MATLAB functions with examples of the analyses in ter Braak & Schaffers (2004).

Reference
ter Braak, C.J.F and Schaffers, A.P. (2004).  Co-correspondence analysis: a new
ordination method to relate two community compositions. Ecology, 85, 834-846.


1. Symmetric co-correspondence analysis:
SymCoCa.m              - function for Symmetric Co-Correspondence analysis
                           using Coinertia analysis (coinertia.m) [equation (7)]
SymCoCa_u1u2rescale.m  - rescales the U1 and U2 scores from SymCoCa 
                          by the quarter root of the eigenvalues lamda
                          to form a biplot of Q1' R0 Q2,
                          the R0-based covariance between the Q-matrices
Example_SymCoCa.m      - example for using SymCoCa 
                          produces scores, fits and ordination diagram
Example_SymCoCa.txt    - output of Example_SymCoCa
CoCa_trans.m           - function that implements the transition formulae (3)- (6)
                         Gives the first axis of co-correspondence analysis
Example_CoCa_trans.m   - example for using CoCa_trans and SymCoCa.m 
Example_CoCa_trans.txt - output of Example_CoCa_trans.m; 
                         Shows that transition formulae can also be solved
                         by coinertia analysis on Q1 and Q2  [equation (7)]



2. Predictive co-correspondence analysis:
2.1 With PLS_Toolbox 2.1 (www.eigenvector.com)

PredCoCa.m            - function for Predictive Co-Correspondence analysis
                        using the simpls function of PLS_Toolbox 2.1 
Example_PredCoCa.m    - example for using PredCoCa, 
                        produces percentage crossvalidatory fit, scores, fitted values 
							        	and ordination diagram
Example_PredCoCa.txt  - output of Example_PredCoCa

2.2 Without PLS_Toolbox 2.1
PredCoCa_eig.m             - function for Predictive Co-Correspondence analysis
									           SIMPLS version using the formulae of ter Braak & de Jong 1998, 
									           Journal of Chemometrics 12 41-54.
									           It solves the eigenvalue problem posed by equations (1) and (2)
									           and the simpls contraints for further axes.  
Example_PredCoCa_eig.m    - example for using PredCoCa_eig, 
                            produces scores and ordination diagram
Example_PredCoCa_eig.txt  - output of Example_PredCoCa_eig

2.3 Testing axes of Predictive co-correspondence analysis
          
Test_axes_CoCa.m            - function testing axis of Predictive Co-Correspondence analysis
									            by permutation using the formulae of ter Braak & Smilauer 2002;
									            and the NIPALS version of Predictive Co-Correspondence analysis
Example_Test_axes_CoCa.m    - example for using Test_axes_CoCa 
Example_Test_axes_CoCa.txt  - output of Example_Test_axes_CoCa


3. Crossvalidatory Canonical Correspondence analysis(CCA) and CCA-PLS
   using simpls from PLS_Toolbox 2.1 (www.eigenvector.com)
Example_crossvalCCA.m    - example for crossvalidatory CCA-PLS and CCA;
                       produces percentage crossvalidatory fit
Example_crossvalCCA.txt  - output of Example_crossvalCCA




Utility functions:
coinertia.m     -  does coinertia analysis according to Doledec & Chessel (1996) via
                   a (generalized) singular value decomposition (svd)
coinertia_I.m   -  coinertia analysis in the identity metric based on svd of X'Y
                   = Tucker's interbattery factor analysis (used in permtest.m)
rcsum.c         -  row, colum and total sums of a matrix
scale_chi       -  forms chi-square residuals with predefined row and column metrics R0 and K1
scale_lin       -  centers and standardizes wrt to a prefined mean and standard deviation
mncn_chi        -  forms chi-square residuals in the metric R0; it implements equation 8
                   This transformation can be seen as mean-centering in the chi-square metric
mncn_lin        -  centers the colums of a matrix in the R0-metric and pre-multiplies with
                   the square-root of the metric R0
permtest        -  permutation test
residualmatrix  -  function to substract the fit of an ordination axis (deflation)


Functions that require a licence for PLS_Toolbox 2.1  (www.eigenvector.com)
simpls.m           - basic function for SIMPLS  [NOT included in this archive!]
crossval.m         - function for crossvalidation using PLS, PCA and regression [not used itself here]
crossval_chi.m     - modification of crossval.m by Cajo ter Braak for 
                       crossvalidation of simpls-based predictive co-correspondence analysis
crossval_chi_lin.m   - modification of crossval.m by Cajo ter Braak for 
                       crossvalidation of canonical correspondence analysis (CCA) and CCA-PLS;
                       Uses simpls.