skbio.stats.ordination.ca

skbio.stats.ordination.ca(X, scaling=1)

Compute correspondence analysis.

Correspondence analysis is a multivariate statistical technique for ordination. In general, rows in the data table will correspond to samples and columns to features, but the method is symmetric. In order to measure the correspondence between rows and columns, the \(\chi^2\) distance is used, and those distances are preserved in the transformed space. The \(\chi^2\) distance doesn’t take double zeros into account, and so it is expected to produce better ordination that PCA when the data has lots of zero values.

It is related to Principal Component Analysis (PCA) but it should be preferred in the case of steep or long gradients, that is, when there are many zeros in the input data matrix.

Parameters:

Xpd.DataFrame

Samples by features table (n, m). It can be applied to different kinds of data tables but data must be non-negative and dimensionally homogeneous (quantitative or binary). The rows correspond to the samples and the columns correspond to the features.

scaling{1, 2}

Scaling type 1 maintains \(\chi^2\) distances between rows. Scaling type 2 preserves \(\chi^2\) distances between columns. For a more detailed explanation of the interpretation, check notes below and Legendre & Legendre 1998, section 9.4.3.

Returns:

OrdinationResults

Object that stores the computed eigenvalues, the transformed sample coordinates, the transformed features coordinates and the proportion explained.

Raises:

NotImplementedError

If the scaling value is not either 1 or 2.

ValueError

If any of the input matrix elements are negative.

See also

cca

rda

OrdinationResults

Notes

The algorithm is based on [1], S 9.4.1., and is expected to give the same results as cca(X) in R’s package vegan.

In Scaling type 1, the euclidean distances between rows in the transformed space equal their \(\chi^2\) distances in the original space. Rows (samples) near a column (features) indicate high contributions from that feature.

In Scaling type 2, the euclidean distances between columns in the transformed space equal their \(\chi^2\) distances in the original space. Columns (features) near a row (sample) indicate higher abundance in that sample. Other types of scalings are currently not implemented, as they are less used by ecologists (Legendre & Legendre 1998, p. 456).

Features far from the center of the biplot and far from its edges often exhibit better relationships than features either in the center (may represent multimodal features, not related to the shown ordination axes) or the edges (sparse features).

References

[1]

Legendre P. and Legendre L. 1998. Numerical Ecology. Elsevier, Amsterdam.