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

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 notmore » hold in general for isotopic mixing lines. Here, 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.« less

Authors:
 [1];  [1]
  1. Univ. of Arizona, Tucson, AZ (United States)
Publication Date:
Research Org.:
Univ. of Arizona, Tucson, AZ (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1337893
Alternate Identifier(s):
OSTI ID: 1347529
Grant/Contract Number:
SC0006741
Resource Type:
Journal Article: Published Article
Journal Name:
Biogeosciences (Online)
Additional Journal Information:
Journal Name: Biogeosciences (Online); Journal Volume: 14; Journal Issue: 1; Journal ID: ISSN 1726-4189
Publisher:
European Geosciences Union
Country of Publication:
United States
Language:
English
Subject:
59 BASIC BIOLOGICAL SCIENCES; 58 GEOSCIENCES

Citation Formats

Wehr, Richard, and Saleska, Scott R. The long-solved problem of the best-fit straight line: Application to isotopic mixing lines. United States: N. p., 2017. Web. doi:10.5194/bg-14-17-2017.
Wehr, Richard, & Saleska, Scott R. The long-solved problem of the best-fit straight line: Application to isotopic mixing lines. United States. doi:10.5194/bg-14-17-2017.
Wehr, Richard, and Saleska, Scott R. Tue . "The long-solved problem of the best-fit straight line: Application to isotopic mixing lines". United States. doi:10.5194/bg-14-17-2017.
@article{osti_1337893,
title = {The long-solved problem of the best-fit straight line: Application to isotopic mixing lines},
author = {Wehr, Richard and Saleska, Scott R.},
abstractNote = {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. Here, 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.},
doi = {10.5194/bg-14-17-2017},
journal = {Biogeosciences (Online)},
number = 1,
volume = 14,
place = {United States},
year = {Tue Jan 03 00:00:00 EST 2017},
month = {Tue Jan 03 00:00:00 EST 2017}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.5194/bg-14-17-2017

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Cited by: 3works
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