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13.8. Some Other Types of Data
Apley and Shi (2001) assume that a vector of p measured features from
a process or product can be modelled as in the probabilistic PCA model of
Tipping and Bishop (1999a), described in Section 3.9. The vector therefore
2
has covariance matrix BB + σ I p , where in the present context the q

columns of B are taken to represent the effects of q uncorrelated faults
on the p measurements. The vectors of principal component coefficients
(loadings) that constitute the columns of B thus provide information about
the nature of the faults. To allow for the fact that the faults may not
be uncorrelated, Apley and Shi suggest that interpreting the faults may
be easier if the principal component loadings are rotated towards simple
structure (see Section 11.1).
13.8 Some Other Types of Data

In this section we discuss briefly some additional types of data with special
features.
Vector-valued or Directional Data—Complex PCA
Section 12.2.3 discussed a special type of complex PCA in which the series
x t + ix H is analysed, where x t is a p-variate time series, x H is its Hilbert
√
t t
transform 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 this
general 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) data
matrix. This is then a special case of three-mode data (Section 14.5) for
which the index for the third mode takes only two values.
One situation in which such data arise is for landmark data (see Sec-
tion 13.2). 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 a
vector whose elements are the zonal (x or easterly) and meridional (y or
northerly) components.
Suppose that X is an (n × p) data matrix whose (h, j)th element is
x hj + iy hj . A complex covariance matrix is defined as
1
S = X X,
†
n − 1
†
where X is the conjugate transpose of X. Complex PCA is then done
by finding the eigenvalues and eigenvectors of S. Because S is Hermitian
the eigenvalues are real and can still be interpreted as proportions of total
variance accounted for by each complex PC. However, the eigenvectors are
complex, and the PC scores, which are obtained as in the real case by mul-
tiplying the data matrix by the matrix of eigenvectors, are also complex.