Virtual Observation Method and Precision Estimation for Ill-Posed Partial EIV Model

This paper presents a method for solving the ill-posed problem of the partial error-in-variables (EIV) model and a precision estimation method based on a second-order derivative approximate function method. Since precision estimation in the ill-posed partial EIV model has not been analyzed well, the partial EIV model and virtual observation equation were combined based on the virtual observation method to further study the problem. The formulas of the bias and covariance of parameter estimation were derived based on the nonlinear theory. In addition, the mean square error (MSE) of parameter estimation was calculated according to the covariance of the second-order approximation of the precision estimation. Compared with the existing methods for addressing ill-posed problems, this method obtains the same results of parameter estimation and can be used for precision estimation; it also provides a method to determine the ridge parameter for the ill-posed partial EIV model. In addition, this method is more widely applicable since it inherits the advantages of the partial EIV model. The first-order approximate variance, the second-order approximate variance, and mean square error of parameter estimation were compared. The feasibility of the method proposed in this paper has been verified through experiments.