Residuals rescaling

If we want to work with residuals in a comparative way—if we want to go beyond naive and qualitative (but effective ) diagnosis—we need to rescale our residuals to make them really of comparable size. 

Two lessons are to be drawn from this Rescaling is likely to be essential and residuals that have to be heavily rescaled are less informative about the actual disturbance at the point concerned because a larger fraction of them comes from the propagated effects of other observed values. In drawing inferences about disturbance distributions,... 

A Statistician s Burden If we are to have satisfactory methods for inferring shapes of disturbance distributions from sets of residuals, we need to implement the sorts of rescaling and careful analysis just hinted at, making the implementation easy to use. While this can presumably be done without too much effort for the plain least squares case, it does not seem, in fact, to have been even considered for the diagnostically satisfactory procedures. Clearly, statisticians have an obligation to proceed apace with this implementation. 

The applied researcher, then, needs to remove extreme values that are truly nonrepresentational and include extreme values that are representational. The researcher must also discover the phenomena contributing to these values. Rescaling the residuals can be very valuable in helping to identify outliers. Rescaling procedures include standardizing residuals, studentizing residuals, and jackknife residuals. 

The reader is directed to Appendix II for a review of matrices and application of matrix algebra. Once that is completed, we will look at examples of Studentized and jackknifed residuals applied to data from simple linear regression models and then discuss rescaling of residuals as it applies to model leveraging due to outliers. 

Suppose, however, that the residues a 2 and a i are not known. It may be the case for more complicated systems than those studied thus far that the D 1 limit of the Schrodinger equation is very difficult to solve. Even if the solution is possible it would be useful to have a quick procedure that could yield a reasonably accurate result from the expansion coefficients alone. A particularly simple method for doing this is the rescaled expansion, which was suggested several years ago by Herschbaeh [13]. Suppose that we multiply and divide the partial sums by (1 — S) as follows ... 

Figure 10. Acctiracy of various summation methods for the expansion of the ground state energy of with R = 1. The curves are labeled as follows o, partial sums, Eq. (11) x, shifted expansion, Eq. (46) , conventional Fade approximants a, rescaled approximants, Eq. (41) v, weighted truncation, Eq. (44) o, hybrid Fade approximants with residues from Eqs. (45). The dotted curve corresponds to biased hybrid Fade summation with the exact value for the residue a 2. The number of accmate digits is defined as — logjo K S n — E)/E, where E is the exact energy.

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