Supplementary Mathematical Arguments - Electromagnetic Brain Imaging - page 249

∂ L
∂ ʱ

= ʠ xy ʲ − ʽ 1 ʱ = 0 , (C.36)

∂ L
∂ ʲ

xy

= ʠ ʱ − ʽ 2 ʲ = 0 , (C.37)

ʠ xy = ʣ − 1 / 2 ʣ xy ʣ − 1 / 2

T

where . Using the constraint ʱ ʱ = 1, and left multiplying

xx yy

T

ʱ to Eq. ( C.36 )gives

T

ʱ ʠ xy ʲ = ʽ 1 .

T T

Using the constraint ʲ ʲ = 1, and left multiplying ʲ to Eq. ( C.37 )gives

T xy

ʲ ʠ ʱ = ʽ 2 .

ʽ 1 = ʽ 2 = ˁ c .

Since the left-hand sides of the two equations above are equal, we have

Also, using Eq. ( C.37 ), we get

1
ˁ c ʠ

T

ʲ =

xy ʱ ,

and substituting the equation above into Eq. ( C.36 ), we get

xy

T 2

ʠ xy ʠ ʱ = ˁ

c ʱ .

(C.38)

c

This equation indicates that the squared canonical correlation ˁ is obtained as the

xy , and the corresponding eigenvector is the solution

eigenvalues of a matrix ʠ xy ʠ

xy

for the vector ʱ . Note that ʠ xy ʠ is a real symmetric matrix, the eigenvectors ʱ

and ʲ are real-valued, and therefore a and b are real-valued because ʣ xx and ʣ yy

are real-valued matrices.
Denoting the eigenvalues in Eq. ( C.38 )as

μ j where j = 1 ,..., d and d =

min { p , q } , the canonical correlation between the two sets of random variables

x 1 ,..., μ 1 , which is the

best overall measure of the association between x and y . However, other eigenvalues
may provide complementary information on the linear relationship between those
two sets of random variables. The mutual information described in Sect. C.3.2 is a
measure that can take all the eigenvalues into account.
Also, it is worth mentioning that the matrices,

x p and y 1 ,..., y q is obtained as the largest eigenvalue

T

xy = ʣ − 1 / 2

ʣ xy ʣ − 1

T

xy ʣ − 1 / 2

ʠ xy ʠ

yy ʣ xx xx

and

ʣ − 1

xx ʣ xy ʣ − 1

T
xy yy ʣ

(C.39)