An algorithm has been developed and a program has been compiled in the Python programming language for calculating numerical values of the optimal lagged filtering operator for an L-Markov process with quasi-rational spectral density, which is a generalization of the Markov process with a rational spectrum. The construction of an optimal delayed filtering operator is based on the spectral theory of random processes. The calculation formula of the filtration operator was obtained using the theory of L-Markov processes, methods for calculating stochastic integrals, the theory of functions of a complex variable, and methods of trigonometric regression. An example of an L-Markov process (signal) with a quasi-rational spectrum is considered, which is interesting from the point of view of controlling complex stochastic systems. The trigonometric model was used as the basis for constructing a mathematical model of the optimal delayed filtration operator. It is shown that the values of the delayed filtering operator are represented by a linear combination of the values of the received signal at certain time points and the values of the sinusoidal and cosine functions at the same time points. It is established that the numerical values of the filtering operator significantly depend on the parameter β of the joint spectral density of the received and transmitted signals, and therefore three different tasks of signal transmission through different physical media were considered in the work. It is established that the absolute value of the real part of the filtration operator at all three intervals of the delay period change and in all three media exceeds the absolute value of the imaginary part by an average of two or more times. Graphs of the dependence of the real and imaginary parts of the filtration operator on the delay period t are constructed, as well as three-dimensional graphs of the dependence of the filtration operator itself with a delay on the delay period. The physical justification of the obtained results is given.

Keywords: random process, L-Markov process, noise, delayed filtering, spectral characteristic, filtering operator, trigonometric trend, standardized approximation error

A mathematical model has been constructed, an algorithm has been developed, and a program has been written in the Python programming language for calculating the numerical values of the optimal filtering operator with a forecast for an L-Markov process with a quasi-rational spectrum. The probabilistic model of the filtering operator formula has been obtained based on the spectral analysis of L-Markov processes using methods for calculating stochastic integrals, the theory of analytical functions of a complex variable, and methods for correlation and regression analysis. Considered an example of L-Markov process, the values of the optimal filtering operator with a forecast for which it was possible to express in the form of a linear combination of the values of the process at some moments of time and the sum of numerical values of cosines and sines at the same moments. The basis for obtaining the numerical values of the filtering operator was the mathematical model of trigonometric regression with 16 harmonics, which best approximates the process under study and has a minimum

Keywords: random process, L-Markov process, prediction filtering, spectral characteristics, filtering operator

A stochastic model of the optimal functional extrapolator of a fractal L–Markov process with a quasi-rational spectrum is constructed. When developing the model, methods of spectral and fractal analysis of random processes, the theory of functions of a complex variable, methods for calculating stochastic integrals, and the theory of stochastic differential–difference equations connecting processes with a quasi-rational spectrum with processes with a rational spectrum were used.; as well as an original technique for constructing spectral characteristics of extrapolation, developed by the famous mathematician A. Yaglom. Using the Levinson–McKean theorem, it is established that the random processes studied in this paper are L–Markovian in nature. The fulfillment of the conditions of Mandelbrot's theorem on the shape of the spectral density of fractal random processes, as well as the values of the Hearst exponents and the fractality index, suggest that the random process under study is fractal and, moreover, persistent. It is proved that the optimal extrapolator constructed over the entire past of the process can be represented as the sum of a linear combination of the values of the process itself at three time points in the case of 0 < τ < 1 and at two time points in the case of 1 < τ < 2 and an integral with an exponentially decaying weight function extended to (– ∞; ∞). In the first case, the L – boundary of the L–Markov process under study consists of three points L = {t; t – 2; t + τ – 2}, and in the second case it consists of two points L = {t; t + τ – 2}, where τ is the lead time.

Keywords: extrapolation, L –Markov process, fractality, trend tolerance, spectral characteristic, optimal extrapolator