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where
s
refers to the set of nonlinear parameters that are optimized to minimize the
data-fit cost, and the Frobenius norm of a matrix is the square root of the sum of the
squares of all the elements of the matrix [80, 6]. The nonlinear parameters of the
i
th
dipole are its position vector,
r
(
i
)
=(
T
,
which specify the dipole's location and orientation. Thus, the cost is minimized in a
space of dimension 5
d
d
(or 4
d
d
for the MEG single sphere head model), where
d
d
is the number of dipoles in the model (i.e., the order of the model).
B
is the part of the data explained by the ECD generative model:
B
T
, and its angle vector,
x
i
,
y
i
,
z
i
)
ω
(
i
)
=(
φ
i
,
θ
i
)
L
s
J
s
,
where
L
s
is the lead field or gain matrix containing
d
d
d
b
-dimensional column vec-
tors called gain vectors. They are computed for
d
d
dipoles of unit amplitude with
parameters specified in
s
. The estimated
d
d
by
d
v
current matrix,
J
s
=
=
L
s
B
, contains
the moments of the
d
d
dipoles, where
L
s
is the pseudoinverse of
L
s
[23]. Thus, the
i
th row vector of
J
s
contains the moments of the dipole located at position
r
(
i
)
with
.
I
is the
d
b
-dimensional identity matrix, and
P
L
s
is the orthogonal
projection operator onto the null space of
L
s
. Note that the gain matrix needs to be
recomputed at each iteration for every new
s
.
Alternatively, the orientations of the dipoles can be obtained linearly if only the
positions are optimized by including the gain vectors of all three orthogonal dipole
components pointing in the
orientation
ω
(
i
)
directions, so that
L
s
is a
d
b
by 3
d
d
matrix and
J
s
is a 3
d
d
by
d
v
matrix. For this rotating dipole model, the cost function exists in a
space of 3
d
d
dimensions.
This least-squares approach is equivalent to maximum likelihood estimation of
the parameters that maximize the Gaussian likelihood defined by:
(
x
,
y
,
z
)
−
d
b
d
v
/
2
exp
ϒ
)=
2
B
LJ
s
1
2
F
J
s
,
2
2
ϒ
p
(
B
|
s
,
d
d
,
σ
πσ
−
−
,
(8.2)
2
ϒ
2
σ
2
ϒ
where noise is assumed to be Gaussian and uncorrelated with scalar variance
.
The parameters
s
(
ml
)
and
J
s
(
ml
)
that maximize the likelihood or equivalently mini-
mize the negative log likelihood at convergence are the maximum likelihood esti-
mates of the dipole positions, orientations, and amplitudes.
σ
8.5.2 Correlated Noise Model
In the presence of correlated noise, a modified cost function can be minimized:
Σ
−
1
/
2
ϒ
L
s
J
s
tr
B
LJ
s
B
LJ
s
T
B
2
F
=
Σ
−
1
ϒ
min
s
−
−
−
,
(8.3)
Σ
−
1
/
2
ϒ
where
is a whitening matrix obtained by taking the square root inverse of
the noise covariance matrix,
Σ
ϒ
[78]. This solution again is equivalent to a maxi-
mum likelihood estimate of the parameters using a Gaussian likelihood noise model
defined by:
