[1] Parameter estimation by least-squares methods

Abstract

This chapter presents an overview of least-squares methods for the estimation of parameters by fitting experimental data. Least-squares methods produce the estimated parameters with the highest probability (maximum likelihood) of being correct if several critical assumptions are warranted. The chapter discusses several least-squares parameter estimation procedures and methods for the evaluation of confidence intervals for the determined parameters. It discusses the practical aspects of applying least-squares techniques to experimental data. The chapter provides an overview of several least-squares methods that can be applied to the evaluation of constants from experimental data. It outlines some of the least-squares methods available for evaluating the set of parameters with the highest probability of being correct in given a set of experimental data. Nonlinear least-squares analysis actually comprises a group of numerical procedures that can be used to evaluate the optimal values of the parameters in vector a for the experimental data. The chapter discusses inherent assumptions of least-squares methods. The chapter reviews several of the more common algorithms—the Gauss–Newton method and derivatives and the Nelder-Mead simplex method. The Gauss–Newton least-squares method is formulated as a system of Taylor series expansions of the fitting function. The Marquardt method is the most commonly used procedure for improving the convergence properties of the Gauss–Newton method.