Maximum entropy properties of discrete-time first-order stable spline kernel

The first order stable spline (SS-1) kernel is used extensively in regularized system identification. In particular, the stable spline estimator models the impulse response as a zero-mean Gaussian process whose covariance is given by the SS-1 kernel. In this paper, we discuss the maximum entropy properties of this prior. In particular, we formulate the exact maximum entropy problem solved by the SS-1 kernel without Gaussian and uniform sampling assumptions. Under general sampling schemes, we also explicitly derive the special structure underlying the SS-1 kernel (e.g. characterizing the tridiagonal nature of its inverse), also giving to it a maximum entropy covariance completion interpretation. Along the way similar maximum entropy properties of the Wiener kernel are also given

Maximum Entropy Kernels for System Identification

A new nonparametric approach for system identification has been recently proposed where the impulse response is modeled as the realization of a zero-mean Gaussian process whose covariance (kernel) has to be estimated from data. In this scheme, quality of the estimates crucially depends on the parametrization of the covariance of the Gaussian process. A family of kernels that have been shown to be particularly effective in the system identification framework is the family of Diagonal/Correlated (DC) kernels. Maximum entropy properties of a related family of kernels, the Tuned/Correlated (TC) kernels, have been recently pointed out in the literature. In this paper we show that maximum entropy properties indeed extend to the whole family of DC kernels. The maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the DC kernel. In particular, we prove that the DC kernel admits a closed-form factorization, inverse and determinant. These results can be exploited both to improve the numerical stability and to reduce the computational complexity associated with the computation of the DC estimator.Comment: Extends results of 2014 IEEE MSC Conference Proceedings (arXiv:1406.5706

Robust EM kernel-based methods for linear system identification

Recent developments in system identification have brought attention to regularized kernel-based methods. This type of approach has been proven to compare favorably with classic parametric methods. However, current formulations are not robust with respect to outliers. In this paper, we introduce a novel method to robustify kernel-based system identification methods. To this end, we model the output measurement noise using random variables with heavy-tailed probability density functions (pdfs), focusing on the Laplacian and the Student's t distributions. Exploiting the representation of these pdfs as scale mixtures of Gaussians, we cast our system identification problem into a Gaussian process regression framework, which requires estimating a number of hyperparameters of the data size order. To overcome this difficulty, we design a new maximum a posteriori (MAP) estimator of the hyperparameters, and solve the related optimization problem with a novel iterative scheme based on the Expectation-Maximization (EM) method. In presence of outliers, tests on simulated data and on a real system show a substantial performance improvement compared to currently used kernel-based methods for linear system identification.Comment: Accepted for publication in Automatic

Efficient Multidimensional Regularization for Volterra Series Estimation

This paper presents an efficient nonparametric time domain nonlinear system identification method. It is shown how truncated Volterra series models can be efficiently estimated without the need of long, transient-free measurements. The method is a novel extension of the regularization methods that have been developed for impulse response estimates of linear time invariant systems. To avoid the excessive memory needs in case of long measurements or large number of estimated parameters, a practical gradient-based estimation method is also provided, leading to the same numerical results as the proposed Volterra estimation method. Moreover, the transient effects in the simulated output are removed by a special regularization method based on the novel ideas of transient removal for Linear Time-Varying (LTV) systems. Combining the proposed methodologies, the nonparametric Volterra models of the cascaded water tanks benchmark are presented in this paper. The results for different scenarios varying from a simple Finite Impulse Response (FIR) model to a 3rd degree Volterra series with and without transient removal are compared and studied. It is clear that the obtained models capture the system dynamics when tested on a validation dataset, and their performance is comparable with the white-box (physical) models

Design of nonlinear controllers through the virtual reference method and regularization

This work proposes a new extension for the nonlinear formulation of the data-driven control method known as the Nonlinear Virtual Reference Feedback Tuning. When the process to be controlled contains a significant quantity of noise, the standard Nonlinear VRFT approach – that uses the Least Squares method – yield estimates with poor statistical properties. These properties may lead the control system to undesirable closed loop performances and even instability. With the intention to improve these statistical properties and controller sparsity and hence, the system’s closed loop performance, this work proposes the use of ℓ1 regularization on the nonlinear formulation of the VRFT method. Regularization is a component that has been extensively employed and researched in the Machine Learning and System Identification communities lately. Furthermore, this technique is appropriate to reduce the variance in the estimates. A detailed analysis of the noise effect on the estimate is made for the Nonlinear VRFT method. Finally, three different regularization methods, the third one proposed in this work, are compared to the standard Nonlinear VRFT.Este trabalho propõe uma nova extensão para a formulação não linear do método de controle orientado por dados conhecido como Método da Referência Virtual Não Linear, ou Nonlinear Virtual Reference Feedback Tuning – denominado aqui somente como VRFT. Quando o processo a ser controlado contém uma quantidade significativa de ruído, a abordagem padrão do VRFT – que usa o método dos Mínimos Quadrados – fornece estimativas com propriedades estatísticas pobres. Essas propriedades podem levar o sistema de controle a desempenhos indesejáveis em malha fechada. Com a intenção de melhorar essas propriedades estatística, identificar um controlador simples em quantidade de parâmetros e melhorar o desempenho em malha fechada do sistema, este trabalho propõe o uso da regularização ℓ1 na formulação não linear do método VRFT. A regularização é uma técnica que tem sido amplamente empregada e pesquisada nas comunidades de Aprendizagem de Máquina e Identificação de Sistemas ultimamente. Além disso, esta técnica é apropriada para reduzir a variância das estimativas. Uma análise detalhada do efeito do ruído na estimativa é feita para o método VRFT não linear. Finalmente, três diferentes métodos de regularização, o terceiro proposto neste trabalho, são comparados com o VRFT

Implementation of Algorithms for Tuning Parameters in Regularized Least Squares Problems in System Identification

There has been recently a trend to study linear system identification with high order finite impulse response (FIR) models using the regularized least-squares approach. One key of this approach is to solve the hyper-parameter estimation problem that is usually nonconvex. Our goal here is to investigate implementation of algorithms for solving the hyper-parameter estimation problem that can deal with both large data sets and possibly ill-conditioned computations. In particular, a QR factorization based matrix-inversion-free algorithm is proposed to evaluate the cost function in an efficient and accurate way. It is also shown that the gradient and Hessian of the cost function can be computed based on the same QR factorization. Finally, the proposed algorithm and ideas are verified by Monte-Carlo simulations on a large data-bank of test systems and data sets