Digital Signal Processing Reference
In-Depth Information
(
)
(
)
(4)
Pzt H
() 1|
=
=
0
=
1
−
Pzt H
()
=
0|
=
0
=
EP
[
]
k
k
f
If the cell selected at time
t
is
i
(
t
), the corresponding search result
z
(
t
) can be predicted
as
1
(
)
(
)
(5)
=
P
z
(
t
)
|
i
(
t
)
=
P
z
(
t
)
|
H
=
j
,
i
(
t
)
⋅
P
(
H
=
j
)
i
(
t
)
i
(
t
)
j
0
Now the information gain obtained is analyzed when the cell
i
(
t
) is determined for
next search. The measure function of information gain is expressed as
f
(
p
k
,
q
k
) which
can be defined according to Shannon entropy, Kullback-Leibler entropy, Renyi entro-
py etc. The parameters
q
k
and
p
k
are the probabilities of the target existing before and
after the sensor search in the cell k respectively. So the total information gain consi-
dering all cells and possible detection results is expressed as
1
M
(6)
(
)
(
)
(
)
(
)
Δ
Dtit
,()
=
f PH zt
=
1| ()
=
jit
,() ,
π
()
t Pzt
⋅
()
=
j it
| ()
k
k
j k
==
01
The search process is that: first the total information gain Δ
D
(
t
,
i
(
t
)) for each cell
i
(
t
) is
predicted, and then the cell with the maximal information gain is selected and used for
next search, which can decrease uncertainty of the target furthest.
3
Cued Search with Unknown Detection Performance
When the probabilities of detection and false alarm are unknown, the mathematics of
the previous section can no longer be straightforwardly applied. Rather, probability
densities must be used to model unknown probabilities of detection and false alarm,
which may be a random value between 0 and 1. In this paper, beta distribution is se-
lected, and the reasons are: (1) beta distribution is the natural conjugate prior for a
binomial process, and its posterior probability density is also a beta distribution; (2)
beta distribution can simulate most of probability density functions such as uniform
distribution, normal distribution, Rayleigh distribution, Log-normal distribution and
so on.
The beta distribution used to model the probability density function of
P
d
is ex-
pressed as follows
1
(7)
a
−
1
b
−
1
fPab
(,,)
=
⋅
P
⋅
1
−
P
),0
<
P
<
1
d
d
d
d
Bab
(,)
where
B
(
a,b
) is the beta function. Beta distribution has two positive parameters
a
and
b
with mean
a
/(
a+b
) and variance
ab
/((
a+b
)
2
(
a+b+
1)). The model of the false alarm
probability is the same to that of detection probability. So the modeling process for
the false alarm probability is omitted.
According to the beta distribution model, (3) can be rewritten as
(
)
(
)
(8)
Pzt H
() 1|
=
=
1
=
1
−
Pzt H
()
=
0|
=
1
=
a a b
/(
+
)
k
k
