Topics in descriptive Principal Component Analysis.

Principal Component Analysis (PCA) is viewed as a descriptive multivariate method for a set of n observations on p variables.

Geometric considerations in the inner product spaces associated with such nxp data sets play a central role throughout this thesis and provide the motivation for the main results. Among these spaces are spaces of matrices, whose geometry is meaningfully discussed in terms of PCA’s kej' concepts.

It is argued that the conventional interpretation of Principal Components, which is based on the magnitude of each variable's loading for that PC. can be misleading. Alternative approaches based on multiple regression are discussed.

The effects on PCA of linear transformations of the data are discussed in general terms, for non-singular and projective transformations. Specific applications are analyzed and a new solution to the problem of removing isometric size from morphometric data is suggested.

Indicators measuring the degree of similarity between the PCA of a data matrix and the PCA of some transformation of that matrix are provided, for various concepts of ‘similarity’. Methods for joint multiple comparisons of several such transformations are also suggested, discussed and exemplified.

Finally, a truljr scale-invariant alternative to PCA is suggested. At the core of Most Correlated Component Analysis (MCCA) lies a result by Hotelling in his 1933 pioneering paper on PCA. The new method and its performance relative to PCA are discussed in detail. This discussion provides new insights into the information provided by covariance and correlation matrices, as well as a new optimal criterion for PCA.