A Mathematical Comparison of the Lachance - Traill Matrix Correction Procedure with Statistical Multiple Linear Regression Analysis in XRF Applications

Abstract

AbstractIt will be shown that the Lachance-Traill XRF matrix correction equations can be derived from the statistical multiple linear regression model. By selecting and properly transforming the independent variables and then applying the statistical multiple linear regression model, the following form of the matrix correction equation is obtained:Furthermore, it will be shown that the Lachance-Traill influence coefficients have a deeper mathematical meaning. They can be related to the multiple regression coefficients of the transformed system:Finally, it will be proposed that the Lachance-Traill model is equivalent to the statistical multiple linear regression model with the transformed independent variables. Knowing these facts will simplify correction subroutines in Quantitative/Empirical XRF Analysis programs. These mathematical facts have already been implemented and presented in a paper: “Automated Quantitative XRF Analysis Software in Quality Control Applications” (Pacific-International Congress on X-ray Analytical Methods, Hawaii, 1991).This demonstrates that the Lachance-Traill model has a strong mathematical foundation and is naturally justified mathematically.