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Principal components analysis (PCA)
Principal components analysis is a technique for finding a set of weighted linear composites of original variables such that each composite (a principal component) is uncorrelated with the others. It was originally devised by Pearson (1901) though it is more often attributed to Hotelling (1933) who proposed it independently. The first principal component is such a weighted linear composite of the original variables with weights chosen so that the composite accounts for the maximum variation in the original data. The second component accounts for the maximum variation that is not accounted for the first. The third component likewise accounts for the maximum given the first two components and so on. These weights are found by a matrix analysis technique called eigen-decomposition which produces eigenvalues (which represent the amount of variation accounted for by the composite) and eigenvectors (which give the weights for the original variables).

The technique is thus a useful device for representing a set of variables by a much smaller set of composite variables that account for much of the variance among the set of original variables. For this reason the technique has often been used as an approximation to factor analysis, particularly if the components are rotated. The two models differ however in that factor analysis partitions the variation into that which is common among the variables and that which is unique to a given variable (and finds factors of the common variation) while principal components analysis regards all variation as common.
Principal components is also related to the technique of singular-value-decomposition which is the more widely used method of analysing repertory grid data.


  • Hotelling, H. (1933) Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417-441.
  • Pearson, K. (1901) On lines and planes of closest fit to systems of points in space. Philosophical Magazine, 2, 559-572.

Richard C. Bell

Establ. 2003
Last update: 15 February 2004