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Soft Bayesian Context Tree Models for Real-Valued Time Series

Shota Saito, Yuta Nakahara, Toshiyasu Matsushima

TL;DR

The Soft-BCT considers soft (probabilistic) splits of the context space, instead of hard splits of the context space as in the previous BCT for real-valued time series, which is a novel BCT model for real-valued time series.

Abstract

This paper proposes the soft Bayesian context tree model (Soft-BCT), which is a novel BCT model for real-valued time series. The Soft-BCT considers soft (probabilistic) splits of the context space, instead of hard (deterministic) splits of the context space as in the previous BCT for real-valued time series. A learning algorithm of the Soft-BCT is proposed based on the variational inference. For some real-world datasets, the Soft-BCT demonstrates almost the same or superior performance to the previous BCT.

Soft Bayesian Context Tree Models for Real-Valued Time Series

TL;DR

The Soft-BCT considers soft (probabilistic) splits of the context space, instead of hard splits of the context space as in the previous BCT for real-valued time series, which is a novel BCT model for real-valued time series.

Abstract

This paper proposes the soft Bayesian context tree model (Soft-BCT), which is a novel BCT model for real-valued time series. The Soft-BCT considers soft (probabilistic) splits of the context space, instead of hard (deterministic) splits of the context space as in the previous BCT for real-valued time series. A learning algorithm of the Soft-BCT is proposed based on the variational inference. For some real-world datasets, the Soft-BCT demonstrates almost the same or superior performance to the previous BCT.
Paper Structure (18 sections, 5 theorems, 56 equations, 3 figures, 1 table)

This paper contains 18 sections, 5 theorems, 56 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Soft Bayesian Context Tree Model
  3. Data generative model
  4. Soft splits of context space
  5. Data generative model of real-valued time series
  6. Prior distributions
  7. Variational Inference of Soft-BCT
  8. Problem setup
  9. Variational inference
  10. Update formula of (7) and ((8))
  11. Update formula of (9)
  12. Experiments
  13. Experiment 1: MAP estimation
  14. Experiment 2: prediction
  15. Concluding Remark
  16. ...and 3 more sections

Key Result

Proposition 1

The posterior $q^*(\bm U)$ can be factorized as $q^*(\bm U) = \prod_{t=1}^{n} q^*(\bm U_t)$, and each $q^*(\bm U_t)$ is where $\pi'_{t, s, s_m} \coloneqq \rho_{t, s, s_m}/\sum_{m=1}^M \rho_{t, s, s_m}$, and $\rho_{t, s, s_m}$ is given as and $(\star)$ is given as where $\psi(\cdot)$ denotes the digamma function and $\bm \Lambda'_{s_m}$, $\bm \mu'_{s_m}$, $a'_{s_m}$, $b'_{s_m}$, $g'_{s_m}$ are g

Figures (3)

  • Figure 1: The graphical model of our proposed model. We denote observed variables by shading the corresponding nodes.
  • Figure 2: An example of $\bm U_t$.
  • Figure 3: The MAP estimated model and parameters for unemp

Theorems & Definitions (11)

  • Remark 1
  • Example 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • Proposition 2
  • Remark 5
  • Lemma 1
  • Lemma 2
  • ...and 1 more