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practical applications, dimensionality is such that reasonably small sample sizes are
su
cient.
Monte Carlo integration was used to estimate
V
ˆ
by generating
m
uniformly
distributed random sample points and calculating an estimate of
V
ˆ
as
∑
m
()
x
i
i
1
S
VV
S
m
S
=
where
x
i
,
i
1, ...,
m
, is the sequence of the sample points. h e interval of confi -
dence of such an estimate can be calculated for a given allowed error. Particularly,
an interval of confi dence equal to 0.998 was considered; therefore, the correspond-
ing allowed error was calculated from the right-hand side of the inequality in the
following equation:
2
VV
S
S
P
V
V
3
0 998
.
m
S
S
Once an estimate of the volume of self-space has been calculated, and an estimation
of the eff ective coverage (volume) of a detector has also been computed, an estimation
of the number of detectors necessary to cover the nonself space can be obtained.
Simulated annealing was used to fi nd a good distribution of the detectors. h e
set of initial detectors is generated at random; subsequently, detectors are iteratively
redistributed to approach the optimal distribution. h is process was done to opti-
mize an “objective function”
C
(
D
) that measured the coverage of a set of detectors
D
. h us,
C
(
D
) is defi ned as
=
+
×
C
(
D
)
Overlapping
(
D
)
β
SelfCovering
(
D
),
=
where
D
{
d
1
, …,
d
num
ab
} is a set of detectors (antibodies);
num
ab
is the number of
detectors in
D
;
overlapping
(
D
) is a function used to calculate the overlap between
the detectors in
D
defi ned as
2
dd
r
i
j
≠
∑
2
…
Overlapping D
()
e
, ,
i j
1
num
ab
ab
ij
and “SelfCovering” is a function that is used to “penalize” a detector when it
matches any self-sample, and it is calculated as
2
ds
∑
∈
∑
2
((
r
r
)/
2
)
SelfCovering
()
D
e
ab
self
∈
sS
dD
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