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Identification of non-causal systems with random switching modes (Extended Version)

Yanxin Zhang, Chengpu Yu, Filippo Fabiani

TL;DR

This work tackles the identification of non-causal switching systems with random switching modes (NCS-RSM), where outputs depend on both historical and future states through two independent switching sequences. It introduces an expectation-maximization framework in which the E-step employs a bidirectional, modified Kalman filter to estimate causal and non-causal states along with MAP-based switching sequences, and the M-step uses a switching least-squares approach to update all subsystem parameters and noise covariances. The authors establish that the EM iterations monotonically increase the likelihood and derive a data-dependent convergence bound with rate $O\left(\sqrt{\log(T)/T}\right)$, under mild stability and martingale-difference noise assumptions. Numerical experiments, including an academic NCS-RSM and a Department Store Inventory Price Index, demonstrate accurate state and mode recovery, competitive parameter estimates, and improved smoothing performance as the number of subsystems grows, highlighting the method’s practical applicability to complex, bidirectional dynamic systems.

Abstract

We consider the identification of non-causal systems with random switching modes (NCSRSM), a class of models essential for describing typical power load management and department store inventory dynamics. The simultaneous identification of causal-andanticausal subsystems, along with the presence of random switching sequences, however, make the overall identification problem particularly challenging. To this end, we develop an expectation-maximization (EM) based system identification technique, where the E-step proposes a modified Kalman filter (KF) to estimate the states and switching sequences of causal-and-anticausal subsystems, while the M-step consists in a switching least-squares algorithm to estimate the parameters of individual subsystems. We establish the main convergence features of the proposed identification procedure, also providing bounds on the parameter estimation errors under mild conditions. Finally, the effectiveness of our identification method is validated through two numerical simulations.

Identification of non-causal systems with random switching modes (Extended Version)

TL;DR

This work tackles the identification of non-causal switching systems with random switching modes (NCS-RSM), where outputs depend on both historical and future states through two independent switching sequences. It introduces an expectation-maximization framework in which the E-step employs a bidirectional, modified Kalman filter to estimate causal and non-causal states along with MAP-based switching sequences, and the M-step uses a switching least-squares approach to update all subsystem parameters and noise covariances. The authors establish that the EM iterations monotonically increase the likelihood and derive a data-dependent convergence bound with rate , under mild stability and martingale-difference noise assumptions. Numerical experiments, including an academic NCS-RSM and a Department Store Inventory Price Index, demonstrate accurate state and mode recovery, competitive parameter estimates, and improved smoothing performance as the number of subsystems grows, highlighting the method’s practical applicability to complex, bidirectional dynamic systems.

Abstract

We consider the identification of non-causal systems with random switching modes (NCSRSM), a class of models essential for describing typical power load management and department store inventory dynamics. The simultaneous identification of causal-andanticausal subsystems, along with the presence of random switching sequences, however, make the overall identification problem particularly challenging. To this end, we develop an expectation-maximization (EM) based system identification technique, where the E-step proposes a modified Kalman filter (KF) to estimate the states and switching sequences of causal-and-anticausal subsystems, while the M-step consists in a switching least-squares algorithm to estimate the parameters of individual subsystems. We establish the main convergence features of the proposed identification procedure, also providing bounds on the parameter estimation errors under mild conditions. Finally, the effectiveness of our identification method is validated through two numerical simulations.

Paper Structure

This paper contains 16 sections, 5 theorems, 45 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Summary of contribution
  4. Paper organization
  5. Mathematical formulation
  6. System model description
  7. Problem statement
  8. The EM method for identifying NCS-RSM
  9. Implementation details of the EM algorithm
  10. The E-step
  11. The M-step
  12. Numerical examples
  13. Example 1: Academic NCS-RSM
  14. Example 2: The Department Store Inventory Price Index
  15. Conclusion
  16. ...and 1 more sections

Key Result

Lemma 1

Given a dataset $\bm{y}$, let $\{\theta^k\}_{k\in\mathbb{Z}}$ be the sequence generated by Algorithm alg:EM. Then, the likelihood function in eq:mdf, evaluated along $\{\theta^k\}_{k\in\mathbb{Z}}$, is non-decreasing, thereby yielding $\ln \mathbb P_{\theta^{k+1}}[\bm{y}]\geq\ln \mathbb P_{\theta^k}

Figures (4)

  • Figure 1: The true (blue cross) and estimated (red circle) mode sequences over a certain time window of length $100$.
  • Figure 2: Dynamical evolution of the true state variables $\bm{x}_c$ and $\bm{x}_a$ (solid blue line), and of the estimated ones $\hat{\bm x}_c$, and $\hat{\bm x}_a$ (dashed red lines).
  • Figure 3: Match rates obtained by the proposed algorithm for different noise levels.
  • Figure 4: The smoothing estimated prices and the true prices with different numbers of the subsystem. (a) $\bm{s}_c=\bm{s}_a$ and $m_c=m_a=1$; (b) $\bm{s}_c\neq \bm{s}_a$ and $m_c=m_a=1$; (c) $\bm{s}_c\neq \bm{s}_a$ and $m_c=m_a=2$

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Remark 3
  • Definition 1
  • Theorem 1
  • Remark 4
  • Lemma 3
  • Lemma 4