Maximum likelihood estimation via the extended covariance and combined square-root filters
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.
scientific article; zbMATH DE number 5526322
scientific article