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BASTION: A Bayesian Framework for Trend and Seasonality Decomposition

This paper introduces BASTION, a flexible Bayesian framework that uniquely identifies and robustly decomposes time series into trend and multiple seasonality components while effectively handling abrupt changes, outliers, and time-varying volatility, offering superior uncertainty quantification compared to existing methods.

Original authors: Jason B. Cho, David S. Matteson

Published 2026-01-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jason B. Cho, David S. Matteson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are listening to a complex piece of music. It has a steady bassline (the trend), a repeating melody that comes back every few bars (the seasonality), and sometimes there are sudden screeches from a violin or a drum hit that doesn't fit the rhythm (outliers and noise).

For a long time, statisticians have tried to separate these sounds to understand the song better. But existing tools often struggle: they might smooth out the sudden screeches too much, get confused when the melody changes speed, or fail to tell you how sure they are about their guess.

The authors of this paper, Jason B. Cho and David S. Matteson, introduce a new tool called BASTION (Bayesian Adaptive Seasonality and Trend DecompositION). Think of BASTION as a highly intelligent, flexible audio engineer who can separate the music into its purest parts with incredible precision.

Here is how BASTION works, using simple analogies:

1. The "Smart Filter" (Global-Local Shrinkage)

Imagine you are trying to clean a muddy window.

Old methods are like using a giant, stiff brush. If you scrub too hard, you wipe away the beautiful picture behind the glass (the real signal). If you scrub too lightly, the mud stays (the noise).
BASTION uses a "smart filter." It has a global setting that keeps the whole window clean, but it also has local sensors. If it detects a sudden, sharp smudge (an abrupt change in the trend), it knows to scrub harder right there without ruining the rest of the picture. If it sees a tiny speck of dust (noise), it gently ignores it. This allows it to keep the smooth curves of the song while still capturing sudden, dramatic shifts.

2. The "Math Detective" (Solving the Puzzle of Identity)

When you try to separate the bassline from the melody, sometimes they look so similar that you can't tell which is which. In math, this is called an "identifiability" problem.

The authors acted like detectives. They wrote down strict rules (mathematical proofs) to ensure that the "bassline" (trend) and the "melody" (seasonality) are truly unique and don't overlap in a way that confuses the computer.
They discovered that if you have multiple melodies playing at different speeds (multiple seasonalities), you need a special combination of rules to keep them distinct. BASTION is the first to formally prove exactly how to set these rules so the answer is always unique.

3. The "Safety Net" (Uncertainty Quantification)

Most old tools give you a single number and say, "This is the trend." They don't tell you if they are guessing or if they are sure.

BASTION is different. It gives you a range of confidence. It's like a weather forecast that doesn't just say "It will rain," but says, "It will rain, and we are 95% sure it will be between 1 and 2 inches." This helps users know exactly how much trust to put in the results.

4. Handling the "Wild Cards" (Outliers and Volatility)

Real-world data is messy. Sometimes a sensor breaks, or a rare event happens (like a pandemic causing airline traffic to crash).

Outliers: BASTION has a special "outlier detector." Instead of trying to force a weird data point to fit the pattern, it recognizes it as a "wild card" and sets it aside, so it doesn't ruin the rest of the analysis.
Changing Noise: Sometimes the background noise gets louder or quieter over time (like a radio that gets more static during a storm). BASTION can detect these changing levels of "static" (volatility) and adjust its hearing accordingly, whereas other tools assume the static is always the same.

How They Tested It

The authors didn't just talk about the theory; they put BASTION to the test:

The Simulation Lab: They created fake time-series data with known "secret ingredients" (like sudden jumps, weird noise, and multiple repeating patterns). BASTION consistently found the hidden ingredients more accurately than other popular tools (like TBATS, MSTL, and STR).
Real-World Tests:
- Airline Traffic: They looked at US airline data. When the pandemic hit in 2020, traffic dropped sharply. BASTION correctly identified this as a sudden, sharp break in the trend, while other tools tried to smooth it out as a slow decline.
- Electricity Demand: They analyzed New York's power usage. BASTION not only separated the daily and yearly patterns but also figured out that the "noise" in the data gets more volatile (unpredictable) in the summer and winter, providing extra insights for energy managers.

The Bottom Line

BASTION is a new, flexible framework that breaks down time-series data into its trend and seasonal parts. It is better at handling sudden changes, ignoring weird outliers, and dealing with changing noise levels than previous methods. Crucially, it tells you how confident it is in its answers. The authors have made this tool available as a free software package for others to use.

