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 <b>User Guide.pdf</b><b><br> Polynomial_RDA-CCA_PC.zip<br> Polynomial_RDA-CCA_Mac.sit </b> 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. <br> <br> 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. <br> <br> The program is also available on the web site of Pierre Legendre. <br> <br> See the User's Guide for more information. <br> <br> 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.

文件列表：<b>用户指南.pdf</b>、<b>Polynomial_RDA-CCA_PC.zip</b> 与 <b>Polynomial_RDA-CCA_Mac.sit</b>

数据集说明：本批次文件分别为用户手册，以及本文中用于执行线性与多项式冗余分析（Redundancy Analysis, RDA）、典型对应分析（Canonical Correspondence Analysis, CCA）的多项式RDA-CCA程序的两个适配版本（PC版与Mac版）。

该程序可实现四种典型分析范式：线性冗余分析、多项式冗余分析、线性典型对应分析与多项式典型对应分析。经典线性冗余分析（Rao，1964）与典型对应分析（ter Braak，1986、1987）通过多元线性回归计算，并对拟合值矩阵执行直接特征分析，其计算方法详见Legendre与Legendre（1998）专著的第11章。作为线性分析形式的推广，多项式RDA与CCA采用Makarenkov与Legendre（1999、2001）提出的全新方法实现。

多项式分析方法在RDA与CCA的第一阶段中，使用多元多项式回归替代线性分析形式所采用的多元线性回归。多项式任意项中的解释变量仅限定为其二次形式。本程序可生成绘制线性与多项式RDA或CCA双标图（biplot diagram）所需的输出文件。在多项式RDA或CCA分析中，解释变量可通过两种不同方式在双标图中表征：(1) 将多项式方程的各个单项作为独立变量分别展示；(2) 可选择利用典范排序轴针对该变量的线性与二次形式的多重相关系数（根据所选标准化方法进行重缩放）来表征解释变量。通过置换检验程序，可对线性与多项式两种模型及其差值的显著性进行检验。

该程序亦可在Pierre Legendre的官方网站获取。

更多详细信息请参阅用户指南。

参考文献：Press, W. H., B. P. Flanery, S. A. Teukolsky 与 W. T. Vetterling. 1986. 《Numerical Recipes：The Art of Scientific Computing》（中文译名：《数值计算方法——科学计算的艺术》），剑桥大学出版社，英国剑桥。