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Recursive identification with regularization and on-line hyperparameters estimation

Bernard Vau, Tudor-Bogdan Airimitoaie

TL;DR

The paper addresses online identification under regularization by simultaneously estimating the impulse response and the kernel hyperparameters using a kernel-based prior and Marginal Likelihood (Empirical Bayes) optimization. It couples a recursive, RLS-like update for the impulse response with online gradient-based updates of the hyperparameters, leveraging a TC-inspired prior that becomes a DI kernel after a basis change. The proposed method shows improved early performance and faster convergence compared with standard Recursive Least Squares (RLS) in simulations, demonstrating practical benefits for real-time system identification. The work contributes a feasible online scheme for adapting both model and prior beliefs in real time, with potential impact on ill-conditioned online identification tasks.

Abstract

This paper presents a regularized recursive identification algorithm with simultaneous on-line estimation of both the model parameters and the algorithms hyperparameters. A new kernel is proposed to facilitate the algorithm development. The performance of this novel scheme is compared with that of the recursive least squares algorithm in simulation.

Recursive identification with regularization and on-line hyperparameters estimation

TL;DR

The paper addresses online identification under regularization by simultaneously estimating the impulse response and the kernel hyperparameters using a kernel-based prior and Marginal Likelihood (Empirical Bayes) optimization. It couples a recursive, RLS-like update for the impulse response with online gradient-based updates of the hyperparameters, leveraging a TC-inspired prior that becomes a DI kernel after a basis change. The proposed method shows improved early performance and faster convergence compared with standard Recursive Least Squares (RLS) in simulations, demonstrating practical benefits for real-time system identification. The work contributes a feasible online scheme for adapting both model and prior beliefs in real time, with potential impact on ill-conditioned online identification tasks.

Abstract

This paper presents a regularized recursive identification algorithm with simultaneous on-line estimation of both the model parameters and the algorithms hyperparameters. A new kernel is proposed to facilitate the algorithm development. The performance of this novel scheme is compared with that of the recursive least squares algorithm in simulation.
Paper Structure (14 sections, 1 theorem, 54 equations, 5 figures)

This paper contains 14 sections, 1 theorem, 54 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Classical propagation equations for recursive estimation
  4. Case where the regularization matrix is no longer invariant
  5. Marginal Likelihood in a recursive context
  6. Prior structure and hyperparameters estimation
  7. Prior structure
  8. On-line hyperparameters estimation
  9. Overall algorithm
  10. Update of the adaptation gain an estimated parameters vector
  11. Summary
  12. Simulation results
  13. Concluding remarks
  14. Proof of Lemma \ref{['lem1']}

Key Result

Lemma 1

The derivative of $\log L(\eta(t+1) |Y(t+1))$ with respect to the $k$ entry of $\eta(t+1)$ (denoted as $\eta_k(t+1)$) is

Figures (5)

  • Figure 1: Impulse response of $G_o$.
  • Figure 2: Comparison with recursive least squares with and without regularization ($\eta_1(0)=\log(0.001)$).
  • Figure 3: Evolution of the hyperparameters ($\eta_1(0)=\log(0.001)$).
  • Figure 4: Comparison with recursive least squares with and without regularization ($\eta_1(0)=\log(0.1)$).
  • Figure 5: Evolution of the hyperparameters ($\eta_1(0)=\log(0.1)$).

Theorems & Definitions (1)

  • Lemma 1