Description of Principal Components Analysis

The decomposition according to Eq. 5-16 is performed by a principal axes transformation of the correlation matrix R. The correlation matrix of the raw data is therefore the starting point of the calculations. 

The nontrivial solution of this problem leads to the determinant  

The evolution of this determinant first yields the eigenvalues. The solution of the whole eigenvalue problem provides pairs of eigenvalues and eigenvectors. The mathematical algorithm is described in detail in [MALINOWSKI, 1991]. A simple example, discussed in Section 5.4.2, will demonstrate the calculation. The following properties of these abstract mathematical measures are essential  

The eigenvalues were normalized by dividing by the square root of Xf, they then have variance equal to unity and one therefore gets the matrix of factor loadings A. These factor loadings are the weights of original features in the new variables, the factors. 

Therefore the new synthetic factors are noncorrelated with each other, they have themselves the variance of one and they contain a certain part of the total variance of the data set expressed by their eigenvalues. Because  

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There are several books on pattern recognition and multivariate analysis. An introduction to several of the main techniques is provided in an edited book [19]. For more statistical in-depth descriptions of principal components analysis, books by Joliffe [20] and Mardia and co-authors [21] should be read. An early but still valuable book by Massart and Kaufmann covers more than just its title theme cluster analysis [22] and provides clear introductory material. 

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