nag_mv_canon_corr (g03adc) performs canonical correlation analysis upon input data matrices.

Let there be two sets of variables,  and . For a sample of  observations on  variables in a data matrix  and  variables in a data matrix , canonical correlation analysis seeks to find a small number of linear combinations of each set of variables in order to explain or summarise the relationships between them. The variables thus formed are known as canonical variates.

Let the variance-covariance matrix of the two data sets be

and let

then the canonical correlations can be calculated from the eigenvalues of the matrix . However, nag_mv_canon_corr (g03adc) calculates the canonical correlations by means of a singular value decomposition (SVD) of a matrix . If the rank of the data matrix  is  and the rank of the data matrix  is , and both  and  have had variable (column) means subtracted, then the  by  matrix  is given by:

where  is the first  rows of the orthogonal matrix  either from the  decomposition of  if  is of full column rank, i.e., :

or from the SVD of  if :

Similarly  is the first  rows of the orthogonal matrix  either from the  decomposition of  if  is of full column rank, i.e., :

or from the SVD of  if :

Let the SVD of  be:

then the non-zero elements of the diagonal matrix , , for , are the  canonical correlations associated with the  canonical variates, where .

The eigenvalues, , of the matrix  are given by:

The value of  gives the proportion of variation explained by the th canonical variate. The values of the  give an indication as to how many canonical variates are needed to adequately describe the data, i.e., the dimensionality of the problem.

To test for a significant dimensionality greater than  the  statistic:

can be used. This is asymptotically distributed as a  distribution with  degrees of freedom. If the test for  is not significant, then the remaining tests for  should be ignored.

The loadings for the canonical variates are calculated from the matrices  and  respectively. These matrices are scaled so that the canonical variates have unit variance.

"NE_3_INT_ARG_CONS"

On entry, ,  and . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_CANON_CORR_1"

A canonical correlation is equal to one. This will happen if the  and  variables are perfectly correlated.

"NE_INT_ARG_LT"

On entry, nx must not be less than 1: .

"NE_INTERNAL_ERROR"

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.

"NE_MAT_RANK_ZERO"

The rank of the  matrix or the rank of the  matrix is zero. This will happen if all the  and  variables are constants.

"NE_NEG_WEIGHT_ELEMENT"

On entry, . Constraint: When referenced, all elements of wt must be non-negative.

"NE_OBSERV_LT_VAR"

With weighted data, the effective number of observations given by the sum of weights , while number of variables included in the analysis, . Constraint: Effective number of observations .

"NE_REAL_ARG_LT"

On entry, tol must not be less than : .

"NE_SVD_NOT_CONV"

The singular value decomposition has failed to converge. This is an unlikely error exit.

"NE_VAR_INCL_INDICATED"

The number of variables, nx in the analysis , while the number of  variables included in the analysis via array . Constraint: these two numbers must be the same. The number of variables, ny in the analysis , while the number of  variables included in the analysis via array . Constraint: these two numbers must be the same.

As the computation involves the use of orthogonal matrices and a singular value decomposition rather than the traditional computing of a sum of squares matrix and the use of an eigenvalue decomposition, nag_mv_canon_corr (g03adc) should be less affected by ill conditioned problems.

None.