Source-Space Connectivity Analysis Using Imaginary Coherence - Electromagnetic Brain Imaging - page 151

Using the same derivation mentioned in Sect. C.9, the maximum value is equal to
the largest eigenvalue of the matrix

ʣ − 1

xy aa H

yy ʣ

ʣ xy .

The matrix above is a rank-one matrix and it only has a single nonzero eigenvalue.
According to Sect. C.8 (Property No. 10), this eigenvalue is equal to

a H ʣ xy ʣ − 1

H

yy ʣ xy a

, (7.63)

which is a scalar.
Thus, the optimization

2

b H

H

xy aa H

subject to a H 1 and b H | ˈ | = max

a

ʣ ʣ xy b ʣ xx a = ʣ yy b = 1 ,

, b

(7.64)

is now rewriten as

2

a H ʣ xy ʣ − 1

H

subject to a H | ˈ | = max

a

yy ʣ xy a

ʣ xx a = 1 . (7.65)

Using again the derivation described in Sect. C.9, the solution of this maximization
is obtained as the maximum eigenvalue of the matrix

ʣ − 1

xx ʣ xy ʣ − 1

H

yy ʣ xy .

That is, denoting the eigenvalues of this matrix as ʳ j where j = 1 ,..., d and

d = min { p , q } , the canonical squared magnitude coherence is derived as

2

= S max { ʣ − 1

xx ʣ xy ʣ − 1

H

| ˈ |

yy ʣ xy }= ʳ 1 ,

(7.66)

where the notation indicates the maximum eigenvalue of a matrix between

the parentheses, as is defined in Sect. C.9.
This canonical squared magnitude coherence is considered the best overall mag-
nitude coherence measure between the two sets of multiple spectra x 1 ,...,

S max {·}

x p and

ʳ 1 in Eq. ( 7.66 ). However,
other eigenvalues may have information complementary to

y 1 ,..., y q , and it is equal to the maximum eigenvalue

ʳ 1 , and therefore, a met-
ric that uses all the eigenvalues may be preferable. Let us assume the random vectors
x and y to be complex Gaussian. According to Eq. (C.52), we can then define the
mutual information between x and y such that

d

1

I( x , y ) = log

− ʳ j ,

(7.67)

1

j = 1