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A Local Gaussian Process Regression Approach to Frequency Response Function Estimation

Xiaozhu Fang, Yu Xu, Tianshi Chen

TL;DR

This paper tackles frequency response function estimation in a local setting by introducing Local Gaussian Process Regression (LGPR), which embeds prior knowledge that the local FRF is analytic or resonant. Building on the Local Regularized Polynomial Method (LRPM), LGPR models local FRF blocks as Gaussian processes and learns hyperparameters via empirical Bayes, using kernel designs that enforce analytic decay (dot-product kernel) and resonance features (second-order resonance kernel). Through simulations on a benchmark FRF, LGPR with a DPpR1 kernel (dot-product plus one resonance) achieves the best MSE across varying sample sizes and noise levels, demonstrating robustness to model-order choices and light damping. The work offers a new route to address model-order selection and lightly damped systems in FRF estimation, with potential extensions in kernel design and broader frequency-domain system identification applications.

Abstract

Frequency response function (FRF) estimation is a classical subject in system identification. In the past two decades, there have been remarkable advances in developing local methods for this subject, e.g., the local polynomial method, local rational method, and iterative local rational method. The recent concentrations for local methods are two issues: the model order selection and the identification of lightly damped systems. To address these two issues, we propose a new local method called local Gaussian process regression (LGPR). We show that the frequency response function locally is either analytic or resonant, and this prior knowledge can be embedded into a kernel-based regularized estimate through a dot-product kernel plus a resonance kernel induced by a second-order resonant system. The LGPR provides a new route to tackle the aforementioned issues. In the numerical simulations, the LGPR shows the best FRF estimation accuracy compared with the existing local methods, and moreover, the LGPR is more robust with respect to sample size and noise level.

A Local Gaussian Process Regression Approach to Frequency Response Function Estimation

TL;DR

This paper tackles frequency response function estimation in a local setting by introducing Local Gaussian Process Regression (LGPR), which embeds prior knowledge that the local FRF is analytic or resonant. Building on the Local Regularized Polynomial Method (LRPM), LGPR models local FRF blocks as Gaussian processes and learns hyperparameters via empirical Bayes, using kernel designs that enforce analytic decay (dot-product kernel) and resonance features (second-order resonance kernel). Through simulations on a benchmark FRF, LGPR with a DPpR1 kernel (dot-product plus one resonance) achieves the best MSE across varying sample sizes and noise levels, demonstrating robustness to model-order choices and light damping. The work offers a new route to address model-order selection and lightly damped systems in FRF estimation, with potential extensions in kernel design and broader frequency-domain system identification applications.

Abstract

Frequency response function (FRF) estimation is a classical subject in system identification. In the past two decades, there have been remarkable advances in developing local methods for this subject, e.g., the local polynomial method, local rational method, and iterative local rational method. The recent concentrations for local methods are two issues: the model order selection and the identification of lightly damped systems. To address these two issues, we propose a new local method called local Gaussian process regression (LGPR). We show that the frequency response function locally is either analytic or resonant, and this prior knowledge can be embedded into a kernel-based regularized estimate through a dot-product kernel plus a resonance kernel induced by a second-order resonant system. The LGPR provides a new route to tackle the aforementioned issues. In the numerical simulations, the LGPR shows the best FRF estimation accuracy compared with the existing local methods, and moreover, the LGPR is more robust with respect to sample size and noise level.
Paper Structure (20 sections, 7 theorems, 32 equations, 1 figure, 1 table)

This paper contains 20 sections, 7 theorems, 32 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background and Problem Formulation
  3. Frequency Response Function Estimation
  4. Existing Local Methods
  5. Problem Statement
  6. Local Regularized Polynomial Method
  7. Bayesian/Regularized Estimate
  8. Kernel Design for the LRPM
  9. Finding True $\theta_k$
  10. Diagonal Kernel
  11. What if Assumption \ref{['as:analytic']} is invalid?
  12. Local Gaussian Process Regression
  13. The LGPR: an Extension of the LRPM
  14. Kernel Design from the LRPM to the LGPR
  15. Kernel Design for Lightly Damped Systems
  16. ...and 5 more sections

Key Result

Proposition 3.1

Consider eq:linear_regre_model with eq:prior_V, where $V_k$ is independent of $\theta_k$. Given the observation $\widetilde{Y}_k =[Y_k^T\ Y_k^H]^T$, the MAP estimate of $\widetilde{\theta}_k = [\theta_k^T\ \theta_k^H]^T$ is where $H$ denotes the Hermitian transpose.

Figures (1)

  • Figure 1: The residual of the FRF estimates (upper panels) and the noise variance estimates (bottom panels), where the dashed lines are the true FRF and true noise variance.

Theorems & Definitions (7)

  • Proposition 3.1: MAP Estimate for $\theta_k$
  • Theorem 3.1
  • Corollary 3.1
  • Proposition 4.1: MAP Estimate for $G_k$
  • Proposition 4.2: Dot-Product Kernel
  • Proposition 4.3: DCpR1 Kernel
  • Proposition 4.4: DPpR1 Kernel