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Understanding Least Squares through Monte Carlo Calculations

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Joel Tellinghuisen

Department of Chemistry, Vanderbilt University, Nashville, TN 37235

J. Chem. Educ., 2005, 82 (1), p 157

DOI: 10.1021/ed082p157

Publication Date (Web): January 1, 2005

Abstract

The mathematics of linear least squares (LLS) is summarized in a compact and easy-to-remember matrix notation and Monte Carlo computations are used to illustrate its fundamental properties: Gaussian (normal) distributions for parameters, χ² distributions for variances, and t-distributions for parameters when their variances have to be estimated from the fit. Deviations from these forms are illustrated for cases where the data do not possess random normal error, are biased, or are not weighted properly in the LS fit. The distinction between the a priori and a posteriori variance–covariance matrices is emphasized. The former is appropriate when the error structure of the data is known a priori, as in Monte Carlo calculations; it is exact. The latter assumes ignorance about at least the scale of the data error, which is then estimated from the fit. Importantly, the a posteriori estimates of parameter standard errors are unreliable unless the data are given correct relative weights. The exact equation for error propagation of correlated variables is presented and illustrated. The matter of confidence limits and joint confidence limits on parameters and functions of parameters is discussed and illustrated, with reference to the properties of χ².

Keywords (Audience):

Upper-Division Undergraduate

Keywords (Domain):

Physical Chemistry

Keywords (Subject):

Chemometrics

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This article has been cited by 4 ACS Journal articles (4 most recent appear below).

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