Maximum Likelihood Estimation Using Square Root Information Filters

The method of maximum likelihood has been previously applied to the problem of determining the parameters of a linear dynamical system model. Calculation of the maximum likelihood estimate may be carried out iteratively by means of a scoring equation which involves the gradient of the negative log likelihood function and the Fisher information matrix. Evaluation of the latter two requires implementation of a Kalman filter (and its derivative with respect to each parameter) which is known to be unstable. In this paper, we derive equations which can be used to obtain the maximum likelihood estimate iteratively but based upon the Square Root Information Filter (SRIF). Unlike the conventional Kalman filter, the SRIF avoids numerical instabilities arising from computational errors. Thus, our new algorithm should be numerically superior to a Kalman filter mechanization.