Identification and validation of periodic autoregressive model with additive noise: finite-variance case

TL;DR

This paper develops a comprehensive framework for identifying and validating periodic autoregressive models with additive noise, focusing on order selection, period estimation, and residual analysis using characteristic functions, applicable to real-world non-stationary data.

Contribution

It introduces novel methods for optimal model order and period identification and residual validation in PAR models with additive noise, extending previous work to more realistic noisy scenarios.

Findings

Effective residual analysis using characteristic functions.

New procedures for model order and period selection.

Framework applicable to non-stationary, noisy data.

Abstract

In this paper, we address the problem of modeling data with periodic autoregressive (PAR) time series and additive noise. In most cases, the data are processed assuming a noise-free model (i.e., without additive noise), which is not a realistic assumption in real life. The first two steps in PAR model identification are order selection and period estimation, so the main focus is on these issues. Finally, the model should be validated, so a procedure for analyzing the residuals, which are considered here as multidimensional vectors, is proposed. Both order and period selection, as well as model validation, are addressed by using the characteristic function (CF) of the residual series. The CF is used to obtain the probability density function, which is utilized in the information criterion and for residuals distribution testing. To complete the PAR model analysis, the procedure for estimating the coefficients is necessary. However, this issue is only mentioned here as it is a separate task (under consideration in parallel). The presented methodology can be considered as the general framework for analyzing data with periodically non-stationary characteristics disturbed by finite-variance external noise. The original contribution is in the selection of the optimal model order and period identification, as well as the analysis of residuals. All these findings have been inspired by our previous work on machine condition monitoring that used PAR modeling