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Variable Splitting Binary Tree Models Based on Bayesian Context Tree Models for Time Series Segmentation

Yuta Nakahara, Shota Saito, Kohei Horinouchi, Koshi Shimada, Naoki Ichijo, Manabu Kobayashi, Toshiyasu Matsushima

TL;DR

The paper tackles offline time-series segmentation with an unknown number of changes by introducing Variable Splitting Binary Tree (VSBT) models, where time-domain interval partitioning is encoded as a binary rooted tree with adjustable split points realized through recursive logistic regression. It combines a CTW-inspired prior over trees with local variational inference for the logistic components, enabling simultaneous estimation of split locations and tree depth, as well as uncertainty quantification for change points. Key contributions include the VSBT model formulation, a variational Bayesian algorithm with explicit update rules for $q(\mathbf u)$, $q(\mathbf z, T)$, and $q(\boldsymbol{\theta}, \boldsymbol{\tau}, \boldsymbol{\pi}, \boldsymbol{\beta})$, and a practical initialization scheme validated on synthetic data showing more compact representations and credible change-point uncertainty. The approach offers a principled, interpretable framework for time-domain segmentation with quantified uncertainty, useful for applications in finance, biology, and climatology.

Abstract

We propose a variable splitting binary tree (VSBT) model based on Bayesian context tree (BCT) models for time series segmentation. Unlike previous applications of BCT models, the tree structure in our model represents interval partitioning on the time domain. Moreover, interval partitioning is represented by recursive logistic regression models. By adjusting logistic regression coefficients, our model can represent split positions at arbitrary locations within each interval. This enables more compact tree representations. For simultaneous estimation of both split positions and tree depth, we develop an effective inference algorithm that combines local variational approximation for logistic regression with the context tree weighting (CTW) algorithm. We present numerical examples on synthetic data demonstrating the effectiveness of our model and algorithm.

Variable Splitting Binary Tree Models Based on Bayesian Context Tree Models for Time Series Segmentation

TL;DR

The paper tackles offline time-series segmentation with an unknown number of changes by introducing Variable Splitting Binary Tree (VSBT) models, where time-domain interval partitioning is encoded as a binary rooted tree with adjustable split points realized through recursive logistic regression. It combines a CTW-inspired prior over trees with local variational inference for the logistic components, enabling simultaneous estimation of split locations and tree depth, as well as uncertainty quantification for change points. Key contributions include the VSBT model formulation, a variational Bayesian algorithm with explicit update rules for , , and , and a practical initialization scheme validated on synthetic data showing more compact representations and credible change-point uncertainty. The approach offers a principled, interpretable framework for time-domain segmentation with quantified uncertainty, useful for applications in finance, biology, and climatology.

Abstract

We propose a variable splitting binary tree (VSBT) model based on Bayesian context tree (BCT) models for time series segmentation. Unlike previous applications of BCT models, the tree structure in our model represents interval partitioning on the time domain. Moreover, interval partitioning is represented by recursive logistic regression models. By adjusting logistic regression coefficients, our model can represent split positions at arbitrary locations within each interval. This enables more compact tree representations. For simultaneous estimation of both split positions and tree depth, we develop an effective inference algorithm that combines local variational approximation for logistic regression with the context tree weighting (CTW) algorithm. We present numerical examples on synthetic data demonstrating the effectiveness of our model and algorithm.
Paper Structure (18 sections, 4 theorems, 57 equations, 4 figures)

This paper contains 18 sections, 4 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proposed Model
  3. Data generative model
  4. Variable splitting
  5. Assignment of autoregressive model
  6. Generation of data points
  7. Prior distributions
  8. Learning Algorithm
  9. Problem setup
  10. Variational inference with local variational approximation
  11. Update formula of (13)
  12. Update formula of (14)
  13. Update formula of (15)
  14. Initialization
  15. Numerical Experiments
  16. ...and 3 more sections

Key Result

Lemma 1

The posterior $q(\bm u)$ can be factorized as $\prod_{t=1}^{n} q(\bm u_t)$, and each $q(\bm u_t)$ is given as where and

Figures (4)

  • Figure 1: The graphical model of our proposed model. We denote observed variables by shading the corresponding nodes.
  • Figure 2: The segmentation under the FSBT model
  • Figure 3: The segmentation under the VSBT model
  • Figure 4: The segmentation and the posterior probabilities of the changes

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4