Modelling finite time-series signals using closed form solutions for first zero crossing of the Sample Autocorrelation

Abstract

<div>Order identification is an important criterion for modelling any signal. The present paper aims to derive a closed-form solution of the 1st zero crossing of the sample Autocorrelation Function (ACF) of a finite time-series signal and identify the order of the signal directly from it. For this, the zero crossing condition of the standard sample ACF has been derived for a sampled deterministic (power law) time function of the form aiti. This method is then extended to the stochastic power law and the deterministic polynomial cases, where additive random noise and coefficient interactions respectively are present. Thereafter, an algorithm has been developed for determining the order of a time-series signal. This has been applied to obtain the order and also the models of real differential dermal potentials. It is established that for the functions aiti, 2 distinct lag values are obtained as closed form solutions for which the autocorrelation is zero. Of these, the 1st zero crossing value is related to the order i of the signal, while the 2nd zero crossing value is the final instant N. This primary result has been extended to estimate the orders of the stochastic, polynomial as well as real time-series signals. The models of the real-time potentials obtained using this approach have been analyzed and also compared with those obtained using standard PACF based order detection approach. The findings establish that the proposed approach provides an effective method to obtain characteristic models of real, finite, time-series signals.</div>