Page 436 - Jolliffe I. Principal Component Analysis

14.6. Miscellanea
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• Linear Approximation Asymmetric PCA. This leads to an equation
that is equivalent to (9.3.2). Hence the technique is the same as re-
dundancy analysis, one form of reduced rank regression and PCA of
instrumental variables (Sections 9.3.3, 9.3.4, 14.3).
• Cross-correlation Asymmetric PCA. This reduces to finding the SVD
of the matrix of covariances between two sets of variables, and so is
equivalent to maximum covariance analysis (Section 9.3.3).
• Constrained PCA. This technique finds ‘principal components’ that
are constrained to be orthogonal to a space defined by a set of con-
straint vectors. It is therefore closely related to the idea of projecting
orthogonally to the isometric vector for size and shape data (Sec-
tion 13.2) and is similar to Rao’s (1964) PCA uncorrelated with
instrumental variables (Section 14.3). A soft-constraint version of this
technique, giving a compromise between constrained PCA and or-
dinary PCA, is discussed in Diamantaras and Kung (1996, Section
7.3).
• Oriented PCA. In general terms, the objective is to find a 1 , a 2 ,...,

a k ,... that successively maximize a S 1 a k , where S 1 , S 2 are two covari-
k

a S 2 a k
k
ance matrices. Diamantaras and Kung (1996, Section 7.2) note that
special cases include canonical discriminant analysis (Section 9.1) and
maximization of a signal to noise ratio (Sections 12.4.3, 14.2.2).
Xu and Yuille (1992) describe a neural network approach based on statis-
tical physics that gives a robust version of PCA (see Section 10.4). Fancourt
and Principe (1998) propose a network that is tailored to find PCs for
locally stationary time series.
As well as using neural networks to find PCs, the PCs can also be
used as inputs to networks designed for other purposes. Diamantaras and
Kung (1996, Section 4.6) give examples in which PCs are used as inputs
to discriminant analysis (Section 9.1) and image processing. McGinnis
(2000) uses them in a neural network approach to predicting snowpack
accumulation from 700 mb geopotential heights.

14.6.2 Principal Components for Goodness-of-Fit Statistics

The context of this application of PCA is testing whether or not a (uni-
variate) set of data y 1 ,y 2 ,...,y n could have arisen from a given probability
distribution with cumulative distribution function G(y); that is, we want a
goodness-of-fit test. If the transformation

x i = G(y i ), i =1, 2,... ,n
is made, then we can equivalently test whether or not x 1 ,x 2 ,...,x n are
from a uniform distribution on the range (0, 1). Assume, without loss of