The long-solved problem of the best-fit straight line: application to isotopic mixing lines
AffiliationUniv Arizona, Dept Ecol & Evolutionary Biol
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PublisherCOPERNICUS GESELLSCHAFT MBH
CitationThe long-solved problem of the best-fit straight line: application to isotopic mixing lines 2017, 14 (1):17 Biogeosciences
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AbstractIt 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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SponsorsUS Department Of Energy, Office of Science, Terrestrial Ecosystem Science program [DE-SC0006741]; National Science Foundation Macrosystems Biology program ; Agnese Nelms Haury Program in Environment and Social Justice at the University of Arizona
Except where otherwise noted, this item's license is described as © Author(s) 2017. This work is distributed under the Creative Commons Attribution 3.0 License.