Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis

Abstract

The weighted total least squares (TLS) method has been developed to deal with observation equations, which are functions of both unknown parameters of interest and other measured data contaminated with random errors. Such an observation model is well known as an errors-in-variables (EIV) model and almost always solved as a nonlinear equality-constrained adjustment problem. We reformulate it as a nonlinear adjustment model without constraints and further extend it to a partial EIV model, in which not all the elements of the design matrix are random. As a result, the total number of unknowns in the normal equations has been significantly reduced. We derive a set of formulae for algorithmic implementation to numerically estimate the unknown model parameters. Since little statistical results about the TLS estimator in the case of finite samples are available, we investigate the statistical consequences of nonlinearity on the nonlinear TLS estimate, including the first order approximation of accuracy, nonlinear confidence region and bias of the nonlinear TLS estimate, and use the bias-corrected residuals to estimate the variance of unit weight.

Adjustment of non-typical errors-in-variables models

“…In fact the old works by Deming (1931) treat very special cases which we might now classify as non-linear type of EIV models, but he does not treat what we call non-typical EIV model in all his generality. The suggested paper by Xu et al (2012) offers a presentation of preceding work plus two new results in Sects. 2.2 and 2.3.…”

“…(7) differ from the partial EIV model proposed in Xu et al (2012) because they merely partitioned the matrix A into a stochastic and a deterministic parts while in our case the observed vector t in A(t) [Eq. (4)] cannot be separated from design matrix before linearization since its elements are arbitrary non-linear functions.…”

On the errors-in-variables model with equality and inequality constraints for selected numerical examples

“…Total Least-Squares (TLS) is a method of fitting that is appropriate when there are errors in both the observation vector and in the design matrix in computational mathematics (Golub and Van Loan 1980) and geodesy (Teunissen 1988;Schaffrin and Wieser 2008;AmiriSimkooei and Jazaeri 2012;Grafarend and Awange 2012;Xu et al 2012;Chang 2015;Shi et al 2015), which is also referred as errors-in-variables (EIV) modelling or orthogonal regression in the statistical community. The TLS/EIV principle was studied by Adcock (1878) and Pearson (1901) already more than one century ago.…”

“…Later, Fang (2014b) solved the weighted TLS (WTLS) problem with inequality constraints within the standard optimization framework. Recently, Zeng et al (2015) used the partial EIV model proposed by Xu et al (2012) to treat the inequality constrained WTLS problem.…”

“…Except being compatible with equality and inequality constraints, the EICTLS solution should be numerically efficient as well as familiar to geodesists. Furthermore, statistical aspects of a (unconstrained or constrained) TLS estimate have not been thoroughly investigated, as most works on TLS focus on theoretical methods and algorithms for numerically finding the TLS estimates (Xu et al 2012). Therefore, we provide the quality description of the corresponding constrained TLS estimates in the case of finite measurements.…”