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Deep Switching State Space Model (DS$^3$M) for Nonlinear Time Series Forecasting with Regime Switching

Xiuqin Xu, Hanqiu Peng, Ying Chen

TL;DR

Experimental results reveal that DS$^3$M outperforms several state-of-the-art models in terms of forecasting accuracy, while providing meaningful regime identifications.

Abstract

Modern time series data often display complex nonlinear dependencies along with irregular regime-switching behaviors. These features present technical challenges in modeling, inference, and in offering insightful understanding into the underlying stochastic phenomena. To tackle these challenges, we introduce a novel modeling framework known as the Deep Switching State Space Model (DS$^3$M). This framework is engineered to make accurate forecasts for such time series while adeptly identifying the irregular regimes hidden within the dynamics. These identifications not only have significant economic ramifications but also contribute to a deeper understanding of the underlying phenomena. In DS$^3$M, the architecture employs discrete latent variables to represent regimes and continuous latent variables to account for random driving factors. By melding a Recurrent Neural Network (RNN) with a nonlinear Switching State Space Model (SSSM), we manage to capture the nonlinear dependencies and irregular regime-switching behaviors, governed by a Markov chain and parameterized using multilayer perceptrons. We validate the effectiveness and regime identification capabilities of DS$^3$M through short- and long-term forecasting tests on a wide array of simulated and real-world datasets, spanning sectors such as healthcare, economics, traffic, meteorology, and energy. Experimental results reveal that DS$^3$M outperforms several state-of-the-art models in terms of forecasting accuracy, while providing meaningful regime identifications.

Deep Switching State Space Model (DS$^3$M) for Nonlinear Time Series Forecasting with Regime Switching

TL;DR

Experimental results reveal that DSM outperforms several state-of-the-art models in terms of forecasting accuracy, while providing meaningful regime identifications.

Abstract

Modern time series data often display complex nonlinear dependencies along with irregular regime-switching behaviors. These features present technical challenges in modeling, inference, and in offering insightful understanding into the underlying stochastic phenomena. To tackle these challenges, we introduce a novel modeling framework known as the Deep Switching State Space Model (DSM). This framework is engineered to make accurate forecasts for such time series while adeptly identifying the irregular regimes hidden within the dynamics. These identifications not only have significant economic ramifications but also contribute to a deeper understanding of the underlying phenomena. In DSM, the architecture employs discrete latent variables to represent regimes and continuous latent variables to account for random driving factors. By melding a Recurrent Neural Network (RNN) with a nonlinear Switching State Space Model (SSSM), we manage to capture the nonlinear dependencies and irregular regime-switching behaviors, governed by a Markov chain and parameterized using multilayer perceptrons. We validate the effectiveness and regime identification capabilities of DSM through short- and long-term forecasting tests on a wide array of simulated and real-world datasets, spanning sectors such as healthcare, economics, traffic, meteorology, and energy. Experimental results reveal that DSM outperforms several state-of-the-art models in terms of forecasting accuracy, while providing meaningful regime identifications.

Paper Structure

This paper contains 21 sections, 1 theorem, 22 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and related works
  3. Deep Switching State Space Model (DS$^3$M)
  4. Model
  5. Estimation of parameters
  6. The ELBO and the approximated posterior
  7. Factorization and approximation of the ELBO
  8. Gradients of the ELBO
  9. Predictive distributions
  10. Theoretical analysis of the DS$^3$M
  11. Experiments
  12. Simulations
  13. Toy example
  14. Lorenz attractor
  15. Real data analysis
  16. ...and 6 more sections

Key Result

Theorem 1

The DS$^3$M is stable, i.e. the latent variable $z_t$ and the observed variable $y_t$ are globally mean-square stable if the 2-norms of all weight matrices and activation scaling matrices are upper bounded by 1.

Figures (3)

  • Figure 1: Deep Switching State Space Model (DS$^3$M)
  • Figure 2: Plots for the two simulated datasets
  • Figure 3: Predictions of the testing data for different datasets

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Theorem 1