Information Technology Reference
In-Depth Information
Suppose that the T-index values
T
ij
between a pair
of electrode sites are
given by the elements of an
nxn
real-valued matrix
Q
, and let the selection of the crit-
ical electrodes be represented by an
n
-dimensional vector
x
(
i
,
j
)
n
=(
x
1
, ··· ,
x
n
)
∈{
0
,
1
}
, where the selection of the cortical site
i
corresponds to
x
i
=
0 indi-
cates that the channel
i
is not selected. By adding a linear constraint on the number
k
1 , while
x
i
=
(
1
≤
k
≤
n
)
of selected channels,
∑
1
≤
i
,
j
≤
n
x
i
=
k
,
(20.5)
the problem of determining
k
critical electrodes sites involved in transition into the
ictal state based on the matrix of T-index values
Q
can be formulated as the follow-
ing quadratic 0-1 knapsack problem:
min
x
T
Qx
∑
∑
1
≤
i
,
j
≤
n
n
,
s
.
t
x
i
=
k
,
x
∈{
0
,
1
}
.
(20.6)
By introducing the penalty term to guarantee that the optimal solution satis-
fies the constraint (20.5) the problem (20.6) can be equivalently reformulated as
a quadratic 0-1 programming problem:
c
k
2
min
x
T
Qx
∑
1
≤
i
,
j
≤
n
n
+
x
i
−
,
s
.
tx
∈{
0
,
1
}
,
(20.7)
n
i
,
j
=
where the penalty constant
c
is computed from
Q
=(
q
ij
)
1
as
2
∑
c
=
1
≤
i
,
j
≤
n
|
q
ij
|
+
1
.
(20.8)
There are several computational approaches to solving problem (20.7), including
a branch and bound procedure (B&B) with dynamical rule for fixing variables, a lin-
earization approach to reformulate (20.7) as an integer programming (IP) problem
by introducing additional variables to represent
x
i
×
x
j
, and utilizing Karush-Khun-
Tucker optimality conditions to obtain mixed integer linear programming (MILP)
reformulation.
20.4 Two Main Components of the Seizure Monitoring and Alert
System
The proposed seizure monitoring and alert system (SMAS) consists of two main
components, the algorithm for generating automatic seizure warnings, and the
seizure susceptibility index (SSI).
