Parameter estimation using splines

Abstract

In this paper 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 modeled 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, a nonreproducing density. Therefore, exact realization of the nonlinear parameter estimators requires immense computational effort and storage capacity. To alleviate these difficulties, splines functions are used for the approximate realization of the Bayesian parameter estimation algorithm. Specifically, variation-diminishing splines are used, which are pieces of smoothly tied polynomials 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. A convergence theorem is obtained for the convergence of the spline algorithm to the optimal algorithm, as well as two implementable error bounds. Extensive numerical simulation indicates that the spline algorithm performs well, yielding parameter estimates very close to the true values. In summary, the salient characteristics of the proposed spline-based estimator are as follows: 1. (a) it is readily implementable, its realization requiring essentially a bank of Kalman filters and integration of piece-wise polynomial functions; 2. (b) the spline algorithm is based essentially on an imbedded quantization algorithm, namely, the one corresponding to the bank of Karman filters used; 3. (c) as such, the spline estimator has greater computational requirements than the associated quantization algorithm; but 4. (d) the spline algorithm is more accurate than the quantization algorithm. In view of the above, the spline-based estimator should prove useful in practical applications of parameter estimation as well as of general nonlinear estimation.