Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)
Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s) Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)
11.3. Simplified Approximations to Principal Components 295are given that show how the optimum threshold changes as sample sizeincreases (it decreases) for unrotated, orthogonally rotated and obliquelyrotated PCs. The precision of the results is indicated by boxplots. Thepaper provides useful guidance on choosing a threshold in the spatial atmosphericscience setting if truncation is to be done. However, it largelyignores the fact stressed above that neither the size of loadings nor thatof correlations are necessarily reliable indicators of how to interpret a PC,and hence that truncation according to a fixed threshold should be usedwith extreme caution.Interpretation can play a rôle in variable selection. Section 6.3 defines anumber of criteria for choosing a subset of variables; these criteria are basedon how well the subset represents the full data set in one sense or another.Often there will be several, perhaps many for large p, subsets that do almostas well as the best in terms of the chosen criterion. To decide between themit may be desirable to select those variables that are most easily measuredor, alternatively, those that can be most easily interpreted. Taking this trainof thought a little further, if PCs are calculated from a subset of variables,it is preferable that the chosen subset gives PCs that are easy to interpret.Al-Kandari and Jolliffe (2001) take this consideration into account in astudy that compares the merits of a number of variable selection methodsand also compares criteria for assessing the value of subsets. A much fullerdiscussion is given by Al-Kandari (1998).11.3.1 Principal Components with Homogeneous, Contrastand Sparsity ConstraintsChipman and Gu (2001) present a number of related ideas that lead toresults which appear similar in form to those produced by the techniquesin Section 11.2. However, their reasoning is closely related to truncationso they are included here. Components with homogeneous constraints havetheir coefficients restricted to −1, 0 and +1 as with Hausmann (1982), butinstead of solving a different optimization problem we start with the PCsand approximate them. For a threshold τ H a vector of PC coefficients a kis replaced by a vector a H kwith coefficientsa H kj ={ sign(akj ) if a kj ≥ τ H0 if a kj 0and ∑ pj=1 aC kj = 0. Again a threshold τ C determines
296 11. Rotation and Interpretation of Principal Componentsthe coefficients. We have⎧⎪⎨ −c 1 if a kj ≤−τ Ca C kj = 0 if |a kj |
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296 11. Rotation and Interpretation of <strong>Principal</strong> <strong>Component</strong>sthe coefficients. We have⎧⎪⎨ −c 1 if a kj ≤−τ Ca C kj = 0 if |a kj |
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