Mathematical models of non-stationary random processes in the SVVP representation

Abstract

The work examines methods and mathematical models that provide the possibility of researching the statistical characteristics of complex and non-stationary random processes describing a wide class of physical phenomena. The proposed models can be used to study the processes observed in various fields of human activity, namely, to analyze the trajectories of unmanned aerial vehicles, their acoustic signals, meteorological processes reflecting the state of the atmosphere. Real and simulated non-stationary random processes considered in the work are represented by the complex vector random process (CVRP) model. In this case, the length of the subvector is equal to the period of the seasonal component. In fact, in such a representation, the time series readings are replaced by their aggregate, i.e. subvectors. Statistical relationships are analyzed for subvectors, and not, as usual, for process counts. If the length of the subvector is equal to one, all operations in the SVVP representation are equivalent to the usual operations for time series. The mathematical apparatus developed in the article was used to analyze changes in time series of atmospheric temperature observed over a long period of time; average annual temperatures were estimated with subsequent smoothing with a low-pass filter. The results obtained can be used to analyze medium-term and long-term changes in atmospheric conditions, refine the results obtained by traditional methods of mathematical statistics, analyze and predict data flows in mobile communication networks, as well as in other areas of human activity.