Magdolna Hargittai, Istvan Hargittai's Advances in Molecular Structure Research, Volume 3, First PDF

By Magdolna Hargittai, Istvan Hargittai

ISBN-10: 0080545904

ISBN-13: 9780080545905

ISBN-10: 0762302089

ISBN-13: 9780762302086

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Additional resources for Advances in Molecular Structure Research, Volume 3, First Edition (Advances in Molecular Structure Research)

Example text

In order^ to ^ obtain least-squares estimates of the variables 13and their covariance matrix 0([3) which are optimum in the Gauss-Markov sense, the positive definite n x n covariance matrix M must first be reduced by the congruent transformation p'r to the unit matrix, pTMp= 1; then the transformed covariance matrix I~)(~1) = ~21 is, as 42 J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH required, that of uncorrelated unity-weighted errors, a property which is denoted by the subscript 1. By the transformation pT in the n-dimensional space of the observations and their errors, the vectors y and ~ and the n x p design matrix X are transformed accordingly: pTy = Yl, PTe = e l, and pTx = XI.

Epple et al. [66] have proposed a method to assign weights which is hopefully more appropriate in this situation. They assign to each inertial moment I~, (i) an additional "model-induced" error: A(1 ~. go ( i )")j = ~A . (i))2 3(I~ ~--,(g,()) Io i 2 for g = a,b,c, all/ (31) = a,b,c The errors of the inertial moments could be increased up to the values given by Eq. 31 representing "apparent" (model-induced) errors, before the standard deviation of the fit would decrease to the expected value of approximately unity.

The "jackknifed" residuals defined as, A A o ~i t(i) = ti~(i ) - ~(i)ql hii (40) where ~(i) is the estimated standard deviation of the fit when the ith measurement is dropped. The last diagnostic is as easy to calculate as the first two ones and it is the most sensitive to errors. The variables t i and t(i) follow a t-distribution if the errors are Gaussian, and they have a near t-distribution under a wide range of circumstances. In least-squares fitting, it is desirable that none of the parameters is determined by a single experimental datum or a small subset of them because the disproportionate influence of only one or a few observations on the parameter would diminish the balancing effect of a regression analysis.

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Advances in Molecular Structure Research, Volume 3, First Edition (Advances in Molecular Structure Research) by Magdolna Hargittai, Istvan Hargittai


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