Iterative Parameter Estimate with Batched Binary-Valued Observations: Convergence with an Exponential Rate

Abstract

This paper considers the linear system identification with batched binary-valued observations. An iterative parameter estimate algorithm is constructed to achieve the Maximum Likelihood (ML) estimate. The first interesting result is that there exists at most one finite ML solution for this specific maximum likelihood problem, which is induced by the fact that the Hessian matrix of the log-likelihood function is negative definite under binary data and Gaussian system noises. The global concave property and local strongly concave property of the log-likelihood function are obtained. Under mild conditions on the system input, the ML function can be proved to have unique maximum point. The second main result is that the proposed iterative estimate algorithm converges to a fixed vector with an exponential rate which are proved by constructing a Lyapunov function. The more interesting result is that the limit of the iterative algorithm achieves the maximization of the ML function. Numerical simulations are illustrated to support the theoretical results obtained in this paper well.