Page 405 - Jolliffe I. Principal Component Analysis

13. Principal Component Analysis for Special Types of Data
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Hence both the PC scores and their vectors of loadings have real and imag-
inary parts that can be examined separately. Alternatively, they can be
expressed in polar coordinates, and displayed as arrows whose lengths and
directions are defined by the polar coordinates. Such displays for loadings
are particularly useful when the variables correspond to spatial locations,
as in the example of wind measurements noted above, so that a map of the
arrows can be constructed for the eigenvectors. For such data, the ‘obser-
vations’ usually correspond to different times, and a different kind of plot
is needed for the PC scores. For example, Klink and Willmott (1989) use
two-dimensional contour plots in which the horizontal axis corresponds to
time (different observations), the vertical axis gives the angular coordinate
of the complex score, and contours represent the amplitudes of the scores.
The use of complex PCA for wind data dates back to at least Walton
and Hardy (1978). An example is given by Klink and Willmott (1989) in
which two versions of complex PCA are compared. In one, the real and
imaginary parts of the complex data are zonal (west-east) and meridional
(south-north) wind velocity components, while wind speed is ignored in the
other with real and imaginary parts corresponding to sines and cosines of
the wind direction. A third analysis performs separate PCAs on the zonal
and meridional wind components, and then recombines the results of these
scalar analyses into vector form. Some similarities are found between the
results of the three analyses, but there are non-trivial differences. Klink and
Willmott (1989) suggest that the velocity-based complex PCA is most ap-
propriate for their data. Von Storch and Zwiers (1999, Section 16.3.3) have
an example in which ocean currents, as well as wind stresses, are considered.
One complication in complex PCA is that the resulting complex eigenvec-
tors can each be arbitrarily rotated in the complex plane. This is different
in nature from rotation of (real) PCs, as described in Section 11.1, be-
cause the variance explained by each component is unchanged by rotation.
Klink and Willmott (1989) discuss how to produce solutions whose mean
direction is not arbitrary, so as to aid interpretation.
Preisendorfer and Mobley (1988, Section 2c) discuss the theory of
complex-valued PCA in some detail, and extend the ideas to quaternion-
valued and matrix-valued data sets. In their Section 4e they suggest that
it may sometimes be appropriate with vector-valued data to take Fourier
transforms of each element in the vector, and conduct PCA in the fre-
quency domain. There are, in any case, connections between complex PCA
and PCA in the frequency domain (see Section 12.4.1 and Brillinger (1981,
Chapter 9)).
PCA for Data Given as Intervals
Sometimes, because the values of the measured variables are imprecise or
because of other reasons, an interval of values is given for a variable rather
than a single number. An element of the (n × p) data matrix is then an
interval (x ij , x ij ) instead of the single value x ij . Chouakria et al. (2000)