Information Technology Reference
In-Depth Information
4.6.3.1
Estimation of Detector Volume and Overlap
=
[0, 1]
n
be the system state space and
A
⊆
X
a subset of
X
, whose volume
needs to be computed. Also, a typical assumption is that it is hard to compute the
volume of
A
analytically. If
x
is drawn from a uniform distribution on
X
, then
P
(
x
Let
X
=
=
1. h en the problem
of computing
V
(
A
) can be seen as the problem of estimating
P
(
x
∈
A
)
volume of
A
, denoted as
V
(
A
) because
V
([0, 1]
n
)
∈
A
).
h erefore, let
U
be a random variable uniformly distributed on
X
, denoted by
∼
U
(0, 1). Let
U
1
,
U
2
, …,
U
n
be a sequence of “independent and identically
distributed (iid)”
U
(0, 1) random variables. h en, the sequence
X
1
,
X
2
, …,
X
N
of
random variables, generated as follows, are uniformly
iid
in [0, 1]
n
, denoted by
X
i
∼
U
U
([0, 1]
n
).
X
1
=
(
U
1
, …,
U
n
)
=
X
2
(
U
n
+
1
, …,
U
2
n
)
…
=
(
U
(N
−
1
)n
+
1
, …,
U
nN
)
To estimate the volume of
A
, generate a sequence
X
1
,
X
2
, …,
X
N
, as defi ned earlier.
h en, an estimation of the volume of
A
may be computed as
X
N
{:
iX
∈
A
}
()
i
VA
N
where |·| denotes the number of points in a set. In other words, the volume of
A
is
estimated as the fraction of points that lie in
A
. An estimate of the volume of
A
can
also be expressed as
N
∑
Y
i
()
1
i
VA
N
=
with
Y
i
I
A
(
X
i
), where
I
A
(·) denotes the indicator function of set
A.
Y
1
,
Y
2
, …,
Y
N
is a sequence of independent Bernoulli random variables, with
P
(
Y
i
=
=
1)
∫
h e main advantage of this method is that it is possible to calculate a confi dence
interval for the estimated volume
V
ˆ
(
A
) as follows. To estimate the volume with a
confi dence of (1
∫
P X
()
……
1
dx
,
,
dx
V A
( .
i
n
A
−
α
), using the “Chernoff bound,” it can be shown that if
32
2
ln(
/ )
()
,
N
then
VA
However, as the dimensionality increases
V
(
A
) approaches zero exponentially
quickly, which will require a sample size exponentially large. Nevertheless, in many
PVA
(()
VA
()
VA
( )
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