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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)

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5Graphical Representation of DataUsing <strong>Principal</strong> <strong>Component</strong>sThe main objective of a PCA is to reduce the dimensionality of a set ofdata. This is particularly advantageous if a set of data with many variableslies, in reality, close to a two-dimensional subspace (plane). In this case thedata can be plotted with respect to these two dimensions, thus giving astraightforward visual representation of what the data look like, instead ofappearing as a large mass of numbers to be digested. If the data fall closeto a three-dimensional subspace it is still possible to gain a good visualimpression of the data using interactive computer graphics. Even with afew more dimensions it is possible, with some degree of ingenuity, to geta ‘picture’ of the data (see, for example, Chapters 10–12 (by Tukey andTukey) in Barnett (1981)) although we shall concentrate almost entirelyon two-dimensional representations in the present chapter.If a good representation of the data exists in a small number of dimensionsthen PCA will find it, since the first q PCs give the ‘best-fitting’q-dimensional subspace in the sense defined by Property G3 of Section 3.2.Thus, if we plot the values for each observation of the first two PCs, weget the best possible two-dimensional plot of the data (similarly for threeor more dimensions). The first section of this chapter simply gives examplesillustrating this procedure. We largely defer until the next chapter theproblem of whether or not two PCs are adequate to represent most of thevariation in the data, or whether we need more than two.There are numerous other methods for representing high-dimensionaldata in two or three dimensions and, indeed, the book by Everitt (1978)is almost entirely on the subject, as are the conference proceedings editedby Wang (1978) and by Barnett (1981) (see also Chapter 5 of the book

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