Using derivative regularization in parameter estimation

The condition of the final set of equations in any parameter estimating method is a critical feature and can often determine the success or failure of the method used. This aspect of parameter estimation becomes especially critical when sensitivity coefficients for parameters being estimated are correlated, which is a characteristic of ill-posed problems. Various forms of regularization have been used in the past in the interest of enhancing stability in various parameter estimation methods. This research presents a new general method, called derivative regularization, which can be used to provide improved structure to any ill-posed time dependent parameter estimation problem. Most methods impart structure to the square sensitivity matrix ((XX)-X-T) by adding a regularization term. This research, introducing derivative regularization, uses the principle of matrix pre-multiplication to reduce the ill-conditioned nature of the matrix structure. While the additive forms of regularization generally introduce bias, the derivative regularization method presented here carries the advantage of being unbiased at the cost of larger estimation errors. The method adds structure by approximating the first and second derivatives of the sensitivity coefficients and the measured data. The gain in ability to compute parameters resulting from the use of this method is shown in quantitative form by comparing the condition number of the (XX)-X-T matrix for various cases.