Abstract
The periodic behavior of real data can be manifested in the time series or in its characteristics. One of the characteristics that often manifests the periodic behavior is the sample autocovariance function. In this case, the periodically correlated (PC) behavior is considered. One of the main models that exhibits PC property is the periodic autoregressive (PARMA) model that is considered as the generalization of the classical autoregressive moving average (ARMA) process. However, when one considers the real data, practically the observed trajectory corresponds to the “pure” model with the additional noise which is a result of the noise of the measurement device or other external forces. Thus, in this paper we consider the model that is a sum of the periodic autoregressive (PAR) time series and the additive noise with finite-variance distribution. We present the main properties of the considered model indicating its PC property. One of the main goals of this paper is to introduce the new estimation method for the considered model’s parameters. The novel algorithm takes under consideration the additive noise in the model and can be considered as the modification of the classical Yule–Walker algorithm that utilizes the autocovariance function. Here, we propose two versions of the new method, namely the classical and the robust ones. The effectiveness of the proposed methodology is verified by Monte Carlo simulations. The comparison with the classical Yule–Walker method is presented. The approach proposed in this paper is universal and can be applied to any finite-variance models with the additive noise.
Highlights
One of the main models that exhibits periodically correlated (PC) property is the periodic autoregressive (PARMA) model that is considered as the generalization of the classical autoregressive moving average (ARMA) process
We have shown that the considered model is still PC; it does not satisfy the periodic autoregressive moving average (PARMA) equation
The additive noise included in the model makes that the classical estimation methods for PAR model’s parameters are not effective in the considered case
Summary
We consider the model described by Eq (1) that is the classical PAR (periodic autoregressive) time series disturbed by the additive noise with finitevariance distribution. The classical estimation method useful for the PARMA models is based on the so-called Yule–Walker approach [4] It utilizes the autocovariance function of the time series, and at the final step, the classical measure of dependence is replaced by its empirical counterpart. When the model under consideration is described by the process given in Eq (1), the classical Yule–Walker algorithm seems to be not effective, especially for the additional noise with large variance. The main goal of this paper is to introduce the general model that is a sum of the ‘‘pure’’ PAR time series and the additive noise and demonstrate its main properties. The last section concludes the paper and presents the future study
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