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|Table of Contents Multivariate Data Modeling PCA PCA|
|See also: eigenvectors, PC loadings and scores, applications of PCA, factor analysis, Exercise - Detection of mixtures of two different wines by PCA, Exercise - Classification of unknown wine samples by PCA|
The problem with multivariate data is that it cannot be displayed on 2-dimensional paper or computer screens. For more than two dimensions, we have to project the data onto a plane. This projection changes with its direction; or, in other words, the projected image changes if the data points are rotated in the n-dimensional space. One might now ask how to find a rotation of the data (or of the axes - which is quite the same) which displays a maximum of information in the projected image.
The process described above is generally called principal component analysis (PCA) and results in a rotation of the coordinate system in such a way that the axes show a maximum of variation along their directions. This somewhat simplified picture can be mathematically condensed to a so-called eigenvalue problem. The eigenvectors of the covariance matrix constitute the principal components. The corresponding eigenvalues give a hint to how much "information" is contained in the individual components.
The following shows a three-dimensional data set and the corresponding principal components. Note that the principal components are orthogonal to each other, and the correlation between any two principal components is zero.
Last Update: 2006-Jšn-17