Cannonical Correlation Analysis

Question

I have just started working using CCA in Matlab. I have two vectors X and Y of dimension 60x1920 and 60x1536 with the number of samples being 60 and variables in the different set of vectors being 1920 and 1536 respectively. I want to know do CCA for reducing them to the subspace and then do feature matching.

I am using this commands.

%% DO CCA
[A,B,r,U,V] = canoncorr(X,Y);

The output I get is this :

  Name         Size             Bytes  Class     Attributes

  A         1920x58            890880  double              
  B         1536x58            712704  double              
  U           60x58             27840  double              
  V           60x58             27840  double              
  r            1x58               464  double       

Can anyone please tell me what these variables mean. I have gone over the documentation several times and still is unclear about them. As I understand CCA finds two linear projection matrices Wx and Wy such that the projection of X and Y on Wx and Wy are maximally correlated.

1) Could anyone please tell me which of the following matrices are these?

2) Also how can I find the projected vectors in the learned subspace of CCA?

Any help will be appreciated. Thanks in advance.

Answer 1

As I understand it, with X and Y being your original data matrices, A and B are the sets of coefficients that perform a change of basis to maximally correlate your original data. Your data is represented in the new bases as the matrices U and V.

So to answer your questions:

1. The projection matrices you are looking for would be A and B since they transform X and Y into the new space.

2. The resulting projections of X and Y into the new space would be U and V, respectively. (The r vector represents the entries of the correlation matrix between U and V, which is a diagonal matrix.)

The The MATLAB documentation says this transformation can be done with the following formulae, where N is the number of observations:

U = (X-repmat(mean(X),N,1))*A
V = (Y-repmat(mean(Y),N,1))*B

This page lays out the process nicely so you can see what each coefficient means in the transformation process.