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Probabilistic analysis of two-dimensional vector random processes on the phase plane

Introduction: The models of vector random processes are characterized by a variety of practical applications and, at the same time, by the complexity of studying the detailed probabilistic structure. Such features often lead to considering only separate components of vector functions, forcing the shift to scalar process analysis. All this significantly reduces the overall information content of the research, increasing the relevance of the search for promising approaches to probabilistic analysis of vector processes. Purpose: To present vector random processes in a phase space of states, and to apply the general theory of level crossings of random functions to the study of probabilistic structure of phase trajectories. Results: We have identified the characteristic features of displaying random processes on a phase plane, and introduced numerical characteristics for the description and analysis of probabilistic behavior of sample functions. A probabilistic analysis has been performed for the characteristics of the «level crossing» type for vector processes with various definitions of the areas of acceptable values. We used the typical models of two-dimensional Gaussian processes and «signal plus noise» models to demonstrate how the probabilistic structure of the phase trajectories depends on the given threshold levels, major distribution parameters and spectral-correlation properties of the studied processes. Practical relevance : The article contributes to the research information content and its visualization in the analysis of probabilistic behavior of vector random processes. It combines general methods of phase space of vector processes, methods of phase plane in the analysis of scalar processes and the theory of level crossings of random functions. The classical approaches to the visual description of phase trajectories are supplemented by the methods of quantitative analysis of detailed probabilistic structure of random functions.