Instrumental variables system identification with $L^p$ consistency

2025-11-12

AI Summary

To address the inherent bias in least-squares identification of dynamic systems under noise and the lack of exogenous instruments for nonlinear time series, this paper proposes a data-driven endogenous instrumental variable (IV) estimator that constructs instruments solely from lagged observations—requiring no external excitation. Theoretically, we establish, for the first time, finite-sample (L^p) consistency for all (p geq 1) in both discrete- and continuous-time settings, recovering the nonparametric (sqrt{n}) convergence rate. Methodologically, the estimator supports both linearly identifiable modeling and sparse dynamical system learning. Evaluated on a forced Lorenz system, it reduces parameter estimation bias by 200–500× and achieves up to a tenfold reduction in RMSE compared to standard least squares, demonstrating substantial empirical superiority.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Instrumental variables (eliminate the bias that afflicts least-squares identification of dynamical systems through noisy data, yet traditionally relies on external instruments that are seldom available for nonlinear time series data. We propose an IV estimator that synthesizes instruments from the data. We establish finite-sample $L^{p}$ consistency for all $p ge 1$ in both discrete- and continuous-time models, recovering a nonparametric $sqrt{n}$-convergence rate. On a forced Lorenz system our estimator reduces parameter bias by 200x (continuous-time) and 500x (discrete-time) relative to least squares and reduces RMSE by up to tenfold. Because the method only assumes that the model is linear in the unknown parameters, it is broadly applicable to modern sparsity-promoting dynamics learning models.

Problem

• Developing instrumental variables estimator synthesizing instruments from data
• Establishing finite-sample L^p consistency for dynamical systems identification
• Reducing parameter bias and RMSE in nonlinear time series modeling

Innovation

• Synthesizes instruments from noisy data
• Achieves L^p consistency for all p
• Reduces parameter bias by 200-500x