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$L_1$-Based Adaptive Identification with Saturated Observations

Xin Zheng, Lei Guo

TL;DR

This work addresses adaptive identification for systems with saturated observations by leveraging a weighted $\ell_1$-norm to enhance robustness. A two-step weighted LAD algorithm (TSWLAD) is proposed, with a projection-based scheme and density-informed gains to cope with saturation and non-PE conditions, and it achieves global convergence of the parameter estimates. Theoretical results establish almost-sure parameter convergence under weak excitation, along with asymptotic upper bounds on averaged regrets and prediction errors, without requiring iid assumptions. Empirical validation via a numerical example and real judicial sentencing data demonstrates that the $\ell_1$-based approach outperforms traditional $\ell_2$-based methods in robustness and predictive accuracy, highlighting the practical impact forcontrol, signal processing, and legal analytics.

Abstract

It is well-known that saturated output observations are prevalent in various practical systems and that the $\ell_1$-norm is more robust than the $\ell_2$-norm-based parameter estimation. Unfortunately, adaptive identification based on both saturated observations and the $\ell_1$-optimization turns out to be a challenging nonlinear problem, and has rarely been explored in the literature. Motivated by this and the need to fit with the $\ell_1$-based index of prediction accuracy in, e.g., judicial sentencing prediction problems, we propose a two-step weighted $\ell_1$-based adaptive identification algorithm. Under certain excitation conditions much weaker than the traditional persistent excitation (PE) condition, we will establish the global convergence of both the parameter estimators and the adaptive predictors. It is worth noting that our results do not rely on the widely used independent and identically distributed (iid) assumptions on the system signals, and thus do not exclude applications to feedback control systems. We will demonstrate the advantages of our proposed new adaptive algorithm over the existing $\ell_2$-based ones, through both a numerical example and a real-data-based sentencing prediction problem.

$L_1$-Based Adaptive Identification with Saturated Observations

TL;DR

This work addresses adaptive identification for systems with saturated observations by leveraging a weighted -norm to enhance robustness. A two-step weighted LAD algorithm (TSWLAD) is proposed, with a projection-based scheme and density-informed gains to cope with saturation and non-PE conditions, and it achieves global convergence of the parameter estimates. Theoretical results establish almost-sure parameter convergence under weak excitation, along with asymptotic upper bounds on averaged regrets and prediction errors, without requiring iid assumptions. Empirical validation via a numerical example and real judicial sentencing data demonstrates that the -based approach outperforms traditional -based methods in robustness and predictive accuracy, highlighting the practical impact forcontrol, signal processing, and legal analytics.

Abstract

It is well-known that saturated output observations are prevalent in various practical systems and that the -norm is more robust than the -norm-based parameter estimation. Unfortunately, adaptive identification based on both saturated observations and the -optimization turns out to be a challenging nonlinear problem, and has rarely been explored in the literature. Motivated by this and the need to fit with the -based index of prediction accuracy in, e.g., judicial sentencing prediction problems, we propose a two-step weighted -based adaptive identification algorithm. Under certain excitation conditions much weaker than the traditional persistent excitation (PE) condition, we will establish the global convergence of both the parameter estimators and the adaptive predictors. It is worth noting that our results do not rely on the widely used independent and identically distributed (iid) assumptions on the system signals, and thus do not exclude applications to feedback control systems. We will demonstrate the advantages of our proposed new adaptive algorithm over the existing -based ones, through both a numerical example and a real-data-based sentencing prediction problem.

Paper Structure

This paper contains 11 sections, 13 theorems, 71 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The main results
  3. Problem formulation
  4. Notations and assumptions
  5. Adaptive identification algorithm
  6. The main theorems
  7. Proofs of the main theorems
  8. Simulation and Experiment
  9. Numerical example
  10. Real-data-based judicial sentencing problem
  11. Conclusion

Key Result

Lemma 1

(Armerin2014). If $\mathbb{E}[|x_{k+1}| ] < \infty$, then the $\mathcal{F}_k$-measurable random variable $\rho$ that minimizes $\mathbb{E}[| x_{k+1} - \rho| ]$ is the conditional median of $x_{k+1}$ specified in Definition mediand.

Figures (6)

  • Figure 1: Comparison of parameter estimation errors for the two algorithms under different $q$ values, $\tilde{\theta}_{10000}$ is the parameter estimation error computed at time $k=10000$.
  • Figure 2: A trajectory of $\frac{1}{n}\sum\limits_{k=1}^{n}R_k$.
  • Figure 3: A trajectory of $E_n$.
  • Figure 4: Comparison of adaptive prediction accuracy in minor injury cases.
  • Figure 5: Comparison of adaptive prediction accuracy in serious injury cases.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Definition 1
  • Remark 3
  • Lemma 1
  • Remark 4
  • Definition 2
  • Remark 5
  • Theorem 1
  • Remark 6
  • ...and 11 more