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

翻译：本文研究了具有周期自回归（PAR）时间序列和加性噪声的建模问题。在大多数情况下，数据通常假设为无噪声模型（即不含加性噪声），这在现实应用中并不合理。PAR模型辨识的前两个步骤是阶次选择与周期估计，因此本文重点关注这些问题。最后，模型需经过验证，为此提出了一种残差分析方法，将残差视为多维向量进行处理。阶次与周期选择以及模型验证均基于残差序列的特征函数（CF）实现。利用特征函数获取概率密度函数，并用于信息准则计算及残差分布检验。为完成PAR模型分析，还需建立系数估计流程，但该问题仅作简要提及（作为并行研究的独立任务）。所提出的方法可视为对受有限方差外部噪声干扰的周期非平稳数据进行分析的通用框架。本文创新点在于最优模型阶次与周期的选择，以及残差分析的研究。所有成果均受我们之前在基于PAR建模的机器状态监测研究启发。