A general partial errors-in-variables model and a corresponding weighted total least-squares algorithm

The partial errors-in-variables (PEIV) model is an structured errors-in-variables (SEIV) model reformulated by collecting all the random elements of the coefficient matrix into an auxiliary random error vector to better treat the case where the coefficient matrix (or ‘data matrix’) A and observation vector y share common elements. In the usual PEIV model, the case of A and y sharing common elements has not been treated explicitly. However, in some cases, A and y do have common elements, taking the autoregressive problem as an example. To apply the PEIV model to SEIV problems with correlated matrix A and vector y, we propose an extended model named general PEIV (GPEIV). We also provide a corresponding algorithm for the total least-squares solution within the GPEIV model. A weight-matrix construction method, which is suitable for PEIV and GPEIV models, is also presented. The GPEIV model, the algorithm and the weight-matrix construction method are tested using two groups of numerical examples. The results show that the proposed model and corresponding algorithm can solve both uncorrelated and correlated SEIV problems, and the weight-matrix construction method is simple and efficient.