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TSB-HB: A Hierarchical Bayesian Extension of the TSB Model for Intermittent Demand Forecasting

Zong-Han Bai, Po-Yen Chu

TL;DR

TSB-HB addresses intermittent demand forecasting by integrating the intuitive TSB multiplicative structure with hierarchical Bayesian priors. It models demand occurrence via a Beta-Binomial layer and conditional demand sizes on the log scale via a Normal distribution, enabling partial pooling across items through empirical Bayes updates. The approach yields closed-form updates with linear-time per-epoch forecasting and fully probabilistic forecasts, combining interpretability with calibrated uncertainty. Empirical results on Online Retail and M5 show competitive or superior accuracy and stronger calibration for sparse, lumpy series, demonstrating the practical value of hierarchical shrinkage without sacrificing scalability or transparency.

Abstract

Intermittent demand forecasting poses unique challenges due to sparse observations, cold-start items, and obsolescence. Classical models such as Croston, SBA, and the Teunter-Syntetos-Babai (TSB) method provide simple heuristics but lack a principled generative foundation. Deep learning models address these limitations but often require large datasets and sacrifice interpretability. We introduce TSB-HB, a hierarchical Bayesian extension of TSB. Demand occurrence is modeled with a Beta-Binomial distribution, while nonzero demand sizes follow a Log-Normal distribution. Crucially, hierarchical priors enable partial pooling across items, stabilizing estimates for sparse or cold-start series while preserving heterogeneity. This framework yields a fully generative and interpretable model that generalizes classical exponential smoothing. On the UCI Online Retail dataset, TSB-HB achieves lower RMSE and RMSSE than Croston, SBA, TSB, ADIDA, IMAPA, ARIMA and Theta, and on a subset of the M5 dataset it outperforms all classical baselines we evaluate. The model provides calibrated probabilistic forecasts and improved accuracy on intermittent and lumpy items by combining a generative formulation with hierarchical shrinkage, while remaining interpretable and scalable.

TSB-HB: A Hierarchical Bayesian Extension of the TSB Model for Intermittent Demand Forecasting

TL;DR

TSB-HB addresses intermittent demand forecasting by integrating the intuitive TSB multiplicative structure with hierarchical Bayesian priors. It models demand occurrence via a Beta-Binomial layer and conditional demand sizes on the log scale via a Normal distribution, enabling partial pooling across items through empirical Bayes updates. The approach yields closed-form updates with linear-time per-epoch forecasting and fully probabilistic forecasts, combining interpretability with calibrated uncertainty. Empirical results on Online Retail and M5 show competitive or superior accuracy and stronger calibration for sparse, lumpy series, demonstrating the practical value of hierarchical shrinkage without sacrificing scalability or transparency.

Abstract

Intermittent demand forecasting poses unique challenges due to sparse observations, cold-start items, and obsolescence. Classical models such as Croston, SBA, and the Teunter-Syntetos-Babai (TSB) method provide simple heuristics but lack a principled generative foundation. Deep learning models address these limitations but often require large datasets and sacrifice interpretability. We introduce TSB-HB, a hierarchical Bayesian extension of TSB. Demand occurrence is modeled with a Beta-Binomial distribution, while nonzero demand sizes follow a Log-Normal distribution. Crucially, hierarchical priors enable partial pooling across items, stabilizing estimates for sparse or cold-start series while preserving heterogeneity. This framework yields a fully generative and interpretable model that generalizes classical exponential smoothing. On the UCI Online Retail dataset, TSB-HB achieves lower RMSE and RMSSE than Croston, SBA, TSB, ADIDA, IMAPA, ARIMA and Theta, and on a subset of the M5 dataset it outperforms all classical baselines we evaluate. The model provides calibrated probabilistic forecasts and improved accuracy on intermittent and lumpy items by combining a generative formulation with hierarchical shrinkage, while remaining interpretable and scalable.

Paper Structure

This paper contains 33 sections, 10 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. Setup, Notation, and Assumptions
  5. Assumptions.
  6. TSB-HB: A multiplicative forecast with hierarchical shrinkage
  7. Occurrence (Beta--Binomial, empirical Bayes).
  8. Conditional size on the log scale (Normal--Normal EB).
  9. Relation to TSB.
  10. Deterministic shrinkage properties
  11. Monotonicity and limits.
  12. Bracketing.
  13. Bias--variance decomposition on the log scale.
  14. Identifiability and large-sample guarantees
  15. Computation and Complexity
  16. ...and 18 more sections

Figures (1)

  • Figure 1: Visualization of the hierarchical shrinkage effect. Each point corresponds to a single item (SKU), plotting its per-item Maximum Likelihood Estimate (MLE) on the x-axis against the model's hierarchical Bayesian (HB) posterior estimate on the y-axis. (a) For demand probability $p$, where data is ample, the estimates are highly correlated, indicating weak shrinkage. (b) For conditional demand size, where data is sparser, the shrinkage is more pronounced, pulling unreliable MLEs towards a robust global mean.