar and vector, stationary and nonstationary, linear and nonlinear StSHA is given in Introduction. Synthesis is based on in mean square and Bayes criteria for the typical loss functions. In Section 1 the problem of synthesis of StSHA described by Pugachev nonlinear equations for FBC case in terms of conditional mathematical expectation of loss function is stated. It is supposed noise in observation is additive and Gaussian. Special case when observable stochastic process in defined by nonlinear differential equation. Section 2 is devoted to 4 stage algorithms for scalar nonstationary nonlinear StSHA with independent noises synthesis based of FBC by wavelet canonical expansions (WLCE) on finite time interval. Four Illustrative examples are given in Section 3. First FBC example is minimum of average integral of square error for finite time interval. Second FBC example is minimum of number of excursions for errors on finite time interval. Example 3 illustrates FBC synthesis of linear extrapolator in case of minimum for mean square error and mean square error first derivative. Experimental software tools and supporting resulting computer experiments for 2 and 4 levels of WLCE are described. For nigh accuracy of algorithm it is enough 8 WLCE members. Example 4 illustrates FBC synthesis based on average mean square error for nonlinear StSHA by WLCE. Corresponding experimental software tools and supporting computer experiments are given. Nigh accuracy of algorithm is confirmed for 8 WLCE members at 2 and 4 WLCE levels. In Conclusion the following directions of future research are formulated: methodological support and software tools for vector nonstationary nonlinear StSHA with independent noises; methodological support and software tools for vector nonstationary nonlinear StSHA with dependent noises; methodological support and software tools for vector nonstationary nonlinear StSHA for compact carrier WLCE.