By I. T. Jolliffe (auth.)
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Example text
However, in practice, it is more common to use PCA as a descriptive, rather than inferential, tool, so that the advantage becomes less crucial. The second advantage of covariance matrices holds in the special case when all elements of x are measured in the same units. It can then be argued that standardizing the elements of x to give correlations is equivalent to making an arbitrary choice of measurement units. This argument of arbitrariness can also be applied more generally to the use of correlation matrices, but when the elements of x are measurements of different types, the choice of measurement units leading to a covariance matrix is even more arbitrary, so that the correlation matrix is preferred.
Assume that x", N(JI, l:), that is x has a p-variate normal distribution with mean JI, and covariance matrix l:. Although JI need not be given, l: is assumed known. e. 4). Therefore, investigation of the sampling properties of the coefficients and variances of the sampie pes is equivalent to looking at sampling properties of eigenvectors and eigenvalues of Wishart random variables. The density function of a matrix V which has the ~(l:, n - 1) distribution is 3. Mathematical and Statistical Properties of Sam pie Principal Components 40 where c- 1 = 2P(n-1)/2IIP(1- p )/411:I(n-1)/2 Ii r(n -2 j), j=l and various properties of Wishart random variables have been thoroughly investigated (see, for example, Srivastava and Khatri, 1979, Chapter 3).
1: We shall deal with one other property, which is concerned with the use of 26 3. Mathematical and Statistical Properties of SampIe Principal Components principal components in regression, in this section. Standard terminology from regression will be used and will not be explained in detail-see, for example, Draper and Smith (1981). A full discussion of the use of principal components in regression is given in Chapter 8. Property A6. 5) where y is the vector of n observations on the dependent variable, again measured about the sampie mean.