[Paper Review] Nonparametric and adaptive modeling of dynamic seasonality and trend with heteroscedastic and dependent errors

Yu‐Chun Chen, Ming−Yen Cheng+1 more|arXiv (Cornell University)|Oct 17, 2012

Genetic and phenotypic traits in livestock48 references6 citations

TL;DR

This paper proposes a nonparametric model using the Synchrosqueezing Transform (SST) to extract time-varying trends, multi-component seasonality with dynamic frequency and amplitude, and heteroscedastic, dependent errors. The method achieves theoretical identifiability and robustness under nonparametric assumptions, enabling accurate decomposition in complex, real-world time series with minimal parametric constraints.

ABSTRACT

Seasonality (or periodicity) and trend are features describing an observed sequence, 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 {\it dynamics} of the seasonality such as time-varying frequency and amplitude, and the {\it adaptivity} of the analysis to such dynamics and robustness to heteroscedastic, dependent errors is 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 in the presence of a trend and heteroscedastic, dependent errors. The identifiability problem of the nonparametric seasonality 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 heteroscedastic, dependent error process in a general nonparametric setup. Results of a series of simulations are provided, and the incidence time series of varicella and herpes zoster in Taiwan and respiratory signals observed from a sleep study are analyzed.

Motivation & Objective

• To address limitations in existing parametric models that assume fixed seasonal periods and are sensitive to non-stationary dynamics.
• To develop a nonparametric framework capable of modeling time-varying frequency and amplitude in multiple seasonal components.
• To ensure robustness and adaptivity in the presence of heteroscedastic and dependent errors, including local bursts.
• To theoretically justify the identifiability of the nonparametric seasonality model up to a controlled bias.
• To provide a general decomposition method for trend, seasonality, and error processes in nonparametric, non-stationary time series.

Proposed method

• Introduces a nonparametric model where each seasonal component has time-varying amplitude and frequency with bounded derivatives.
• Applies the Synchrosqueezing Transform (SST) to extract instantaneous frequency and amplitude from the signal.
• Uses discrete- and continuous-time theoretical frameworks to justify SST’s adaptivity and robustness under heteroscedastic, dependent errors.
• Employs a model bias control to ensure identifiability of the nonparametric seasonality representation.
• Decomposes observed time series into trend, multiple seasonal components, and error process via SST-based signal separation.
• Validates performance using simulations and real data from varicella/Herpes zoster incidence and sleep respiratory signals.

Experimental results

Research questions

• RQ1 Can a nonparametric model accurately represent multi-component seasonality with time-varying frequency and amplitude?
• RQ2 Is the Synchrosqueezing Transform robust to heteroscedastic and dependent errors in time series decomposition?
• RQ3 How does the method perform when local bursts are present in the data?
• RQ4 Can the model distinguish between true seasonal dynamics and noise, especially in the presence of non-Gaussian innovations?
• RQ5 What is the theoretical guarantee of identifiability for the nonparametric seasonality model under bounded derivative constraints?

Key findings

• The nonparametric seasonality model is identifiable up to a controlled model bias, providing the first theoretical justification for such models.
• SST-based decomposition achieves low reconstruction error: RRASE for seasonal components is below 0.16 under standard noise conditions.
• Even with local bursts, SST maintains stable time-frequency representations and preserves trend and seasonal component estimates, though error process estimation degrades.
• The method successfully extracts time-varying frequency and amplitude in real data, such as the correlation between respiratory signal frequency and sleep stage.
• In varicella and herpes zoster incidence data, the model reveals changes in seasonal dynamics post-vaccination, supporting public health insights.
• Average computation time for decomposition is under 10 seconds per realization, indicating practical feasibility.