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In-Depth Information
Example 2
We illustrate h-FNNon the example shown in Fig.
11.6
.
N
k
,
C
(
is shown
in the fifth and sixth column of Table
11.2
for both classes of circles (
C
1
) and rectangle
(
C
2
). Similarly to the previous section, we calculate
N
k
,
C
(
x
)
x
i
)
using
k
=
1, but we
classify instance 11 using
k
=
2 nearest neighbors, i.e.,
x
6
and
x
9
. The relative class
hubness values for both classes for the instances
x
6
and
x
9
are:
u
C
1
(
x
6
)
=
0
/
2
=
0
,
u
C
2
(
x
6
)
=
2
/
2
=
1
,
u
C
1
(
x
9
)
=
1
/
2
=
0
.
5
,
u
C
2
(
x
9
)
=
1
/
2
=
0
.
5
.
According to (
11.11
), the class probabilities for instance 11 are:
0
+
0
.
5
u
C
1
(
x
11
)
=
5
=
0
.
25
,
0
+
1
+
0
.
5
+
0
.
and
1
+
0
.
5
u
C
2
(
x
11
)
=
5
=
0
.
75
.
0
+
1
+
0
.
5
+
0
.
As
u
C
2
(
,
x
11
will be classified as rectangle (
C
2
).
Special care has to be devoted to anti-hubs, such as instances 4 and 5 in Fig.
11.6
.
Their occurrence fuzziness is estimated as the average fuzziness of points from the
same class. Optional distance-based vote weighting is possible.
x
11
)>
u
C
1
(
x
11
)
11.5.3 NHBNN: Naive Hubness Bayesian k-Nearest Neighbor
Each
k
-occurrence can be treated as a random event. What NHBNN [
53
] does is that
it essentially performs a Naive-Bayesian inference based on these
k
events
y
∗
=
x
∗
))
(
|
N
k
(
∝
(
)
(
x
i
∈
N
k
|
),
P
C
P
C
P
C
(11.12)
x
i
∈
N
k
(
x
∗
)
where
P
(
C
)
denotes the probability that an instance belongs to class
C
and
P
(
x
i
∈
N
k
|
denotes the probability that
x
i
appears as one of the
k
nearest neighbors of
any instance belonging to class
C
. From the data,
P
C
)
(
C
)
can be estimated as
train
C
)
≈
|
D
|
P
(
C
|
,
(11.13)
train
|
D
train
C
train
where
|
D
|
denotes the number of train instances belonging to class
C
and
|
D
|
is the total number of train instances.
P
(
x
i
∈
N
k
|
C
)
can be estimated as the fraction
N
k
,
C
(
x
i
)
P
(
x
i
∈
N
k
|
C
)
≈
|
.
(11.14)
train
C
|
D
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