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

Abstract: Time series decomposition into trend, seasonal structure, and residual components is a core primitive for downstream analytics such as anomaly detection, change-point detection, and forecasting. However, most existing seasonal-trend decomposition methods rely on user-specified or estimated season lengths and implicitly assume stable periodic structure. These assumptions limit robustness and deployability in large, heterogeneous collections where recurring patterns may drift, appear intermittently, or exist at multiple time scales. We propose LGTD (Local-Global Trend Decomposition), a season-length-free decomposition framework that represents a time series as the sum of a smooth global trend, adaptive local trends whose recurrence induces implicit (emergent) seasonal structure, and a residual component. Rather than explicitly modeling seasonality through a fixed or estimated period, LGTD treats seasonal structure as an emergent property arising from repeated local trend regimes. Concretely, LGTD first estimates a global trend capturing long-term evolution, then applies AutoTrend, an adaptive error-driven local linear trend inference module, to segment the detrended signal into short-lived piecewise-linear regimes. Residuals are obtained after removing both global and local trends. By eliminating manual season-length specification, LGTD supports automated, low-touch deployment across time series with irregular, drifting, or weakly periodic structure. We analyze computational complexity and show that LGTD scales linearly with series length under mild conditions. Experiments on synthetic benchmarks demonstrate robust and balanced decomposition performance across fixed, transitive, and variable season-length settings, especially where period-based methods degrade.