Interpretation of results from simplified principal components
Linear multivariate statistical methods are widely used for analysing data sets which consist of a large number of variables. These techniques, which include principal component analysis, factor analysis, canonical correlation analysis, redundancy analysis and discriminant analysis, all produce a set of new variables, commonly called 'factors', according to some criterion which differs for different techniques. Among these techniques, principal component analysis is one of the most popular techniques used for reducing the dimensions of the multivariate data set. In many applications, when Principal Component Analysis (PCA) is performed on a large number of variables, the interpretation of the results is not simple. The derived eigenvectors of the sample covariance or correlation matrix are not necessarily in a simple form, with all coefficients either 'large' or 'negligible'. To aid interpretation, it is fairly common practice to rotate the retained set of components, often using orthogonal rotation. The purpose of rotation is to simplify structure, and thus to make it easier to interpret the low-dimensional space represented by the retained set of components. Thus, quantification of simplicity is a two step process. The first set involves the extraction of the feature from the data called components, while the second stage uses a rotation method to simplify the structure. One of the two main purposes of this thesis is to combine into one step these two separate stages of dimension reduction (finding the components) and simplification (rotation). This goal is achieved by combining these two objectives in the form of a single function leading to what we call Simplified Components (SCs). Another objective is to discover which of the many possible criteria suggested in factor analysis can be adopted in the proposed procedure of SCs. Thus, a simplified one-step procedure of SCs is proposed, using four measures of simplicity, namely varimax, quartimax, orthomax and equamax indices.