A Newton algorithm for weighted total least-squares solution to a specific errors-in-variables model with correlated measurements

For a weighted total least-squares (WTLS) solution to errors-in-variables (EIV) model, three issues are essential: (I) how to acquire a perfect description of the variance-covariance matrix of the design matrix; (2) how to find an effective algorithm to deal with all the situations, particularly, when the design matrix and right hand side vector are cross-correlated; (3) how to evaluate the accuracy of the proposed methods. This paper presents a specific EIV model in which each random element of the design matrix is a function of the input vector. In such a model, the error corrupted input and output vectors are treated as the measurements equivalently. A scheme of calculating the variance-covariance matrix of the augmented design matrix based on error propagation laws is presented. It is suitable for achieving the dispersion matrices in more general cases even in the presence of the nonzero cross-covariance of dispersion matrix. It is also proved to be consistent with the rules introduced earlier. A WTLS approach, called Newton iterative WTLS algorithm followed by an approximated accuracy assessment method is developed. Case studies demonstrate that the proposed algorithm can achieve the same accuracy as the existing WTLS or structured TLS methods. The proposed Newton WTLS algorithm aims to provide a complete WTLS adjustment method with a general weight matrix for the specific EIV model.