Technical Summary: BASTION – A Bayesian Framework for Trend and Seasonality Decomposition

Problem Statement
Time-series decomposition is a fundamental analytical tool used to isolate trend, seasonality, and remainder components from univariate data. While widely applied in economics, climate science, and epidemiology, existing decomposition methods often treat the problem as a preprocessing step rather than a rigorous statistical inference task. Current approaches, such as STL, MSTL, STR, and TBATS, face several limitations: they struggle to accommodate multiple seasonal patterns simultaneously, lack principled uncertainty quantification for individual components, are sensitive to abrupt changes in trends, and generally fail to model heteroskedastic noise or outliers explicitly. Furthermore, the literature often lacks formal conditions under which trend and seasonal components are uniquely identifiable, particularly when multiple seasonalities are present.

Methodology
The authors propose BASTION (Bayesian Adaptive Seasonality and Trend DecompositON), a flexible Bayesian framework that reframes time-series decomposition as a penalized nonparametric regression problem.

Penalized Regression Formulation: The decomposition is modeled as an additive equation $y_t = T_t + \sum S_{i,t} + R_t$ . The authors establish that without structural constraints, this high-dimensional problem is not identifiable. They formalize identifiability conditions by imposing linear operators (e.g., second-difference for trend, seasonal recurrence or differencing for seasonality) and analyzing the rank of the resulting block matrix. They prove that unique identifiability requires the kernels of these operators to be linearly independent subspaces. To resolve non-identifiability in multi-seasonal settings, they propose using multiple regularization operators (e.g., combining second-difference and seasonal differencing) to reduce the intersection of null spaces.
Bayesian Framework with Global-Local Shrinkage: BASTION casts the penalized regression into a Bayesian model using global-local shrinkage priors (specifically the horseshoe prior). This allows for:
- Adaptive Smoothing: A global parameter controls overall shrinkage, while time-varying local parameters selectively relax shrinkage to capture abrupt shifts in trends and seasonality without oversmoothing.
- Explicit Outlier Modeling: An outlier component ( $\zeta_t$ ) is modeled using a horseshoe+ prior, which induces aggressive shrinkage for most observations while allowing large deviations at isolated time points.
- Heteroskedasticity Handling: The remainder term incorporates a stochastic volatility (SV) model, allowing the error variance to evolve over time, addressing the common issue of changing volatility in real-world data.
Inference: Due to the high dimensionality of the state variables, the authors employ a Gibbs sampling strategy. This approach utilizes conjugate updates for most parameters and specific algorithms (such as the All Without a Loop algorithm for stochastic volatility and parameter-expanded Gibbs schemes for horseshoe priors) to ensure linear scaling with the time-series length, making it computationally feasible for large datasets.

Key Contributions

Formal Identifiability Theory: The paper provides the first formal conditions for the unique identifiability of trend and seasonal components under penalized regression, specifically addressing the complexities introduced by multiple seasonalities and different operator choices.
Unified Bayesian Framework: BASTION integrates trend, multiple seasonalities, outliers, and time-varying volatility into a single coherent model, a capability lacking in existing methods like MSTL or STR.
Robustness and Uncertainty Quantification: By leveraging global-local shrinkage and stochastic volatility, the method offers robust estimation against outliers and heteroskedasticity while providing calibrated posterior credible intervals for all components.
Software Implementation: The authors release BASTION as an open-source R package, facilitating practical application and further research.

Results
The authors evaluate BASTION against established methods (TBATS, MSTL, STR) using both simulated data and real-world datasets.

Simulation Studies: Across four Data Generating Processes (DGPs) featuring trend discontinuities, multiple seasonalities, heteroskedasticity, and outliers, BASTION consistently achieved the lowest Mean Squared Error (MSE) for trend, seasonality, and signal components. Notably, in scenarios with heteroskedastic noise and outliers (DGP 4), BASTION significantly outperformed competitors.
Uncertainty Quantification: BASTION provided well-calibrated 95% credible intervals, maintaining coverage above the nominal level for both combined signals and individual components. In contrast, the frequentist STR method frequently undercovered, particularly for individual components.
Empirical Applications:
- U.S. Airline Traffic: BASTION successfully captured the sharp, structural break in passenger volume during the COVID-19 pandemic, whereas MSTL and STR oversmoothed the transition, and TBATS produced noisy estimates.
- New York Electricity Demand: The model accurately decomposed complex weekly and yearly seasonal patterns and explicitly estimated time-varying volatility, revealing higher volatility during summer and winter months—a feature no other decomposition model in the comparison provided.

Significance and Claims
The paper claims that BASTION represents a significant advancement by addressing the "informal" treatment of identifiability in existing literature and by providing a unified framework that handles complex dynamics (abrupt changes, heteroskedasticity, outliers) simultaneously. The authors assert that BASTION delivers more nuanced and interpretable decompositions with rigorous uncertainty quantification, making it particularly suitable for downstream tasks like forecasting in environments with structural breaks. They conclude that while the method is robust, future work could explore how these improved decompositions translate directly to forecasting performance and the incorporation of additional latent structures like Bayesian change-point detection.