Dimension reduction

Correspondence analysis

Correspondence analysis (CA) is a multivariate statistical technique proposed by Herman Otto Hartley (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. Its aim is to display in a biplot any structure hidden in the multivariate setting of the data table. As such it is a technique from the field of multivariate ordination. Since the variant of CA described here can be applied either with a focus on the rows or on the columns it should in fact be called simple (symmetric) correspondence analysis. It is traditionally applied to the contingency table of a pair of nominal variables where each cell contains either a count or a zero value. If more than two categorical variables are to be summarized, a variant called multiple correspondence analysis should be chosen instead. CA may also be applied to binary data given the presence/absence coding represents simplified count data i.e. a 1 describes a positive count and 0 stands for a count of zero. Depending on the scores used CA preserves the chi-square distance between either the rows or the columns of the table. Because CA is a descriptive technique, it can be applied to tables regardless of a significant chisquared test. Although the statistic used in inferential statistics and the chi-square distance are computationally related they should not be confused since the latter works as a multivariate statistical distance measure in CA while the statistic is in fact a scalar not a metric. (Wikipedia).

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Related pages

Formal concept analysis | Orange (software) | Canonical correspondence analysis | Dimensionality reduction | Biplot | Examples of vector spaces | Bar chart | Vote counting | Level of measurement | Statistics | Outer product | Origin (mathematics) | Principal component analysis | Dot product | Map (mathematics) | Factor analysis | Diagonal matrix | Vertex (geometry) | Singular value decomposition | Scree plot | Scatter plot | Statistical inference | Binary data | Row and column vectors | Variance | Dimension (vector space) | Jean-Paul Benzécri | R (programming language) | Orthogonality | Centering matrix | Orthonormality | Scalar (mathematics) | Detrended correspondence analysis | Multiple correspondence analysis | Matrix multiplication | Ordination (statistics) | Matrix (mathematics) | Statistical distance | Chi-squared test | Multiplicative inverse