The PCA is amongst the oldest of the multivariate statistical methods of data reduction. It is a technique for simplifying a dataset, by reducing multidimensional datasets to lower dimensions for analysis. It produces a small number of derived variables that are uncorrelated and that account for most of the variation in the original data set.'By reducing the number of variables'in this way, we can understand the underlying structure of the data. 'The derived variables are combinations of the original variables. For example, it might be that students take I0 examinations and some students do well in one examination while other students do better in another. It is difficult to compare one student with another when we have 10 marks to consider. One obvious way of comparing students is to calculate the mean score.
This is a constructed combination of the existing variables. However, one might get a more useful comparison of overall performances by considering other constructed cwbinations of the 10 exam marks. The PCA is one way of constructing such combinations, doing so in such a way as to account fer the maximum possible variation in the original data. We can then compare students' performance by considering this much smaller number of variables.
PCA states and then solves a well-defined statistical problem, and except for special cases always gives a unique solution wi.th some very nice mathematical properties. We can even describe some very artificial practical problems for which PCA provides the exact solution. The difficulty comes in trying to relate PCA to real-life scientific problems; the match is simply not very good. Actually PCA often provides a good approximation to common factor analysis, but that feature is now unimportant since both methods are now easy enough.