Correspondence analysis

Correspondence analysis (CA) is a multivariate statistical technique proposed by Hirschfeld and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data. In a similar manner to principal component analysis, it provides a means of displaying or summarising a set of data in two-dimensional graphical form.

All data should be nonnegative and on the same scale for CA to be applicable, and the method treats rows and columns equivalently. It is traditionally applied to contingency tables — CA decomposes the chi-squared statistic associated with this table into orthogonal factors. Because CA is a descriptive technique, it can be applied to tables whether or not the \chi^2 statistic is appropriate.

Versions

 * Canonical correspondence analysis
 * Detrended correspondence analysis
 * Multiple correspondence analysis

Details
Like principal components analysis, correspondence analysis creates orthogonal components and, for each item in a table, a set of scores (sometimes called factor scores, see Factor analysis). Correspondence analysis is performed on a contingency table, C, of size m×n where m is the number of rows and n is the number of columns.

Preprocessing
From table C, compute a sets of weights for the columns and the rows (sometimes called masses), where row weights are
 * $$w_m = (1C1)^{-1} C1$$

and column weights are
 * $$w_n = (1C1)^{-1} 1C$$.

Next, compute a table S (called the stochastic matrix), where C is divided by the sum of C
 * $$S = (1C1)^{-1} C$$.

Finally, compute a table M from S and the weights as such
 * $$M = S-w_{m}w_{n}^{*}$$

where $$w_{n}^{*}$$ denotes the conjugate transpose of $$w_{n}$$.

Orthogonal Components
The table M is then decomposed with the generalized singular value decomposition where the left and right singular vectors are constrained by weights. The weights are diagonal tables
 * $$W_{m} = diag\{w_{m}\}$$

and
 * $$W_{n} = diag\{w_{n}\}$$

where the diagonal elements of $$W_{n}$$ are $$w_{n}$$ and the off-diagonal elements are all 0.

M is then decomposed via the generalized singular value decomposition
 * $$M = U\Sigma V^* \,$$

where
 * $$U^* W_m U = V^* W_n V = I.$$.

Factor scores
Factor scores for the row items of table C are
 * $$F_{m} = W_{m} U \Sigma$$

and for the column items
 * $$F_{n} = W_{n} V \Sigma$$.

Extensions and Applications
Several variants of CA are available, including detrended correspondence analysis (DCA) and canonical correspondence analysis (CCA). The extension of correspondence analysis to many categorical variables is called multiple correspondence analysis. An adaptation of correspondence analysis to the problem of discrimination based upon qualitative variables (i.e., the equivalent of discriminant analysis for qualitative data) is called discriminant correspondence analysis or barycentric discriminant analysis.

In the social sciences, correspondence analysis, and particularly its extension multiple correspondence analysis, was made known outside France through French sociologist Pierre Bourdieu's application of it.

Implementations

 * The data visualization system Orange include the module: orngCA.
 * The statistical system R includes the packages:,  ,  ,  , and  which perform correspondence analysis and multiple correspondence analysis.
 * A MATLAB program (with a tutorial) for correspondence analysis:.
 * A JavaScript library, under MIT-License on github, which works both on client-side Javascript and server-side (with Node.js) : CorrespondenceAnalysis.