To the Problem of the Optimal Mathematical Model Selection of Stationary Time Series

Abstract

It is proved that the optimal mathematical model of a stationary time series is the autoregression – moving average model having the third order both in the autoregressive component and in the moving average component. The proof uses the fact that the dynamical system, in order to be steered in the stability mode, can not be described by a differential equation of order lower than the third. In going from derivatives to differences of the corresponding order, a third-order differential equation is transformed into a third-order difference equation relative to the samples of the output signal of the dynamic system. It is from this difference equation that, as a consequence, the third order of the autoregressive component follows for the output coordinate of this dynamic system, if the random component is also taken into account in the original signal, which is discretized, and therefore turns into a time series. And to prove that the third order of the moving average component in the mathematical model of autoregression – moving average is optimal, it is used as the fact that any stationary stochastic input signal of a dynamic system can be synthesized using a filter model whose input receives a weighted sum of white noise pulses with a constant spectrum, and the fact proved by the authors of this article that in this weighted sum of white noise pulses with an optimal choice of weights, It must only pulse generated in the observed time and two previous pulses preceding this time moment. And since this weighted sum of three white noise pulses is a moving average model for an autoregressive model of a dynamic object whose input is acted upon by a time series signal, this is the proof that the optimal order of the mathematical model of the sampled output signal of a dynamical system in the form of the autoregression – moving average and the moving average component is the third.