Straight line regression through data with correlated uncertainties in two or more dimensions, with an application to kinetic isotope fractionation

Straight line regression algorithms are used frequently for measured data that contain non-negligible uncertainties in each variable. For the general case of correlated measurement uncertainties between two variables that differ from one analysis to the next, the popular algorithm of York, 1968 calculates the maximum likelihood estimate for the line parameters and their uncertainties. However, it considers only two-dimensional data and omits the uncertainty correlation between the slope and y-intercept, an important term for evaluating confidence intervals away from the origin. This contribution applies the maximum likelihood method to straight line regression through data in any number of dimensions to calculate a vector-valued slope and intercept as well as the covariance matrix that describes their uncertainties and uncertainty correlations. The algorithm is applied to Pb data measured by TIMS with a silica gel activator that define a fractionation line in a three dimensional log-ratio space. While the log-ratios of even mass number Pb isotopes follow the slope predicted by mass-dependent fractionation with a Rayleigh or exponential law within calculated uncertainties, the log-ratio containing the odd mass number isotope 207Pb diverges significantly, exhibiting mass-independent fractionation. The straight line regression algorithm is appropriate for fractionation lines that form linear trends in log-ratio space, but not for isochrons or mixing lines, which are predicted to be linear only when plotted as isotope or compositional ratios.