Principal Component Analysis using R

This tutorial is designed to give the reader a short overview of Principal Component Analysis (PCA) using R. PCA is a useful statistical method that has found application in a variety of fields and is a common technique for finding patterns in data of high dimension. Consider we are confronted with the following situation: The data, we want to work with, are in form of a matrix (xij)i=1...N,j=1...M , where xi,jrepresents the value of the i-th observation of the j-th variable. Thus the N members of the population can be identified with the N rows of the data matrix, each correspondending to a M -dimensional vector denoting the values of the different variables. If M is very large it is often desireable to reduce the number of considered variables and to replace M by a smaller number of variables, while losing as little information as possible. Mathematically spoken, PCA is a linear orthogonal transformation, that transforms the data to a new coordinate system such that the greatest variance by any projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on . . . . The (Principal) Components are linear combinations of the original data and a formally description can be found in the book of Carmona (p. 84ff). The data for this analysis are in the package Rsafd: