Confidence regions of estimated parameters for ecological systems

Abstract

The parameters of ecological models are usually estimated through numerical search algorithms. Determining confidence boundaries for the parameter values obtained in such a way is a problem of great practical importance. In this paper a method is proposed to estimate such regions in two ways, based on either the Hessian matrix or the Fisher Information Matrix (FIM). There is a conceptual difference in the two approximations: the FIM approach is based on the sensitivity trajectories, whereas the Hessian expansion depends on the shape of the error functional. From a comparison between the two approaches, a discriminating method is obtained to detect inaccurate estimation results. The Hessian and FIM approaches differ by the second derivative terms of the output function. This difference is used to assess the success of the estimation, because the two methods yield the same confidence estimate only if the search terminates at the optimal parameter value. The method is demonstrated with reference to a pair of widely used dynamics: the Monod kinetics and the Richards logistic function applied to algal growth. It is shown that in both cases this method compares favourably with the residual correlation analysis and appears to have more discriminatory power.