On the Covariance Matrix of Weighted Total Least-Squares Estimates

Three strategies are employed to estimate the covariance matrix of the unknown parameters in an error-in-variable model. The first strategy simply computes the inverse of the normal matrix of the observation equations, in conjunction with the standard least-squares theory. The second strategy applies the error propagation law to the existing nonlinear weighted total least-squares (WTLS) algorithms for which some required partial derivatives are derived. The third strategy uses the residual matrix of the WTLS estimates applicable only to simulated data. This study investigated whether the covariance matrix of the estimated parameters can precisely be approximated by the direct inversion of the normal matrix of the observation equations. This turned out to be the case when the original observations were precise enough, which holds for many geodetic applications. The three strategies were applied to two commonly used problems, namely a linear regression model and a two-dimensional affine transformation model, using real and simulated data. The results of the three strategies closely followed each other, indicating that the simple covariance matrix based on the inverse of the normal matrix provides promising results that fulfill the requirements for many practical applications.