A Second-Order Generalization of TC and DC Kernels

• 成果类型: Article
• 署名作者: Zorzi, Mattia
• 署名单位: University of Padua
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2023.3337056
• 发表日期: 2024
• 页码: 3835-3848
• 关键词: Kernel; Splines (mathematics); Gaussian Processes; closed-form solutions; Symmetric matrices; predictive models; STANDARDS; Covariance extension; Gaussian process; kernel methods; Maximum entropy; System identification

摘要

Kernel-based methods have been successfully introduced in system identification to estimate the impulse response of a linear system. Adopting the Bayesian viewpoint, the impulse response is modeled as a zero mean Gaussian process whose covariance function (kernel) is estimated from the data. The most popular kernels used in system identification are the tuned-correlated (TC), the diagonal-correlated (DC) and the stable spline (SS) kernel. TC and DC kernels admit a closed-form factorization of the inverse. The SS kernel induces more smoothness than TC and DC on the estimated impulse response, however, the aforementioned property does not hold in this case. In this paper we propose a second-order extension of the TC and DC kernel, which induces more smoothness than TC and DC, respectively, on the impulse response and a generalized-correlated kernel, which incorporates the TC and DC kernels and their second order extensions. Moreover, these generalizations admit a closed-form factorization of the inverse and thus they allow to design efficient algorithms for the search of the optimal kernel hyperparameters. We also show how to use this idea to develop higher oder extensions. Interestingly, these new kernels belong to the family of the so called exponentially convex local stationary kernels: such a property allows to immediately analyze the frequency properties induced on the estimated impulse response by these kernels.