Datum transformations are a fundamental issue in geodesy, Global Positioning System (GPS) science and technology, geographical information science (GIS), and other research fields. In this study, we establish a general total least squares (TLS) theory which allows the errors-in-variables model with different constraints to formulate all transformation models, including affine, orthogonal, similarity, and rigid transformations. Through the adaptation of the transformation models to the constrained TLS problem, the nonlinear constrained normal equation is analytically derived, and the transformation parameters can be iteratively estimated by fixed-point formulas. We also provide the statistical characteristics of the parameter estimator and the unit of precision of the control points. Two examples are given, as well as an analysis of the results on how the estimated quantities vary when the number of constraints becomes larger.

In real estate valuation using the Hedonic Price Model (HPM) estimated via Ordinary Least Squares (OLS) regression, subjectivity and measurement errors in the independent variables violate the Gauss–Markov theorem assumption of a non-random coefficient matrix, leading to biased parameter estimates and incorrect precision assessments. In this contribution, the Errors-in-Variables model equipped with Total Least Squares (TLS) estimation is proposed to address these issues. It fully considers random errors in both dependent and independent variables. An iterative algorithm is provided, and posterior accuracy estimates are provided to validate its effectiveness. Monte Carlo simulations demonstrate that TLS provides more accurate solutions than OLS, significantly improving the root mean square error by over 70%. Empirical experiments on datasets from Boston and Wuhan further confirm the superior performance of TLS, which consistently yields a higher coefficient of determination and a lower posterior variance factor, which shows its more substantial explanatory power for the data. Moreover, TLS shows comparable or slightly superior performance in terms of prediction accuracy. These results make it a compelling and practical method to enhance the HPM.

Total Least Squares (TLS) has been comprehensively explored over the last few decades due to its excellent statistical properties. However, the unsatisfactory efficiency of TLS from the intricate process of inverting large matrices limits its practical applications, especially when processing big data. To address this issue, we propose two Recursive Total Least Squares (RTLS) solutions based on the grouping strategy, where all the observations are partitioned into given independent subsets. Within approximated RTLS, observations in the new subset are recursively integrated with results from preceding subsets. The sequential solution is further extended to be exact RTLS. Both approximated RTLS and exact RTLS can significantly reduce the matrix dimensions in the calculation. The proposed methods are compared with several existing algorithms with the experiments of 2D line fitting, image registration, and reference network transformation. In the three experiments, our solutions improve the efficiency of TLS by more than one order of magnitude.