
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
eigendecomposition
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 singularvaluedecomposition which is the
more widely used method of analysing repertory
grid data. 

References


 Hotelling,
H. (1933) Analysis of a complex of statistical variables into principal
components. Journal of Educational Psychology, 24, 417441.
 Pearson,
K. (1901) On lines and planes of closest fit to systems of points in
space. Philosophical Magazine, 2, 559572.


Richard C. Bell


