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BASTION: A Bayesian Framework for Trend and Seasonality Decomposition

Jason B. Cho, David S. Matteson

TL;DR

BASTION tackles the challenge of robustly decomposing time series into trend and multiple seasonal components with principled uncertainty. It combines a penalized regression foundation for identifiability with a Bayesian framework using global–local shrinkage priors, explicit outlier modeling via horseshoe+ priors, and a stochastic-volatility extension for heteroskedastic residuals. The approach yields accurate decompositions and well-calibrated posterior uncertainty, performing favorably against TBATS, STR, and MSTL in simulations and real-world datasets such as airline traffic and electricity demand. Its practical impact lies in providing a flexible, interpretable tool for complex temporal dynamics, available as an R package for broad use.

Abstract

We introduce BASTION (Bayesian Adaptive Seasonality and Trend DecompositION), a flexible Bayesian framework for decomposing time series into trend and multiple seasonality components. We cast the decomposition as a penalized nonparametric regression and establish formal conditions under which the trend and seasonal components are uniquely identifiable, an issue only treated informally in the existing literature. BASTION offers three key advantages over existing decomposition methods: (1) accurate estimation of trend and seasonality amidst abrupt changes, (2) enhanced robustness against outliers and time-varying volatility, and (3) robust uncertainty quantification. We evaluate BASTION against established methods, including TBATS, STR, and MSTL, using both simulated and real-world datasets. By effectively capturing complex dynamics while accounting for irregular components such as outliers and heteroskedasticity, BASTION delivers a more nuanced and interpretable decomposition. To support further research and practical applications, BASTION is available as an R package at https://github.com/Jasoncho0914/BASTION

BASTION: A Bayesian Framework for Trend and Seasonality Decomposition

TL;DR

BASTION tackles the challenge of robustly decomposing time series into trend and multiple seasonal components with principled uncertainty. It combines a penalized regression foundation for identifiability with a Bayesian framework using global–local shrinkage priors, explicit outlier modeling via horseshoe+ priors, and a stochastic-volatility extension for heteroskedastic residuals. The approach yields accurate decompositions and well-calibrated posterior uncertainty, performing favorably against TBATS, STR, and MSTL in simulations and real-world datasets such as airline traffic and electricity demand. Its practical impact lies in providing a flexible, interpretable tool for complex temporal dynamics, available as an R package for broad use.

Abstract

We introduce BASTION (Bayesian Adaptive Seasonality and Trend DecompositION), a flexible Bayesian framework for decomposing time series into trend and multiple seasonality components. We cast the decomposition as a penalized nonparametric regression and establish formal conditions under which the trend and seasonal components are uniquely identifiable, an issue only treated informally in the existing literature. BASTION offers three key advantages over existing decomposition methods: (1) accurate estimation of trend and seasonality amidst abrupt changes, (2) enhanced robustness against outliers and time-varying volatility, and (3) robust uncertainty quantification. We evaluate BASTION against established methods, including TBATS, STR, and MSTL, using both simulated and real-world datasets. By effectively capturing complex dynamics while accounting for irregular components such as outliers and heteroskedasticity, BASTION delivers a more nuanced and interpretable decomposition. To support further research and practical applications, BASTION is available as an R package at https://github.com/Jasoncho0914/BASTION
Paper Structure (35 sections, 5 theorems, 74 equations, 7 figures, 3 tables)

This paper contains 35 sections, 5 theorems, 74 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Decomposition as a Penalized Regression Problem
  4. Identifiability
  5. Multiple Regularization Operators
  6. Bayesian Perspective
  7. Global-Local Shrinkage Prior
  8. Explicit Outlier Detection
  9. Handling Heteroskedasticity
  10. Implementation
  11. Simulation Study
  12. Set Up
  13. Results
  14. Empirical Data Analysis
  15. U.S. Airline Traffic Data
  16. ...and 20 more sections

Key Result

Theorem 1

The solution in Equation obj1 is uniquely identifiable if and only if Equivalently, $\ker(D_T),\ker(D_{S,1}),\dots,\ker(D_{S,P})$ are linearly independent subspaces.

Figures (7)

  • Figure 1: Observed data (points); estimated trend + seasonality, ($\hat{\boldsymbol{T}} + \hat{\boldsymbol{S}}$), (black) with 95% credible regions; and trend ($\hat{\boldsymbol{T}}$) component alone (red), for monthly U.S. international airline traffic, 2016 - 2023. The sharp decline in early 2020 reflects COVID-19 travel restrictions.
  • Figure 2: Daily average electricity demand in New York State from 2015-07-01 to 2024-06-30.
  • Figure 3: Yearly seasonality estimates, ($\hat{\boldsymbol{S}}$), for daily average electricity demand in New York from July 2015 to June 2017. Figures (a) - (d) display estimates from TBATS, MSTL, STR, and BASTION, respectively.
  • Figure 4: Volatility estimate based on BASTION for daily average electricity demand from 2015-07-01 to 2024-06-30. 95% credible regions are drawn in dark grey.
  • Figure 5: Synthetic time series generated by adding a trend component, $T_{t}$, seasonal components, $S_{t}$, and remainders $R_{t}$ based on the descriptions in Table \ref{['tab:simulation_descriptions']}. $T_{t}$ are drawn in blue and $T_{t} + S_{t}$ are drawn in black. Figures represent one replication of each DGP.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 1: Identifiability Condition
  • proof
  • Corollary 1.1: Single seasonality
  • proof
  • Corollary 1.2: Multiple seasonalities
  • proof
  • Theorem 2: Identifiability with Multiple Penalties
  • proof
  • Corollary 2.1: Mixed differencing 1
  • proof