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

370 13. Principal Component Analysis for Special Types of DataHence both the PC scores and their vectors of loadings have real and imaginaryparts that can be examined separately. Alternatively, they can beexpressed in polar coordinates, and displayed as arrows whose lengths anddirections are defined by the polar coordinates. Such displays for loadingsare particularly useful when the variables correspond to spatial locations,as in the example of wind measurements noted above, so that a map of thearrows can be constructed for the eigenvectors. For such data, the ‘observations’usually correspond to different times, and a different kind of plotis needed for the PC scores. For example, Klink and Willmott (1989) usetwo-dimensional contour plots in which the horizontal axis corresponds totime (different observations), the vertical axis gives the angular coordinateof the complex score, and contours represent the amplitudes of the scores.The use of complex PCA for wind data dates back to at least Waltonand Hardy (1978). An example is given by Klink and Willmott (1989) inwhich two versions of complex PCA are compared. In one, the real andimaginary parts of the complex data are zonal (west-east) and meridional(south-north) wind velocity components, while wind speed is ignored in theother with real and imaginary parts corresponding to sines and cosines ofthe wind direction. A third analysis performs separate PCAs on the zonaland meridional wind components, and then recombines the results of thesescalar analyses into vector form. Some similarities are found between theresults of the three analyses, but there are non-trivial differences. Klink andWillmott (1989) suggest that the velocity-based complex PCA is most appropriatefor their data. Von Storch and Zwiers (1999, Section 16.3.3) havean example in which ocean currents, as well as wind stresses, are considered.One complication in complex PCA is that the resulting complex eigenvectorscan each be arbitrarily rotated in the complex plane. This is differentin nature from rotation of (real) PCs, as described in Section 11.1, becausethe variance explained by each component is unchanged by rotation.Klink and Willmott (1989) discuss how to produce solutions whose meandirection is not arbitrary, so as to aid interpretation.Preisendorfer and Mobley (1988, Section 2c) discuss the theory ofcomplex-valued PCA in some detail, and extend the ideas to quaternionvaluedand matrix-valued data sets. In their Section 4e they suggest thatit may sometimes be appropriate with vector-valued data to take Fouriertransforms of each element in the vector, and conduct PCA in the frequencydomain. There are, in any case, connections between complex PCAand PCA in the frequency domain (see Section 12.4.1 and Brillinger (1981,Chapter 9)).PCA for Data Given as IntervalsSometimes, because the values of the measured variables are imprecise orbecause of other reasons, an interval of values is given for a variable ratherthan a single number. An element of the (n × p) data matrix is then aninterval (x ij , x ij ) instead of the single value x ij . Chouakria et al. (2000)