Principal components analysis (PCA)

Principal Components Analysis (PCA) as a method of multivariate statistics was created before the Second World War. However, the wider application of this method only occurred in the 1960s, during the “Quantitative Revolution” in the Natural and Social Sciences. The main reason for this time-lag was the huge difficulty posed by calculations involving this method. Only with the advent and development of computers did the almost unlimited application of multivariate statistical methods, including principal components, become possible. At the same time, requirements arose for precise numerical methods concerning, among other things, the calculation of eigenvalues and eigenvectors, because the application of principal components to technical problems required absolute accuracy. On the other hand, numerous applications in Social Sciences gave rise to a significant increase in the ability to interpret these nonobservable variables, which is just what the principal components are. In the application of principal components, the problem is not only to do with their formal properties but above all, their empirical origins. The authors considered these two tendencies during the creation of the program for principal components. This program—entitled PCA—accompanies this paper. It analyzes consecutively, matrices of variance-covariance and correlations, and performs the following functions: - the determination of eigenvalues and eigenvectors of these matrices. - the testing of principal components. - the calculation of coefficients of determination between selected components and the initial variables, and the testing of these coefficients, - the determination of the share of variation of all the initial variables in the variation of particular components, - construction of a dendrite for the initial set of variables, - the construction of a dendrite for a selected pattern of the principal components, - the scatter of the objects studied in a selected coordinate system. Thus, the PCA program performs many more functions especially in testing and graphics, than PCA programs in conventional statistical packages. Included in this paper are a theoretical description of principal components, the basic rules for their interpretation and also statistical testing.