Principal Component Analysis, PCA, is one of two multivariate data analysis model main types available in Evince. PCA decomposes the original data matrix, X, into a set of principal components that explain most of the variance found in the data. In this way, the multidimensional variable space is reduced to a more manageable size, which is more easily overviewed. Visualizations of principal components is a powerful way of detecting classes, groupings and outliers within the analyzed data.

The first principal component, PC1, can be visualized as a straight line drawn through the variable space and also through the origin. It is drawn in such a way that it explains as much of the variation as possible. A score value, t, is derived from each observation by projection down to the principal component. A loading value, p, is also obtained from the cosine of the the angle between the principal component and the variable axis.

For each calculated principal component, score values are obtained for the observations, and loading values are obtained for the variables. These values are assembled in the score matrix, T, and in the loading matrix, P.

The decomposition of X into the score and loading matrices can be written:

X = TP' + E = t1p1' + t2p2' + ... + tnpn' + E

where E is a residual matrix. When the optimal number of components has been calculated, the residual matrix, E, is reduced to noise. This is equivalent to that the rank of matrix X has been found.

PCA is related to Singular Value Decomposition, SVD, which is used for factorization of matrix X:

X = USV'

S is a diagonal matrix that holds the singular values of X. Matrices U and V hold the left- and right-singular vectors. The matrix V is identical to loading matrix P while U*S is identical to score matrixT.