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RobustSTL: A Robust Seasonal-Trend Decomposition Algorithm for Long Time Series
Qingsong Wen, Jingkun Gao, Xiaomin Song, Liang Sun, Huan Xu, Shenghuo Zhu
TL;DR
Existing decomposition methods do not adequately handle shifting or long-period seasonality, abrupt trend and remainder changes, anomalies, and noise. RobustSTL combines LAD-based sparse trend extraction with non-local seasonal filtering in an iterative procedure, and experiments on synthetic and real-world data demonstrate its effectiveness and practical usefulness.
Problem
Time-series decomposition needs to handle seasonality shifts and fluctuations, abrupt trend and remainder changes, anomalies, noise, and long seasonality periods for anomaly detection and forecasting.
Method
RobustSTL uses LAD loss with sparse regularization for robust trend extraction, followed by non-local seasonal filtering, iterated to estimate trend and seasonality.
Results
Experiments on synthetic and real-world data demonstrate that RobustSTL separates trend, seasonality, and remainder effectively and is practically useful.
Takeaways & Limitations
RobustSTL is designed to extract time-series insights for anomaly detection while accommodating long, noisy, shifting, and abruptly changing seasonal-trend structure.
Takeaways & Limitations
The decomposition assumes additive structure and treats the remainder as containing all signals other than trend and seasonality.
Abstract
from arXiv · showhide
Decomposing complex time series into trend, seasonality, and remainder components is an important task to facilitate time series anomaly detection and forecasting. Although numerous methods have been proposed, there are still many time series characteristics exhibiting in real-world data which are not addressed properly, including 1) ability to handle seasonality fluctuation and shift, and abrupt change in trend and reminder; 2) robustness on data with anomalies; 3) applicability on time series with long seasonality period. In the paper, we propose a novel and generic time series decomposition algorithm to address these challenges. Specifically, we extract the trend component robustly by solving a regression problem using the least absolute deviations loss with sparse regularization. Based on the extracted trend, we apply the the non-local seasonal filtering to extract the seasonality component. This process is repeated until accurate decomposition is obtained. Experiments on different synthetic and real-world time series datasets demonstrate that our method outperforms existing solutions.
Introduction
Seasonal-trend decomposition is needed to expose time-series structure for anomaly detection and forecasting, but existing methods struggle with shifting or long-period seasonality, abrupt changes, noise, and anomalies. RobustSTL addresses these challenges with robust trend regression and non-local seasonal filtering applied iteratively.
- Motivation: Seasonal-trend decomposition reveals underlying time-series insights and supports anomaly detection and forecasting.Without decomposition, a local spike during an idle period can be masked by much larger values during busy periods.
- Existing Methods: STL can lose seasonal accuracy under long periods, high noise, seasonality shifts, and fluctuations.X-13-ARIMA-SEATS and related methods incorporate calendar effects and regressors but scale only to small or medium datasets and mainly monthly or quarterly data.
- Challenges: Real-world data commonly exhibits seasonality fluctuation and shift, abrupt trend and remainder changes, anomalies, and long seasonality periods.For minute-level data with a daily period, T = 1440, requiring estimation of T −1 seasonal parameters.
- Proposed Method: RobustSTL extracts trend with LAD loss and sparse regularization, then uses non-local seasonal filtering in an iterative decomposition process.The method targets flexible, fractional, and shifted seasonality together with abrupt trend and remainder changes.
- Scope: The paper focuses on additive decomposition, while multiplicative decomposition can be obtained similarly.The remainder is treated as containing signals beyond trend and seasonality, rather than only white noise.
Related Work
Existing decomposition methods trade flexibility, robustness, and computational scalability across seasonality and trend conditions. The comparison is organized around outlier robustness, seasonality shifts, long periods, and abrupt trend changes.
- Classical and State-Space Methods: TBATS and related classical methods suit short seasonality periods but face high computational cost for long periods and cannot handle slowly changing seasonality.These methods are therefore less suitable when seasonal structure is long or changes gradually.
- Regression-Based Methods: The Hodrick-Prescott filter is computationally cheap but cannot decompose long-period seasonality and is prone to spikes, dips, and abrupt trend changes.Its smoothness regularization acts on the second derivative of the fitted curve.
- Regression-Based Methods: STR flexibly handles seasonal shifts and multiple seasonalities, but STR and robust STR cannot follow abrupt trend changes because they regularize second differences for smoothness.STR also provides confidence intervals for predicted components.
