Estimation of straight line parameters with fully correlated coordinates

• Published in: Measurement : journal of the International Measurement Confederation 2014-02, Vol.48, p.378-386
• Main Authors: Amiri-Simkooei, A.R., Zangeneh-Nejad, F., Asgari, J., Jazaeri, S.
• Format: Article
• Language: English
• Subjects: Algorithms; Correlation; Errors-in-variables model; Fully correlated coordinates; Least squares method; Linear regression model; Mathematical models; Matlab; Noise; Regression; Standard least squares; Weighted total least squares
• ISSN: 0263-2241; 1873-412X
• DOI: 10.1016/j.measurement.2013.11.005

•Optimal linear regression fit based on the weighted total least squares for many metrological applications.•Considering a fully populated covariance matrix of measurements in x and y axes.•Presenting a fast, reliable and conceptually simple method for linear regression formulation.•Providing the correct (exact) estimate for the uncertainties of the estimated line parameters. Linear regression problem is a widely used problem in many metrological and measurement systems. This contribution presents a simple and reliable formulation for the linear regression fit using the weighted total least squares (WTLS) problem, when both variables are subjected to different and possibly correlated noise. The formulation is a follow up to four recent research papers in which the method was successfully applied to errors-in-variables models. It is a simple modification of the standard least squares method whose principal result is that the so-called perpendicular offsets are minimized when the full structure of correlated noise among all elements of variable x, y or both variables is supposed to be used. The formulation is rigorous, thus without approximation, and can directly provide the uncertainty of the estimated parameters. In a special case, the general formulation simplifies to the well-known standard linear regression model available in the literature. The effectiveness of the algorithm, which was implemented in MATLAB and is available in Appendix A, is demonstrated using three simulated and experimental data sets. The results indicate that accurate and reliable estimates of line parameters along with their covariance matrix can be provided using the proposed formulation in a relatively small amount of time.