Biseasonal MSTL Model
- Biseasonal MSTL model is a decomposition method that splits time series data into dual seasonal components, trend, and residual using LOESS smoothing.
- It iteratively removes daily and weekly cycles, providing a transparent and additive framework for understanding complex seasonal patterns.
- Its high computational efficiency and accuracy make it ideal for industrial applications such as energy, retail, and traffic forecasting compared to traditional methods.
The biseasonal MSTL model refers to the multiple seasonal-trend decomposition using Loess (MSTL) procedure, specialized for time series exhibiting two strong seasonal patterns (biseasonality)—such as daily and weekly cycles in high-frequency data. This methodology extends traditional STL decomposition to jointly model multiple seasonal components, providing a robust and computationally efficient tool for understanding, forecasting, and adjusting time series with complex seasonal structures. Applications span energy, retail, traffic, industrial forecasting, and large-scale enterprise systems.
1. Model Architecture and Decomposition Strategy
The MSTL algorithm generalizes the classical Seasonal-Trend decomposition using Loess (STL) to handle n > 1 seasonalities. For the biseasonal case (n=2), the model is additive:
where:
- : first seasonal component (e.g., daily)
- : second seasonal component (e.g., weekly)
- : trend component
- : remainder (irregular) component
The algorithm iteratively estimates each seasonal component, removing prior seasonalities at each stage via sequential application of STL, followed by trend estimation, and finally computes the remainder. The seasonal periods must be identified (either user-specified or detected by periodogram/FFT) and sorted in increasing order for proper iterative application (2107.13462). The decomposition is strictly additive and all cycles are extracted via local regression (LOESS) smoothing.
2. Algorithmic Implementation and Computational Properties
MSTL builds upon robust and optimized STL routines, leveraging iterative application for each identified seasonality. The workflow is as follows:
1 2 3 4 5 6 7 |
For each defined seasonality period S_j (sorted ascending):
Remove previously estimated seasonal components from the series.
Apply STL to extract current seasonality S_j.
Update the seasonally adjusted series.
After all iterations:
Estimate trend T_t from final seasonally adjusted series.
Compute remainder R_t as data minus all estimated components. |
Computational efficiency is a key property—MSTL is described as “extremely fast, computationally efficient, and scalable to increasing volumes of time series data.” In benchmarks for 100 long (hourly) electricity load time series, MSTL required 7 seconds total, compared to 612 seconds for STR, 936 for Prophet, and 2521 for TBATS (2107.13462). This scalability makes it suitable for industrial deployment and large-scale batch processing (e.g., spark/parallel computing contexts as per (2412.14718, 2506.08113)).
Implementation is available in the R forecast package (mstl() function, supporting the msts class) as well as in Python via the StatsForecast package. Missing values are imputed (na.interp), and optional Box-Cox transformations are supported for variance stabilization.
3. Empirical Performance and Benchmarking
In synthetic and real-world evaluations, the biseasonal MSTL model demonstrates superior or competitive performance for both trend and seasonality extraction compared to additive regression (STR), TBATS, and Prophet. For example, in bootstrapped hourly electricity data from Australia, MSTL achieved the lowest RMSE for trend (207.6), daily seasonality (149.2), and weekly seasonality (180.5), substantially outperforming TBATS and STR (see summary table from (2107.13462)).
In electricity price forecasting (EPF) benchmarks targeting day-ahead auctions for Germany, France, the Netherlands, Austria, and Belgium, the biseasonal MSTL model consistently ranked among the top three models in Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Symmetric Mean Absolute Percentage Error (SMAPE), and was never statistically outperformed by state-of-the-art pre-trained time series foundation models (TSFMs) such as Chronos-Bolt and Time-MoE, as confirmed by Diebold-Mariano (DM) testing (2506.08113). Similar findings are reflected in complex retail and demand forecasting tasks, where MSTL in ensemble frameworks (e.g., Multi-Stage HiFoReAd) contributed substantially to accuracy and robustness against other ML or statistical base models (2412.14718).
