Reduction of the relative standard deviation in the least-squares fitting of linearized equations by using sensitivity weights

In the simple case of the determination of a single parameter, the sensitivity of the signal with respect to the parameter is used to establish a sensitivity weight. It is demonstrated that the sensitivity weight is implicitly used by the least-squares method. The linearization of a model perturbates the way the sensitivity is used by the least-squares method and increases heteroscedasticity which results in an increase of the standard deviation of the evaluated parameter. However, the perturbing effects of linearization can be eliminated by multiplying the points by the reverse of the sensitivity weights of the linearized equation. The single parameter case is used to develop a sensitivity filter for the linear least-squares determination of several parameters. The two-parameter case is studied using simulated experiments. With the sensitivity filter, both the non-linear fitting of the non-linear model and the linear fittings of all the linearized equations lead to the same values of the relative standard deviations of the evaluated parameters.