Modeling the Evolution of the Sample Distributions of Random Variables Using the Liouville Equation
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ling the Evolution of the Sample Distributions of Random Variables Using the Liouville Equation A. A. Kislitsina and Yu. N. Orlova, * aKeldysh
Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia *e-mail: [email protected] Received June 24, 2019; revised June 24, 2019; accepted September 9, 2019
Abstract—We consider the difference approximation of the one-dimensional Liouville equation for modeling the evolution of the sample distribution density (of the nonstationary time series) estimated by a histogram. It is shown that the change in the sample density of the distribution over a certain period of time can be numerically described as a solution of the Liouville equation if the initial density distribution is strictly positive in the internal class intervals. The algorithm for determining the corresponding rate is constructed and its mechanical and statistical meaning is shown as a semigroup equivalent in the Chernoff sense to the average semigroup, which generates the evolution of the distribution function. Keywords: Liouville equation, nonstationary time series, sample distribution function, Chernoff equivalence DOI: 10.1134/S2070048220050087
INTRODUCTION We construct a numerical scheme to solve the Liouville equation for the sample density of the distribution function of a nonstationary time series. The time series is formed by the values of some one-dimensional random variable, which are fixed with the given constant time step. Such series are readings of biometric sensors, series of electroencephalograms, exchange prices with the given frequency of time counts, etc. If a series is broadly stationary [1], then it can be analyzed by the traditional regression methods [1, 2], which represent models of a series in the form of a discrete time dynamic system (most often linear). Recall that a time series is said to be stationary in the general sense if its mathematical expectation does not depend on time t and the correlation function, which is a mathematical expectation of the product of the deviations of the values of the series from the average value at different moments t1 and t2 , and it depends only on the difference t1 − t2 . For nonstationary series, this approach results in inaccurate estimates. On a short time horizon (much shorter than the sample length), nonstationarity accounting can be carried out by adaptive methods [3], where the coefficients of previously stationary models are considered to be time-dependent within certain heuristic models. In long periods of time, the analysis of kinetic equations for sample distribution functions is required. This approach is proposed and developed in [4] and [5, 6], respectively. Despite the discreteness of the practical task, it is convenient to construct and study the kinetic equation by writing it in the differential form. This form involves grinding the difference grid to an arbitrarily small size, which requires studying the convergence, stability [7, 8], and approximation depending on the grid step. In the event of linear t
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