Arguments
label	type	description
leftVar	variable	Variable with same time dimension as predicand
rightVar	variable	Variable with same time dimension as predictor
weight1	proc	a procedure to be applied as weight for both right and left variables if no other optional procedure is set up, or for the left variable otherwise. (optional)
weight2	proc	a procedure to be applied as weight for right variable. (optional)
Tgrid	grid set	Time grid
pm	number	Number of modes to keep from leftVar SVD.
qm	number	Number of modes to keep from rightVar SVD.
Returns
cca	dataset	Dataset containing CCA outputs

Here is a brief description of what the Canonical Corrlation Analysis does. The inputs of the function are italic and the outputs of the function are bold.

1. Computes the Singular Value Decomposition (SVD) for leftVar (rightVar);
2. reduces the left (right) space where leftVar (rightVar) covariance matrix is diagonal to the pm (qm) eigenvectors - Principal Components (PCs) - with the highest explained variance;
3. computes the correlation matrix between the left and right PCs;
4. finds the basis of orthonormal vectors r and s that diagonalize the correlation matrix into a matrix of which elements are the canonical correlations mu;
5. projects the standardized singular values (representing the percentage of explained variance) of leftVar (rightVar) SVD on r (s) to form the left (right) explained variance exvarl (exvarr) showing how each CCA mode explains each of the original datasets;
6. projects the left (right) PCs on r (s) to form unit variance canonical time series u (v);
7. projects leftVar (rightVar) on the left (right) canonical time series u (v) to form the left (right) canonical patterns g (h).

Properties:

• ev, the dimension of the reduced CCA space is the minimum of pm and qm, which is by construction qm;
• r and s are orthonormal basis;
• u v mul [T]sum is mu;
• u and v are orthogonal basis with unit variance;
• u g mul [ev]sum and v h mul [ev]sum are respectively the reconstruction of the pm first leftVar SVD PCs and qm first rightVar SVD PCs.

Example (note the use of shift data to have time series of different time grid actually match):
 SOURCES .NOAA .NCDC .ERSST .version2 .SST
   X (30E) (180) (80W) RANGE
   Y (30S) (30N) RANGE
   T (Sep-Nov 1961-2000) seasonalAverage
   dup [T]average sub
   T -3 shiftdata
 SOURCES .WCRP .GCOS .GPCC .FDP .version4 .0p5 .prcp
   X (118E) (128E) RANGE
   Y (5N) (20N) RANGE
   T (Dec-Feb 1961-2001) seasonalAverage
   dup [T]average sub
 [T]15 10 cca