Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

13.8. Some Other Types of Data 369Apley and Shi (2001) assume that a vector of p measured features froma process or product can be modelled as in the probabilistic PCA model ofTipping and Bishop (1999a), described in Section 3.9. The vector thereforehas covariance matrix BB ′ + σ 2 I p , where in the present context the qcolumns of B are taken to represent the effects of q uncorrelated faultson the p measurements. The vectors of principal component coefficients(loadings) that constitute the columns of B thus provide information aboutthe nature of the faults. To allow for the fact that the faults may notbe uncorrelated, Apley and Shi suggest that interpreting the faults maybe easier if the principal component loadings are rotated towards simplestructure (see Section 11.1).13.8 Some Other Types of DataIn this section we discuss briefly some additional types of data with specialfeatures.Vector-valued or Directional Data—Complex PCASection 12.2.3 discussed a special type of complex PCA in which the seriesx t + ix H t is analysed, where x t is a p-variate time series, x H t is its Hilberttransform and i = √ −1. More generally, if x t , y t are two real-valued p-variate series, PCA can be done on the complex series x t + iy t , and thisgeneral form of complex PCA is relevant not just in a time series context,but whenever two variables are recorded in each cell of the (n × p) datamatrix. This is then a special case of three-mode data (Section 14.5) forwhich the index for the third mode takes only two values.One situation in which such data arise is for landmark data (see Section13.2). Another is when the data consist of vectors in two dimensions,as with directional data. A specific example is the measurement of wind,which involves both strength and direction, and can be expressed as avector whose elements are the zonal (x or easterly) and meridional (y ornortherly) components.Suppose that X is an (n × p) data matrix whose (h, j)th element isx hj + iy hj . A complex covariance matrix is defined asS = 1n − 1 X† X,where X † is the conjugate transpose of X. Complex PCA is then doneby finding the eigenvalues and eigenvectors of S. Because S is Hermitianthe eigenvalues are real and can still be interpreted as proportions of totalvariance accounted for by each complex PC. However, the eigenvectors arecomplex, and the PC scores, which are obtained as in the real case by multiplyingthe data matrix by the matrix of eigenvectors, are also complex.