Information Technology Reference
In-Depth Information
It is proved in [13] that the T-index statistic
T
ij
(
t
)
is asymptotically distributed
according to the
t
-distribution with
degrees of freedom, where
d
denotes
the total number of STLmax values per channel contained in a moving window
W
i
.
Let
T
ij
(
d
−
1
)
(
t
)
>
t
α
/
2
,
d
−
1
denote
(
1
−
α
/
2
)
×
100% critical value of the
t
-distribution
with
degrees of freedom. Then using the paired
t
-test, the sites
i
and
j
are
considered
disentrained
,if
T
ij
(
d
−
1
)
(
t
)
>
t
α
/
2
,
d
−
1; otherwise, the sites are
entrained
.
20.3.3 Finding Critical Sites by Quadratic Optimization Approach
An interesting application of optimization theory to the problem of determining
critical cortical sites involved in the preictal transition into the seizure state is given
by an analog of the Ising spin glass model [13]. The Ising model is defined via
the Sherrington-Kirkpatrick Hamiltonian, which is used to introduce the mean-field
theory of the spin glasses. The spin glass is represented by a regular lattice, with
the elements at the vertices of the lattice. Furthermore, the magnetic interactions
among the elements hold only for the nearest neighbors, and each element has only
two possible states [27]. The Ising spin glass model is widely utilized to examine
phase transitions in the field of statistical physics. More specifically, the ground state
problem in the Ising model of finding the spin glass configurations of the minimal
energy can be applied to determine phase transitions in dynamical systems.
Since the Ising model is defined on a regular lattice, it admits the following nat-
ural representation in terms of graph theory. Given a graph
G
(
V
,
E
)
wit
h
the vertex
set
V
=
{
v
1
, ··· ,
v
n
}
of size
n
, and t
he
edge set
E
, let us assign a weight
ω
ij
to every
edge
ω
ij
represent the interaction energy between the
elements (vertices)
v
i
and
v
j
of the spin glass (graph
G
(
v
i
,
v
j
)
∈
E
. Here the weights
}
denote a magnetic spin variable associated with a given vertex
v
i
of the spin glass
graph. Then the spin glass configuration
(
V
,
E
)
). Let
σ
∈{
+
1
,−
1
σ
min
with the minimum energy of magnetic
interactions is found by minimizing the Hamiltonian
H
:
(
σ
)=
−
∑
H
1
≤
i
,
j
≤
n
ω
ij
σ
i
σ
j
,
(20.4)
over all possible configurations
.
This problem (20.4) is equivalent to combinatorial formulation of the quadratic
bivalent programming problem [7]. Analogously to the quadratic programming for-
mulation (20.4) for the Ising spin glass model, the problem of finding the cortical
sites critical with respect to the transition of the epileptic brain into the seizure state
is formulated as a quadratic 0-1 programming problem.
Let
x
∈
{
σ
=(
σ
1
, ··· ,
σ
n
)
∈{−
1
,
+
1
}
0
,
1
}
denote the choice between selecting
(
x
i
=
1
)
and disregarding
(
the information from the channel
i
, then by introducing a T-index that
represents a statistical measure of the similarity in the STLmax values between a
pair of EEG channels, the problem is formulated as finding critical electrodes that
minimize the average value of T-index statistic.
x
i
=
0
)
