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where
LT
(
o
,{
o
},X
D
(
o
,
A
)
,
σ )
is the local simple typicality of
o
in its representing
region
D
(
o
,
A
)
and
N
=
D
(
o
,
A
)
∩
D
(
{
o
},
σ )
is the
σ
-neighborhood region of
o
in
its representing region
D
(
o
,
A
)
.
Definition 4.6 (Local representative typicality).
Given an uncertain object
O
,a
neighborhood threshold
be the random
vector generating the samples
O
, the
local representative typicality
of an object
o
σ
and a
reported answer set
A
⊂
O
, let
X
∈
(
−
)
(
,
,X ,
σ)=
(
∪{
},X ,
σ )
−
(
,X ,
σ )
O
A
is
LRT
o
A
LGT
A
o
LGT
A
.
For any instance
o
∈
A
, let
N
(
o
,
A
,
O
)=
{
x
|
x
∈
O
∩
D
(
o
,
A
)
}
be the set of in-
stances in
O
that lie in
D
(
o
,
A
)
, then
LN
(
{
o
},
N
(
o
,
A
,
O
)
,
σ )
is the
σ
-neighborhood
of
o
in
N
(
o
,
A
,
O
)
. The local group typicality
LGT
(
A
,X ,
σ )
can be estimated using
)
,
σ )
·
|
LN
(
{
o
},
N
(
o
,
A
,
O
)
,
σ )
|
|
LGT
(
A
,
O
,
σ)=
∑
o
∈
A
LT
(
o
,{
o
},
N
(
o
,
A
,
O
. Hence, the lo-
O
|
cal representative typicality is estimated by
LRT
(
o
,
A
,
O
,
σ)=
LGT
(
A
∪{
o
},
O
,
σ )
−
LGT
.
Local representative typicality can approximate representative typicality with
good quality, as shown below.
(
A
,
O
,
σ )
Theorem 4.7 (Local representative typicality approximation).
Given an uncer-
tain object O, a neighborhood threshold
σ
, and an answer set A
⊂
O, let
o
=
arg max
o
1
∈
(
O
−
A
)
{
LRT
(
o
1
,
A
,
O
,
σ )
}
be the instance in
(
O
−
A
)
having the largest
local representative typicality value, and o
=
arg max
o
2
∈
(
O
−
A
)
{
RT
(
o
,
A
,
O
)
}
be the
instance in
(
O
−
A
)
having the largest representative typicality value. Then,
2
h
√
2
e
−
σ
2
RT
(
o
,
A
,
O
)
−
RT
(
o
,
A
,
O
)
<
(4.13)
2
h
2
π
∈
(
−
)
Moreover, for any x
O
A
,
1
h
√
2
e
−
σ
2
|
RT
(
x
,
A
,
O
)
−
LRT
(
x
,
A
,
O
,
σ )
|<
(4.14)
2
h
2
π
Proof.
To prove Theorem 4.7, we need the following lemma.
Lemma 4.1 (Local group typicality score approximation).
Given an uncertain
object O, a neighborhood threshold
σ
and a reported answer set A
⊂
O.
1
h
√
2
e
−
σ
2
GT
(
A
,
O
)
−
LGT
(
A
,
O
,
σ )
<
(4.15)
2
h
2
π
Proof.
For each instance
o
∈
A
, let
N
(
o
,
A
,
O
)
be the set of instances in
O
that lie in
(
,
)
D
o
A
, according to Equation 4.1, we have
)
×
|
N
(
o
,
O
,
A
)
|
|
1
T
(
o
,
N
(
o
,
O
,
A
))
·
Pr
(
N
(
o
,
O
,
A
)) =
h
|
N
(
o
,
O
,
A
)
|
∑
x
∈
N
(
o
,
O
,
A
)
G
h
(
x
,
o
O
|
1
h
|
O
|
∑
=
G
h
(
x
,
o
)
x
∈
N
(
o
,
O
,
A
)
Let
N
=
LN
(
{
o
},
N
(
o
,
A
,
O
)
,
σ )
be the
σ
-neighborhood of
o
in
N
(
o
,
A
,
O
)
, then
