Joint Identification of Linear Dynamics and Noise Covariance via Distributional Estimation

In this paper, we propose a novel framework for the joint identification of system dynamics and noise covariance in linear systems, under general noise distributions beyond Gaussian. Specifically, we would like to simultaneously estimate the dynamical matrix $A$ and the noise covariance matrix $\varSigma$ using state transition data. The formulation builds upon a novel parameterization of the state-transition distribution, which enables more effective use of distributional "shape" information for improved identification accuracy. We introduce two practical estimators, namely the maximum likelihood estimator (MLE) and the score-matching estimator (SME), to solve the joint dynamics-covariance identification problem, and provide rigorous analysis of their statistical properties and sample complexity. Simulation results show that the proposed estimators outperform the ordinary least squares (OLS) baseline.