seasonality modeling

This work proposes an adaptive time series decomposition framework that eliminates the need to predefine or estimate seasonal periods, addressing the limitations of traditional methods in handling non-stationary, drifting, or multi-scale seasonal patterns. The approach decomposes a sequence into three components: a global trend, an adaptive local linear trend that implicitly captures seasonality, and a residual. An AutoTrend module dynamically partitions the local trend in an error-driven manner, while global smoothing ensures coherent long-term structure; seasonality emerges automatically as a recurring pattern in the local trends. Operating in linear time, the method demonstrates robust performance across synthetic datasets with fixed, transitioning, and varying seasonal periods, achieving high-quality, low-intervention decomposition even in scenarios where conventional techniques fail.

This work addresses the challenge of non-identifiability and sensitivity to structural breaks, outliers, and time-varying volatility in decomposing trend and multiple seasonal components from time series. The authors propose a Bayesian nonparametric regression framework that, through adaptive regularization and Markov chain Monte Carlo (MCMC) inference, establishes the first rigorous identifiability conditions for trend and multi-seasonal effects. This approach enables robust decomposition even in the presence of abrupt changes, anomalies, and heteroskedasticity, while delivering principled uncertainty quantification. Empirical evaluations on both synthetic and real-world datasets demonstrate superior performance over state-of-the-art methods such as TBATS, STR, and MSTL, yielding more accurate, interpretable, and reliable decompositions. An open-source R package implementing the method is made publicly available to the research community.

This paper addresses two key limitations of the standard Decomposition (Decomp) state-space model for seasonal adjustment: excessive smoothing of the trend component and misattribution of long-term variation to the autoregressive (AR) component when AR eigenvalues lie near the unit circle. To resolve these issues, we propose a novel constrained optimization framework that jointly penalizes the modulus and argument of AR eigenvalues, integrated with an L1/L2 hybrid regularization to enhance identifiability and interpretability between trend and AR components. By embedding these constraints within a state-space formulation, our method achieves precise control over trend smoothness and accurate attribution of persistent dynamics. Empirical evaluation across multiple real-world time series demonstrates substantial improvements in statistical robustness and decomposition reliability of seasonal adjustments compared to conventional approaches.

Existing time-series generation methods lack interpretable decomposition mechanisms, hindering faithful modeling of meaningful trend and seasonal patterns. To address this, we propose Seasonal-Trend Diffusion (STDiffusion), the first framework to deeply integrate learnable sequence decomposition with diffusion modeling. STDiffusion explicitly parameterizes the trend component via MLPs and captures multi-scale seasonality through adaptive wavelet distillation; it further introduces a component correction mechanism to enforce consistency and decoupling between trend and seasonal components. This design significantly enhances both interpretability across multiple resolutions and internal coherence of generated series. Evaluated on eight real-world datasets, STDiffusion achieves state-of-the-art performance across all generation benchmarks. Moreover, it demonstrates strong robustness and generalization in multi-window, long-horizon forecasting tasks.

Residential electricity demand forecasting faces modeling challenges arising from concurrent multiple seasonality, cyclicity, and abrupt fluctuations, which existing methods struggle to capture adequately. This paper proposes SPDNet, an end-to-end deep decomposition network. SPDNet innovatively integrates two core modules: a trend–seasonal decomposition module (STDM) and an FFT-driven frequency-domain periodicity identification module. It further introduces, for the first time, FFT-guided 2D tensor reshaping to enable joint modeling of cross-cycle interactions via 1D-CNN, Transformer, and 2D-CNN components. Evaluated on real-world residential load data, SPDNet achieves up to 18.7% higher prediction accuracy than statistical models and RNN/CNN/Transformer baselines, while accelerating inference by 2.3×. The method thus significantly advances both accuracy and computational efficiency in short-term load forecasting.

This study addresses the challenge of identifying an unknown number of periodic components in functional time series by proposing a novel information criterion with theoretical consistency guarantees. The method integrates least squares fitting with residual process analysis and employs an iterative strategy to adaptively estimate the number of periodicities, making it applicable to a broad class of functional time series models. Extensive numerical simulations demonstrate that the proposed criterion performs exceptionally well in finite samples. Furthermore, its practical utility and effectiveness are corroborated through real-data applications to temperature and sunspot records, where it successfully uncovers statistically significant periodic structures.

This work proposes the Fréchet decomposition problem, which seeks to approximate a collection of univariate time series as Fréchet combinations of a small set of basis curves while minimizing the total Fréchet distance. Analogous to principal component analysis, this is the first effort to incorporate the Fréchet distance into time series decomposition, and it introduces two distinct variants of the problem. For the single-basis setting, the authors design a (1+ε)-approximation algorithm; for an arbitrary number k of basis curves, they present an exact solution method based on projection distance, assuming a given candidate set of basis curves. By delivering both an efficient approximation scheme and a general exact algorithm, this study establishes a novel paradigm for modeling nonlinear structures in time series data.

This study uncovers the mathematical mechanism underlying the structural patterns of extreme values in real-world time series. By integrating additive combinatorics with discrete Fourier analysis, and leveraging Fourier sparsity complexity together with a generalized Chang’s lemma, the authors prove that when a time series exhibits low Fourier complexity, its set of maxima can be exactly reconstructed via integer linear combinations with coefficients in \{-1,0,1\} from a minimal generating set containing only 4–7 elements. This work provides the first rigorous theoretical explanation for the structural regularity of extreme values in time series, establishing a quantitative link between Fourier spectral properties and additive generative capacity. The theory is validated on both raw and mean-centered U.S. inflation and Delhi climate data, highlighting the informational richness and pronounced internal structure of extreme events.

This study addresses the challenge of effectively estimating trend and seasonal components in point process time series by proposing a spatiotemporal doubly stochastic Poisson model based on a log-Gaussian intensity function. The authors develop a computationally efficient M-estimator to jointly extract trend and seasonal patterns, offering the first decomposition framework for irregular event sequences that combines interpretability with theoretical guarantees. They derive the asymptotic distribution of the proposed estimator, establishing its statistical properties. Simulation studies demonstrate favorable finite-sample performance, and the method is successfully applied to Chicago Divvy bike-sharing data, uncovering interpretable spatiotemporal dynamics in user demand.

This study addresses the challenge of modeling nonstationary spherical time series that exhibit intrinsic spherical geometry and contain unknown trend and periodic components, which conventional Euclidean methods fail to capture effectively. The work proposes the first unified geometric framework that introduces a novel nonparametric trend–cycle decomposition based on optimal transport. This approach sequentially extracts smooth trend and periodic components while preserving the spherical topology, followed by fitting a spherical autoregressive model to the residual stationary component. The method achieves both interpretability and predictive accuracy; theoretical analysis establishes its consistency, and extensive simulations alongside real-world applications—such as electricity generation mix and bike-sharing flow data—demonstrate its significant superiority over existing approaches in uncovering structural dynamics and enhancing forecasting performance.