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Modeling time series trajectories using the Liouville equation

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This paper presents algorithm for modeling set of trajectories of non-stationary time series, based on a numerical scheme for approximating the sample density of the distribution function in a problem with fixed ends, when the initial distribution for a given number of steps transforms into a certain final distribution, so that at each step the semigroup property of solving the Liouville equation is satisfied. The model makes it possible to numerically construct evolving densities of distribution functions during random switching of states of the system generating the original time series.

The main problem is related to the fact that with the numerical implementation of the left-hand differential derivative in time, the solution becomes unstable, but such approach corresponds to the modeling of evolution. An integrative approach is used while choosing implicit stable schemes with “going into the future”, this does not match the semigroup property at each step. If, on the other hand, some real process is being modeled, in which goal-setting presumably takes place, then it is desirable to use schemes that generate a model of the transition process. Such model is used in the future in order to build a predictor of the disorder, which will allow you to determine exactly what state the process under study is going into, before the process really went into it. The model described in the article can be used as a tool for modeling real non-stationary time series.

Steps of the modeling scheme are described further. Fragments corresponding to certain states are selected from a given time series, for example, trends with specified slope angles and variances. Reference distributions of states are compiled from these fragments. Then the empirical distributions of the duration of the system’s stay in the specified states and the duration of the transition time from state to state are determined. In accordance with these empirical distributions, a probabilistic model of the disorder is constructed and the corresponding trajectories of the time series are modeled.

Keywords: nonstationary time series, sample distribution function, velocity approximation, kinetic equation, semigroup
Citation in English: Goguev M.V., Kislitsyn A.A. Modeling time series trajectories using the Liouville equation // Computer Research and Modeling, 2024, vol. 16, no. 3, pp. 585-598
Citation in English: Goguev M.V., Kislitsyn A.A. Modeling time series trajectories using the Liouville equation // Computer Research and Modeling, 2024, vol. 16, no. 3, pp. 585-598
DOI: 10.20537/2076-7633-2024-16-3-585-598

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