Statistical modeling for nonstationary time series and functional time series with dynamical periodic structures and covariate effects

Many time series in real life exhibit a periodic component as well as a trend. Sometimes, the period of a time series may be unknown and an accurate estimate is necessary for further estimation, prediction and inference. In addition, the period or amplitude of the periodic component may change along time which makes it harder to study. Moreover, there may be some covariate effects which are important in their own rights and may have an influence on the estimation and inference for the periodic and trend terms. Motivated by examples concerning global warming, monthly total import and export by China and hospital admissions of respiratory and cardiovascular disease patients in Hong Kong, we study time series that contain a periodic component with an unknown period and an unknown changing amplitude, a trending behavior and other complex covariate effects. Specifically, our model allows the amplitude function to change at some unknown change-points, and we add to the classical time series decomposition model some covariate effect functions. The covariates may have different effect mechanisms and their effects may vary with time. Also, there may be functional covariates as functional data are increasingly common in various subject areas. We suggest a three- stage estimation procedure to estimate more accurately the period, change-points, periodic sequence, trend and covariate effects. First, we estimate the period with the trend and covariate effects being approximated by B-splines rather than being ignored. Since there are change-points along the periodic sequence, here we apply a segmented penalized least square method. Then, we suggest a new method inspired by binary segmentation to estimate the change-points. Finally, we estimate the entire periodic sequence, trend and covariate effects with the latter two approximated by B-splines. Asymptotic results will be derived and simulation studies will be conducted to examine asymptotic properties and finite sample performance of the proposed method. We will apply the proposed methods to analyze the above-mentioned motivating datasets to demonstrate its utility. Next, since functional data are increasingly common, we will extend our method to deal with functional time series with a similar dynamical periodicity, trend and covariate effects. While this emerges as an important topic, the relevant literature is limited. Also, such an extension and the theoretical study are nontrivial. We will conduct a simulation study and apply the proposed method to analyze a functional time series concerning climate change.