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Correspondence Analysis may be described as a PCA in a different
metric (the metric replaces the usual Euclidean metric).
Mathematically, it differs from PCA also in that points in
multidimensional space are considered to have a mass (or weight)
associated with them, at their given locations. The percentage
inertia explained by axes
takes the place of the percentage variance of PCA, --- and in the former
case the values can be so small that such a figure of merit assumes less
importance than in the case of PCA. Correspondence Analysis is a
technique in which it is a good deal more difficult to interpret
results, but it considerably expands the scope of a PCA--type analysis
in its ability to handle a wide range of data.
While PCA is particularly suitable for quantitative data, CA is recommendable for the following types of input data, which will subsequently be looked at more closely: frequencies, contingency tables, probabilities, categorical data, and mixed qualitative/categorical data.
In the case of frequencies (i.e. the table entry indicates
the frequency of occurrence of attribute j for object i) the row and
column ``profiles'' are of interest. That is to say, the relative
magnitudes are of importance. Use of a weighted Euclidean distance,
termed the
distance, gives a zero distance for example to the
following 5--coordinate vectors which have identical profiles
of values: (2,7,0,3,1) and (8,28,0,12,4). Probability type values can be
constructed here by dividing each value in the vectors by the sum of
the respective vector values.
A particular type of frequency of occurrence data is the contingency
table, --- a table crossing (usually, two) sets of characteristics of
the population under study. As an example, an contingency
table might
give frequencies of the existence of n different metals in stars of
m different ages. CA allows the study of the two sets of variables
which constitute the rows and columns of the contingency table. In its
usual variant, PCA would privilege either the rows or the columns by
standardizing: if, however, we are dealing with a
contingency table, both rows and columns are equally interesting.
The ``standardizing'' inherent in CA (a consequence of the
distance) treats rows and columns in an identical manner.
One byproduct is that the row and column projections in the new space
may both be plotted on
the same output graphic presentations (--- the lack of an analogous
direct relationship between row projections and column projections
in PCA precludes doing this in the latter technique).
Categorical data may be coded by the ``scoring'' of 1 (presence) or 0 (absence) for each of the possible categories. Such coding leads to complete disjunctive coding. CA of an array of such complete disjunctive data is referred to as Multiple Correspondence Analysis (MCA) (and in fact such a coding of categorical data is, in fact, closely related to contingency table type data).
Dealing with a complex astronomical catalogue may well give rise in practice to a mixture of quantitative (real valued) and qualitative data. One possibility for the analysis of such data is to ``discretize'' the quantitative values, and treat them thereafter as categorical. In this way a set of variables --- many more than the initially given set of variables --- which is homogenous, is analysed.