LGTD: Local-Global Trend Decomposition for Season-Length-Free Time Series Analysis

Abstract

Time series decomposition into trend, seasonal, and residual components is a fundamental primitive in data mining and analytics pipelines, underpinning anomaly detection, change-point analysis, and forecasting. Most existing methods require a user-specified or estimated season length and assume stable periodic structure. In large, heterogeneous collections, where recurring patterns drift, appear intermittently, or operate at multiple nonstationary scales, period selection becomes brittle and per-series tuning does not scale. We propose LGTD (Local-Global Trend Decomposition), a season-length-free decomposition framework that requires no period specification and operates with a single fixed default configuration across datasets. LGTD represents a series as the sum of (i) a smooth global trend capturing long-term evolution, (ii) adaptive local trends inferred by an error-driven local linear segmentation procedure, and (iii) a residual component. Rather than modeling seasonality through an explicit periodic basis, LGTD treats it as an emergent property arising from the recurrence of local trend regimes, decoupling decomposition quality from any estimated season length. We prove that the local trend inference procedure terminates in a bounded number of iterations and runs in linear time in the series length, independent of any seasonal parameter, and confirm this empirically: LGTD scales linearly in runtime and memory and is the fastest method across all tested lengths, while several baselines degrade super-linearly. On synthetic benchmarks LGTD achieves balanced accuracy across fixed, transitive, and variable season-length regimes, particularly where period-based methods degrade, and on real-world data it yields interpretable components and low-structure residuals. Source code and datasets are available at https://github.com/chotanansub/LGTD.