Iterative parameter estimate with batched binary-valued observations

Abstract

In this paper, we consider linear system identification with batched binary-valued observations. We constructed an iterative parameter estimate algorithm to achieve the maximum likelihood (ML) estimate. The first interesting result was that there exists at most one finite ML solution for this specific maximum likelihood problem, which was induced by the fact that the Hessian matrix of the log-likelihood function was negative definite under binary data and Gaussian system noises. The global concave property and local strongly concave property of the log-likelihood function were obtained. Under mild conditions on the system input, we proved that the ML function has a unique maximum point. The second main result was that the ML estimate was consistent under persistent excitation inputs, which infers the effectiveness of ML estimate. Finally, the proposed iterative estimate algorithm converged to a fixed vector with an exponential rate that was proved by constructing a Lyapunov function. A more interesting result was that the limit of the iterative algorithm achieved the maximization of the ML function. Numerical simulations are illustrated to support the theoretical results obtained in this paper well.

摘要

创新点

针对给定二值观测样本数据的参数辨识问题, 我们构造了一种迭代算法并进行了相关的理论研究。 首先, 找出了一个似然函数存在极大值点的充分必要条件, 并通过计算对数似然函数的 Hessian 矩阵, 证明了该问题最多只有一个极大似然估计点。 然后, 通过构建Lyapunov函数, 证明了在持续激励条件下, 该迭代算法能达到指数收敛速度, 而且迭代的极限点就是似然函数唯一的最大值点, 这说明了迭代算法具有极强的稳健性。