Supplement 1. Software to compute nonlinear canonical analysis (program POLYNOMIAL RDACCA: source code, compiled versions for Macintosh and Windows program documentation, and example data files).

File List
User Guide.pdf

Polynomial_RDA-CCA_PC.zip

Polynomial_RDA-CCA_Mac.sit

Description
These files are respectively the
user's manual and two versions (PC and Mac) of the Polynomial RDA-CCA program
used in our article to carry out linear and polynomial RDA and CCA.

This program performs four forms of canonical analysis: linear or polynomial
redundancy analysis (RDA) and linear or polynomial canonical correspondence
analysis (CCA). Classical linear redundancy analysis (Rao, 1964) and canonical
correspondence analysis (ter Braak, 1986, 1987) are computed using multiple
linear regression followed by direct eigenanalysis of the matrix of fitted
values. The method of calculation is described in Chapter 11 of Legendre and
Legendre (1998). Polynomial RDA and CCA, which are generalizations of the
linear forms, are implemented using a new approach proposed by Makarenkov
and Legendre (1999, 2001). The polynomial methods are based on the use of
multiple polynomial regression, during the first stage of RDA and CCA, instead
of the multiple linear regression used in the linear forms. The explanatory
variables are limited to their quadratic form in any term of the polynomial.
The program produces the output required to draw biplot diagrams for linear
and polynomial RDA or CCA. In polynomial RDA or CCA, the explanatory variables
can be represented in biplots in two different ways: (1) the individual terms
of the polynomial equation can be represented as separate variables or (2)
one can choose to represent an explanatory variable using the multiple correlations
(rescaled as required by the selected scaling method) of the canonical ordination
axes against the linear and quadratic forms of the variable. A permutation
procedure allows one to test the significance of the two models (linear and
polynomial) and of the difference between them.

The program is also available on the web site of Pierre Legendre.

See the User's Guide for more information.

Press, W. H., B. P. Flanery, S. A.
Teukolsky, and W. T. Vetterling. 1986. Numerical recipes - The art of scientific
computing. Cambridge University Press, Cambridge, UK.