Maximum likelihood estimation via the extended covariance and combined square-root filters
type: Item
instance of: scholarly article
title: Maximum likelihood estimation via the extended covariance and combined square-root filters
zbMATH Open document ID: 1163.65044
author: M. V. Kulikova
published in: Mathematics and Computers in Simulation
publication date: 2009-03-09T00:00:00Z
review text: The author considers the following discrete-time linear stochastic system, i.e. \[ x_k=F_k(\theta)x_{k-1}+G_k(\theta)w_k \] \[ z_k=H_k(\theta)x_{k}+v_k \] with the system state \(x_k\in\mathbb R^{n}\), the state disturbance \(w_k\in\mathbb R^q\), the observed vector \(z_k\in\mathbb R^m\), and the measurement error \(v_k\in\mathbb R^m\). The processes \(\{w_k \mid k=1,2,\dots\}\) and \(\{v_k \mid k=1,2,\dots\}\) are independent white Gaussian sequences with zero mean and covariance matrices \(Q_k(\theta)\) and \(R_k(\theta)\). The initial state \(x_0\) is \(\sim N\big(\tilde x_0,\Pi_0(\theta)\big)\) and each of the \(w_k,\;v_k\) are mutually uncorrelated. Finally, \(\theta\in\mathbb R^p\) is a vector of the unknown system parameters to be estimated. This paper is devoted to the problem of an efficient evaluation of the log-likelihood function \(J\) and of its gradient \(\partial J/\partial\theta\). For the considered system the log-likelihood gradient evaluation requires the implementation of the Kalman filter and its derivatives with respect to the unknown system parameters. It leads to a set of~\(p\) vector equations (system sensitive equations) and set of~\(p\) matrix equations (Riccati-type sensitivity equations). The paper is organized as follows. After a brief description the likelihood and its gradient (with respect the to unknown parameters) are expressed in terms of the square-root variables. A bew method for the likelihood gradient is presented. The algorithms are derived in the measurement and time updates form in terms of covariance matrices. A comparison with the conventional approach, in terms of sensitivity to round-off errors, and some numerical results are finally given together with some general conclusions.
reviewed by: Jaromír Antoch
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 93E03
Mathematics Subject Classification ID: 93E12
Mathematics Subject Classification ID: 93E25
Mathematics Subject Classification ID: 93E11
zbMATH DE Number: 5526322
zbMATH Keywords: System identification
zbMATH Keywords: maximum likelihood estimation
zbMATH Keywords: Kalman filter
zbMATH Keywords: squared-root filtering algorithms
zbMATH Keywords: discrete-time linear stochastic system
zbMATH Keywords: system sensitive equations
zbMATH Keywords: numerical results
MaRDI profile type: Publication
full work available at URL: j.matcom.2008.08.004
OpenAlex ID: W1988281543
cites work: Maximum likelihood estimation using square root information filters
cites work: Computational aspects of maximum likelihood estimation and reduction in sensitivity function calculations
cites work: Score Evaluation Within the Extended Square-Root Information Filter
cites work: Q3818224
cites work: New square-root algorithms for Kalman filtering
cites work: A new method for evaluating the log-likelihood gradient (score) of linear dynamic systems
cites work: A new method for evaluating the log-likelihood gradient, the Hessian, and the Fisher information matrix for linear dynamic systems
cites work: Numerical aspects of different Kalman filter implementations
DOI: 10.1016/J.MATCOM.2008.08.004
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description: scientific article; zbMATH DE number 5526322