Development of a method for forecasting random events during instability periods

The object of research is random events in the formation of new economic and financial models; in particular; with cardinal changes in economic and social strategies. The scope and variety of methods used in the prediction of random processes is large. Promising mathematical apparatus for solving the problem are statistical methods of analysis. Today; there are many methods for predicting random processes; but most existing models are not suitable for predicting non-stationary processes. One of the most problematic places in forecasting time series is that there is no single methodology by which to analyze the characteristics of a non-stationary random process. Therefore; it is necessary to develop special methods of analysis that can be applied to individual cases of unsteady processes. The optimal solution to the problem may be the approximation of the time series by finely rational functions or the so-called Padé approximation. Such an approach should take advantage of polynomial approximation. In polynomial approximation; polynomial can’t have horizontal asymptotes; which makes it impossible to make long-term forecasts. A rational approximation is guaranteed to tend to horizontal asymptotes; with all the poles of the finely rational function lying on the left side of the p-plane; that is; the Laplace transform plane. A method for predicting non-stationary time series with high accuracy of estimation and flexibility of settings is proposed. To ensure the stability of the method and the stability of the obtained results; it is proposed that the poles of the approximating function be introduced into the stability zone – the unit circle of the z-plane in compliance with the rules of conformal transformation. Namely; by transforming linear dimensions and preserving the angles between the orthogonal coordinates on infinitely small neighborhoods of the coordinate plane (the so-called conservatism of angles). It is shown that; subject to the conformity of the proposed transformation; the dynamic characteristics of the estimation and forecasting system are stored. This method can be especially successfully applied in the presence of non-stationarity of various natures.