Homotopy nonlinear weighted total least squares adjustment

Abstract

Total least squares estimation based on Gauss–Newton method in nonlinear errors-in-variables (NEIV) model will encounter the problems of convergence, correctness and accuracy of solution related to the selected initial parameter values. In this contribution, a new total least squares estimator is introduced to solve NEIV model. This method is an extension of the homotopy nonlinear weighted least square (HNWLS) method, which is used in the nonlinear Gauss–Markov model where only the dependent variables contain random errors. The new estimator is called homotopy nonlinear weighted total least squares (HNWTLS), because it adopts weighted total least squares adjustment criterion and homotopy method to estimate nonlinear model parameters. The homotopy function of HNWTLS is constructed by using the normal equation of weighted total least squares adjustment criterion. By taking the error vector of independent variables as a parameter vector, the NEIV model is transformed into a classical nonlinear adjustment model. Then, according to the conclusion of HNWLS, the calculation formula of HNWTLS is derived, and the corresponding calculation algorithm is developed accordingly, where the standard Euler prediction and Newton correction method are introduced into it to tracks the homotopy curves. Finally, three examples to demonstrate the advantage and efficiency of HNWTLS estimator are given and some conclusions are drawn.