An Empirical Analysis of Growth Regimes: Appendix on Principal Component Analysis Andrea Mario Lavezzi * Matteo Marsili † October 27, 2010 1 Introduction In this Appendix we present the results of the application of Principal Component Analysis (PCA) to the database studied in Lavezzi and Marsili (2010), in order to clarify the relationship between the GM algorithm and this standard procedure for the analysis of multidimensional data, and to compare the results from their application. PCA is a statistical method for the analysis of multidimensional data composed by corre- lated variables (define N the number of data, and D their dimension). By applying PCA to the original dataset, a new set of data composed by uncorrelated variables, called Principal Components (PC), is obtained (see, e. g., Jolliffe (2002), Ch. I). The principal aim of PCA is to reduce the dimensionality of the data, so that they can be expressed by k<D principal com- ponents, while at the same time maximizing the amount of variation in the data that the PCs are able to explain. Solving this problem amounts to finding the eigenvalues and eigenvectors associated to the variance/covariance matrix of the variables in the original dataset ( Jolliffe (2002), p. 5). Each eigenvalue of the variance/covariance matrix of the original database corresponds to the variance of a PC. The PCs, therefore, can be ordered on the basis of the value of the eigenvalues ( Jolliffe (2002), p. 5). Althought D principal components can in principle be obtained, only a number k<D will be retained on the basis of the variance of each PC. Each new observation will be therefore composed by k principal components, whose values (scores ) * Dipartimento di Studi su Politica, Diritto e Societ` a, Universit` a di Palermo. Email: lavezzi@unipa.it. † Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Trieste. Email: mar- sili@ictp.trieste.it. 1