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：本文研究了具有加性噪声的周期自回归时间序列建模问题。大多数情况下，数据基于无噪声模型（即不含加性噪声）进行处理，但这在实际应用中并不现实。周期自回归模型识别的前两个步骤是阶次选择与周期估计，因此本文重点聚焦于这些问题。最后，模型需进行验证，为此提出了一种将残差视为多维向量的分析方法。阶次与周期选择以及模型验证均通过利用残差序列的特征函数实现。特征函数用于获取概率密度函数，进而应用于信息准则及残差分布检验。为完成周期自回归模型分析，系数估计步骤必不可少。然而，本文仅对此提及，因其属于独立任务（并行研究中）。所提方法可视为分析受有限方差外部噪声干扰的周期非平稳特性数据的通用框架。其原创性贡献在于最优模型阶次选择、周期识别及残差分析。所有发现均受我们先前基于周期自回归建模的机器状态监测工作的启发。