Parameter estimation using splines

The estimation of unknown, time-invariant parameters that, if known, completely specify a discrete, linear dynamic model with Gaussian disturbances, is considered. Following the Bayesian approach the unknown parameters are modelled as random variables with known a-priori probability density. Optimal in the mean-square-error sense estimates are desired. However, this requires recursive updating and storage of a non-Gaussian, and more importantly, non-reproducing density. Therefore, exact realization of the nonlinear parameter estimators requires immense computational effort and storage capacity. To alleviate these difficulties, spline functions are used for the approximate realization of the Bayesian parameter estimation algorithm. Specifically, variation diminishing splines are used to approximate the a-posteriori probability density (pdf). This approximation of the pdf is specified in terms of a finite number of parameters, yielding a readily implementable approximation of the exact but unimplementable parameter estimation algorithm. Extensive numerical simulation indicates that the spline algorithm performs well, yielding parameter estimates close to the true values.