- Matrix-Based Methods: SSA is a model-free approach that performs well on short time series but relies on strong assumptions that limit applicability to some real-world datasets.It reconstructs components after arranging sliding arrays into a matrix and applying singular value decomposition.
- Comparison Criteria: Methods are compared by outlier robustness, seasonality shift, long seasonal periods, and abrupt trend changes.Table 1 presents the complete algorithm comparison using Y/N indicators.
Robust STL Decomposition
RobustSTL decomposes a denoised series by robustly estimating trend differences and extracting seasonality with non-local neighborhoods. Its design preserves abrupt changes, accommodates shifted seasonality, and reduces the influence of outliers.
- Pipeline: RobustSTL first denoises the series with bilateral filtering, then estimates trend, extracts seasonality non-locally, and iterates these steps.The bilateral filter uses neighboring observations with similar values and is intended to preserve abrupt trend changes and spike-and-dip anomalies.
- Trend Extraction: Seasonal differencing reduces seasonal effects before recovering the trend difference from the denoised signal.The seasonal difference is defined as ∇T x_t = x_t − x_{t−T}, while the first-order difference is ∇x_t = x_t − x_{t−1}.
- Trend Extraction: The trend objective uses LAD loss with first- and second-order difference regularization to resist outliers while allowing abrupt level shifts and piecewise-linear trends.The first-order constraint favors slowly changing trend differences, while sparse second differences encode smooth, piecewise-linear structure.
- Seasonality Extraction: Non-local seasonal filtering uses neighborhoods around previous seasonal positions and weights observations by temporal distance and value similarity.Similar seasonal patterns receive larger weights, while abnormal observations receive smaller weights.
- Seasonality Extraction: When the seasonal shift is Δt, the extracted season can follow it provided the neighborhood half-width satisfies H > Δt.The filter also remains unaffected by an outlier at the current point or near the corresponding historical seasonal point.
- Final Adjustment: The extracted seasonality is centered by removing its mean so that all seasonal components within a period sum to zero, which also estimates the initial trend point.The remainder may still contain residual seasonal and trend components after these adjustments.
Experiments
Experiments on synthetic and real-world datasets compare RobustSTL with STL, TBATS, and STR under shifts, anomalies, noise, and long seasonal periods. RobustSTL separates components more accurately and efficiently across these settings.
- Experimental setup: Synthetic data combine seasonal shifts, abrupt trend changes, anomalies, and Gaussian noise to simulate complex real-world scenarios.The seasonal period is 50 across 15 periods, with 10 abrupt level changes and 14 spike-and-dip anomalies.
- Synthetic results: RobustSTL separates synthetic trend, seasonality, and remainders close to the original signals, whereas each baseline exhibits specific decomposition failures.STL produces rough seasonality and smooths abrupt trends; TBATS is affected by anomalies; STR cannot handle abrupt trend changes.
- Synthetic results: RobustSTL achieves much better MSE and MAE results than STL, STR, and TBATS on the synthetic dataset.The comparison evaluates errors for extracted trend and seasonality against the true components.
- Efficiency: RobustSTL is observed to compute significantly faster than TBATS and STR, while standard STL remains sensitive to anomalies and trend changes.The algorithm is formulated as an ℓ1-norm-regularized optimization problem; STR’s result is unavailable for dataset 2 because of long computation time.
- Overall findings: Across synthetic and real-world data, RobustSTL outperforms existing solutions in handling abrupt trend changes, irregular and shifted seasonality, spike-and-dip anomalies, and efficiency.The experiments include comparisons with Standard STL, TBATS, and STR.
Conclusion
The paper addresses decomposition of complex long time series into trend, seasonality, and remainder components under anomalies, abrupt changes, noise, and seasonal shifts. It proposes RobustSTL and reports effectiveness and practical usefulness on synthetic and real-world data.
- Conclusion: RobustSTL combines LAD with ℓ1-norm regularization and non-local seasonal filtering for complex long-time-series decomposition.The method targets anomalies, abrupt trend changes, seasonal shifts and fluctuations, noise, and computational efficiency.
- Conclusion: Experiments on synthetic and real-world datasets demonstrate the effectiveness and practical usefulness of RobustSTL.The conclusion identifies direct integration of decomposition with anomaly detection as future work.