Total least norm solution for linear structured EIV model

Abstract

Structured total least norm (STLN) and weighted total least squares (WTLS) have been proposed for structured EIV (errors-in-variables) models. STLN is a principle minimizing the Lp norm of the perturbation parts of an EIV model, in which p=1, 2 or ∞. STLN permits affine structure of the matrix A or [A|y] such as Toeplitz. STLN has advantages over WTLS on having ∞-norm and robust 1-norm. However, only Hankel or Toeplitz structure was discussed explicitly in STLN, and weight of errors was not discussed. While in some applications, the matrix [A|y] has arbitrary linear structure, taking linear regression and coordinate transformation as examples. This paper aims at extending STLN to L-STLN (linear structured total least norm), which can deal with EIV models having linear structures other than Toeplitz or Hankel in [A|y]. Additionally, weighted estimation is discussed. A simulated numerical example is computed by STLN and L-STLN under 1-, 2-, and ∞-norm, the results shown that L-STLN can preserve arbitrary linear structure of [A|y]. Also, the estimated correction of [A|y] by WTLS and L-STLN under 2-norm are compared. The results show that weighted L-STLN under 2-norm is consistent with WTLS. The robustness of L-STLN under 1-norm is demonstrated by simulated outlier.