Articles | Volume 14, issue 1
Research article
03 Jan 2017
Research article |  | 03 Jan 2017

The long-solved problem of the best-fit straight line: application to isotopic mixing lines

Richard Wehr and Scott R. Saleska

Abstract. It has been almost 50 years since York published an exact and general solution for the best-fit straight line to independent points with normally distributed errors in both x and y. York's solution is highly cited in the geophysical literature but almost unknown outside of it, so that there has been no ebb in the tide of books and papers wrestling with the problem. Much of the post-1969 literature on straight-line fitting has sown confusion not merely by its content but by its very existence. The optimal least-squares fit is already known; the problem is already solved. Here we introduce the non-specialist reader to York's solution and demonstrate its application in the interesting case of the isotopic mixing line, an analytical tool widely used to determine the isotopic signature of trace gas sources for the study of biogeochemical cycles. The most commonly known linear regression methods – ordinary least-squares regression (OLS), geometric mean regression (GMR), and orthogonal distance regression (ODR) – have each been recommended as the best method for fitting isotopic mixing lines. In fact, OLS, GMR, and ODR are all special cases of York's solution that are valid only under particular measurement conditions, and those conditions do not hold in general for isotopic mixing lines. Using Monte Carlo simulations, we quantify the biases in OLS, GMR, and ODR under various conditions and show that York's general – and convenient – solution is always the least biased.

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Short summary
In 1969, Derek York published a highly general solution to the common problem of how to fit a straight line to points measured with error in both x and y. Unfortunately York's solution is almost unknown outside the geophysical literature, and new studies wrestle with the problem each year. We introduce York's solution and demonstrate it using an example from biogeochemistry: the isotopic mixing line. By Monte Carlo simulation, we show that York’s solution is superior to all popular fit methods.
Final-revised paper