Seasonality (or periodicity) and trend are features describing an observed time series, and extracting these features is an important issue in many scientific fields. However, it is not an easy task for existing methods to analyze simultaneously the trend and dynamics of the seasonality, such as time-varying frequency and amplitude; and the adaptivity of the analysis to such dynamics and robustness to heteroscedastic, dependent errors are not guaranteed. These tasks become even more challenging when there exist multiple seasonal components. We propose a nonparametric model to describe the dynamics of multi-component seasonality, and investigate the recently developed Synchrosqueezing transform (SST) in extracting these features from the observations, in presence of a trend and heteroscedastic, dependent errors. The identifiability problem of the nonparametric model is studied, and the adaptivity and robustness properties of the SST are theoretically justified in both discrete- and continuous-time settings. Consequently we have a new technique for de-coupling the trend, seasonality and non-stationary error process in a general nonparametric setup. We also consider a different model describing the possible non-harmonic oscillatory pattern. A statistical approach to determine the number of components, the possible shape function, and forecasting is provided. Results of a series of simulations are provided. In clinics, we show that based on SST, we are able to predict the first reaction from anesthesia and estimate the anesthetic depth. Another application is the incidence time series of varicella and herpes zoster in Taiwan. We show scientifically the dynamical seasonality and trend introduced by the general application of varicella vaccine.
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