Page 417 - Jolliffe I. Principal Component Analysis

14. Generalizations and Adaptations of Principal Component Analysis
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ter 6), and a shorter description is given by Krzanowski and Marriott (1994,
Chapter 8). The link between non-linear biplots and PCA is somewhat
tenuous, so we introduce them only briefly. Classical biplots are based
on the singular value decomposition of the data matrix X, and provide
a best possible rank 2 approximation to X in a least squares sense (Sec-
tion 3.5). The distances between observations in the 2-dimensional space
of the biplot with α = 1 (see Section 5.3) give optimal approximations to
the corresponding Euclidean distances in p-dimensional space (Krzanow-
ski and Marriott, 1994). Non-linear biplots replace Euclidean distance by
other distance functions. In plots thus produced the straight lines or ar-
rows representing variables in the classical biplot are replaced by curved
trajectories. Different trajectories are used to interpolate positions of ob-
servations on the plots and to predict values of the variables given the
plotting position of an observation. Gower and Hand (1996) give examples
of interpolation biplot trajectories but state that they ‘do not yet have an
example of prediction nonlinear biplots.’
Tenenbaum et al. (2000) describe an algorithm in which, as with
non-linear biplots, distances between observations other than Euclidean
distance are used in a PCA-related procedure. Here so-called geodesic dis-
tances are approximated by finding the shortest paths in a graph connecting
the observations to be analysed. These distances are then used as input to
what seems to be principal coordinate analysis, a technique which is related
to PCA (see Section 5.2).

14.2 Weights, Metrics, Transformations and
Centerings

Various authors have suggested ‘generalizations’ of PCA. We have met ex-
amples of this in the direction of non-linearity in the previous section. A
number of generalizations introduce weights or metrics on either observa-
tions or variables or both. The related topics of weights and metrics make
up two of the three parts of the present section; the third is concerned with
different ways of transforming or centering the data.

14.2.1 Weights
We start with a definition of generalized PCA which was given by Greenacre
(1984, Appendix A). It can viewed as introducing either weights or metrics
into the definition of PCA. Recall the singular value decomposition (SVD)
of the (n × p) data matrix X defined in equation (3.5.1), namely

X = ULA . (14.2.1)