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

Chotanansub Sophaken, Thanadej Rattanakornphan, Piyanon Charoenpoonpanich, Thanapol Phungtua-eng, Chainarong Amornbunchornvej

TL;DR

LGTD tackles nonstationary and irregular seasonality by removing the need for a predefined season length. It decomposes a time series as $y_t = g_t + \ell_t + \varepsilon_t$, where a smooth global trend $g_t$ is complemented by locally inferred, piecewise-linear trends $\ell_t$ whose recurring patterns yield emergent seasonality; residuals $\varepsilon_t$ capture irregular fluctuations. The core is AutoTrend-LLT, an error-driven local trend inference that segments the detrended signal into adaptive regimes, enabling a season-length-free and scalable decomposition with a finite-termination guarantee. Across synthetic and real-world benchmarks, LGTD achieves robust, balanced decomposition under fixed, transitive, and variable season-length settings, outperforming period-based baselines while maintaining linear-time complexity and low manual tuning. These properties make LGTD a practical primitive for large-scale, heterogeneous time-series analytics and downstream investigative tasks.

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.

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

TL;DR

LGTD tackles nonstationary and irregular seasonality by removing the need for a predefined season length. It decomposes a time series as , where a smooth global trend is complemented by locally inferred, piecewise-linear trends whose recurring patterns yield emergent seasonality; residuals capture irregular fluctuations. The core is AutoTrend-LLT, an error-driven local trend inference that segments the detrended signal into adaptive regimes, enabling a season-length-free and scalable decomposition with a finite-termination guarantee. Across synthetic and real-world benchmarks, LGTD achieves robust, balanced decomposition under fixed, transitive, and variable season-length settings, outperforming period-based baselines while maintaining linear-time complexity and low manual tuning. These properties make LGTD a practical primitive for large-scale, heterogeneous time-series analytics and downstream investigative tasks.

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.
Paper Structure (48 sections, 1 theorem, 8 equations, 6 figures, 5 tables, 2 algorithms)

This paper contains 48 sections, 1 theorem, 8 equations, 6 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Classical seasonal trend decomposition.
  4. Robust decomposition under anomalies and complex patterns.
  5. Online, streaming, and systems-oriented decomposition.
  6. Season-length estimation under nonstationarity.
  7. Positioning of LGTD
  8. Methods
  9. Adaptive Local Linear Trend Decomposition (AutoTrend-LLT)
  10. Motivation.
  11. Local Trend Modeling.
  12. Error-Driven Refinement.
  13. Output.
  14. Algorithm.
  15. LGTD: Unified Global--Local Trend Decomposition
  16. ...and 33 more sections

Key Result

proposition 1

Consider Algorithm algo:autotrend_llt with update_threshold enabled, baseline percentile $p_0 \in (0,100]$, and step size $\Delta p > 0$. Assume the percentile operator is clamped so that $\text{percentile}(E,p)=\max(E)$ for all $p \ge 100$. Define If $K_{\max} \ge K^\star$, then AutoTrend-LLT assigns every index $t \in \{w,\dots,T-1\}$ a finite label by iteration $K^\star$ and terminates with an

Figures (6)

  • Figure 1: LGTD global--local decomposition process. After estimating a global trend $g_t$, the detrended series is segmented adaptively by AutoTrend-LLT to discover local trends $\ell_t$, which serve as emergent seasonal components without requiring a predefined season length. Removing both trends yields final residuals $\varepsilon_t$, giving the decomposition $y_t = g_t + \ell_t + \varepsilon_t$.
  • Figure 2: Decomposition on transitive season-length data, where LGTD preserves smooth trends and coherent emergent seasonality under changing cycles, while period-based methods exhibit distortion and residual leakage.
  • Figure 3: Decomposition results on variable season-length data, showing that LGTD robustly captures the evolving global trend and adapts to changing seasonal structure via emergent local trends, while maintaining low-variance residuals compared to period-based baselines that struggle under nonstationary season lengths.
  • Figure 4: Decomposition of the ETTh1 dataset, showing that LGTD extracts a smooth global trend and structured emergent seasonality while leaving low-variance residuals, without requiring a predefined season length.
  • Figure 5: Decomposition of the SILSO sunspot series, showing that LGTD captures long-term solar activity trends and quasi-periodic emergent seasonality with drifting cycles, while producing low-structure residuals without assuming a fixed season length.
  • ...and 1 more figures

Theorems & Definitions (1)

  • proposition 1: Termination Bound for AutoTrend-LLT