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By G Dunn; Brian Everitt
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Additional resources for A introduction to mathematical taxonomy
1. Note that the sum of squares of the coefficients in each component is unity and that the sum of the latent roots is equal to the sum of the diagonal elements in S. The ftrst principal component accounts for nearly all of the variance in the three characters. 31 (height) The size of the turtle shells could be characterized by this single variable with little loss of information since it alone accounts for some 98% of the variation of the three measurements length, width and height. 94(height) both of which appear to be measures of carapace 'shape', being comparisons of length versus width and height, and height versus length and width, respectively.
A1px p = ap1x 1 + ap2x 2 + ... + appxp (b) The coefficients defining each linear transformation are such that the sum of their squares is unity; that is, p L a~ = l,i = 1, ... ,p j= 1 (c) Of all the possible transformations of this type, YI has greatest variance. 2 Geometrical interpretation (d) Of all possible transformations of this type which are uncorrelated with Yl' Y2 has the gi-eatest variance. Similarly Y3 has the greatest variance amongst linear transformations uncorrelated with Yl and Y2' and so on, until the complete set of p transformed variables has been defined.
The real problem with a priori weighting is perhaps not that it is logically invalid, but that it is often very difficult to decide how to weight the characters in practice. After a classification has been arrived at, one may wish to be able to identify new individuals using a diagnostic key or some form of discriminant function analysis (see Chapter 7). Here some characters will almost certainly be more useful than others, and so one uses a posteriori weighting of characters in the construction of such keys.