Source-Space Connectivity Analysis Using Imaginary Coherence - Electromagnetic Brain Imaging - page 154

Following the derivation in Sect. C.3.1, the solution of the above optimization
problem is expressed as the eigenvalue problem,

xy

T 2

ʠ xy ʠ ʱ = ˈ

I ʱ ,

(7.82)

ʠ xy = ʓ − 1 / 2 ʥ xy ʓ − 1 / 2

T

where . In the above equation, since the matrix, ʠ xy ʠ

xy is

xx yy

a real symmetric matrix, the eigenvector ʱ is real-valued. The vector ʲ is obtained

using

1
ˈ I ʥ

T

ʲ =

xy ʱ ,

which is also real-valued.
The matrices

T

ʓ − 1

xx ʥ xy ʓ − 1

T

xy have the same eigenvalues, accord-
ing to Sect. C.8 (Property No. 9). Let us define the eigenvalues of the matrix
ʓ − 1

ʠ xy ʠ

xy and yy ʥ

xx ʥ xy ʓ − 1

T 2

I yy ʥ xy as

ʶ j ( j = 1 ,..., d ). The canonical imaginary coherence ˈ

is derived as

2
I

= S max { ʓ − 1

xx ʥ xy ʓ − 1

T

ˈ

yy ʥ xy }= ʶ 1 .

(7.83)

Using the same arguments in the preceding section, the imaginary-coherence-based
mutual information is given by

d

1

I s ( , ) =

− ʶ j .

x y log (7.84)

1

j = 1

The alternative definition of canonical imaginary coherence, based on I s ( x , y ) ,is

obtained as

d

ˈ

2
I

= 1 − exp [− I s ( x , y ) ]= 1 −

1 (

1 − ʶ j ), (7.85)

j =

which uses all the eigenvalues ʶ 1 ,...,ʶ d . When we can assume ʶ j 1 (where

j = 1 ,..., d ), we have

d

ˈ

2
I

≈

1 ʶ j .

(7.86)

j =

The connectivity metric above has been proposed and called the multivariate inter-
action measure (MIM) in [ 12 ].
The mutual information independent of the sizes of vectors is defined such that,

d

1
d

1

I s (

x , y ) = log

− ʶ j .

(7.87)

1

j = 1