4. Applications and Use Cases
The biseasonal MSTL model finds application wherever multiple periodic cycles structure the data:
- Energy and Utilities: Modeling and forecasting daily/weekly demand in electricity, gas, or water, particularly for high-frequency series with layered periodicities (2107.13462, 2506.08113).
- Retail and Commerce: Decomposing traffic, sales, or call volume data that reflect intraday and intraweek cycles.
- Supply Chain and Operations: Supporting hierarchical reconciliation, e.g., in the Multi-Stage HiFoReAd framework, MSTL acts as a base model within Bayesian ensemble optimization, improving Absolute Percentage Error by 10–40% at lower hierarchy levels (2412.14718).
- Time Series Foundation Model Benchmarking: Establishing a robust baseline against neural and transformer-based TSFMs for structured pattern-dominated domains.
Biseasonal MSTL is particularly effective for exploratory data analysis, interpretable reporting, and as a preprocessing step for downstream learning algorithms that benefit from cycle-adjusted input.
5. Comparative and Methodological Context
Compared to alternative techniques:
- TBATS and Prophet: TBATS models seasonality via trigonometric terms and state space methods; Prophet uses additive (and, in some cases, Fourier) components. MSTL provides faster, more transparent, and more accurate extraction of multiple biseasonalities, with lower computational demand and more interpretable components (2107.13462).
- ARIMA/ARMA with Multiple Seasonality: Model-based AR approaches with manually or spectrum-detected periods (as in (2008.12340) and (2409.18640)) offer full-sample inference within a probabilistic time series framework, and are preferable where statistical modeling of uncertainty and adaptivity to regime change are crucial. MSTL excels in preprocessing and extractive settings, while ARMA-type models are ideal for sequential state-space modeling and Bayesian updating (with consideration of parameter stability, shrinking, and time-varying adaptation).
- Neural and Transformer Models: Modern neural models often struggle to recover strong, domain-specific seasonal cycles unless supplied as explicit features. Benchmarks indicate MSTL’s explicit modeling of daily and weekly structure is often essential for achieving or surpassing neural zero-shot performance (2506.08113, 2412.14718).
Empirical studies corroborate that in domains where seasonality is strong, stable, and dominant, the MSTL approach cannot be statistically outperformed by TSFMs on accuracy metrics.
6. Practical Considerations and Implementation Guidance
MSTL requires user (or pre-processing algorithm) specification of all major seasonal periods. These may be pre-detected by FFT, periodogram, or spectrum analysis (2008.12340, 2412.14718). Smoothing parameters (s.window) for each component influence the degree of smoothing versus adaptation to stochastic variation and are tunable via cross-validation or set to default recommendations [appendix, (2107.13462)]. Extension to more than two seasonality periods is possible and practically tractable.
For deployment at scale (e.g., retail, ads demand, bulk electricity price forecasting), MSTL can be used directly or embedded in ensemble and hierarchical reconciliation structures, where it is often blended with ML and additive-seasonal models to robustly leverage both domain-specific patterns and the flexibility of nonparametric learning.
7. Summary Table: Biseasonal MSTL and Benchmark Models
| Model | Seasonality Handling | Efficiency | Interpretability | Benchmark Standing |
|---|---|---|---|---|
| MSTL | Explicit, multi/biseasonal, LOESS | Very fast | High (decomp. additive) | Top-1 or Top-3 on all datasets |
| TBATS | Trigonometric, multiple | Slower | Moderate | Weaker for biseasonal, slower |
| Prophet | Additive, Fourier, holidays | Slow (multi cycle) | Moderate (trend+season) | Subpar for >1 seasonality |
| ARMA/TVSAR | Flexible, probabilistic | Varies | Varies | Optimal for uncertainty quant. |
| TSFM (Chronos) | Implicit (tokenized, learned) | Fast (zero-shot) | Low | Strong, but not superior for biseasonal structure |
MSTL is thus positioned as the preferred statistical solution for time series decomposition and forecasting tasks where two or more stable seasonal cycles are prominent, and where computational performance and interpretability are as important as predictive accuracy.