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x
∗
)
y
∗
=
u
C
(
∝
ʲ(
x
i
)
·
d
w
(
x
i
)
·
P
k
(
C
|
x
i
).
(11.20)
x
i
∈
N
k
(
x
∗
)
Example 4
Next, we illustrate HIKNN by showing how HIKNN classifies instance
11 of the example shown in Fig.
11.6
.Again,weuse
k
=
2 nearest neighbors to
classify instance 11, but we use
N
1
(
1. The both
nearest neighbors of instance 11 are
x
6
and
x
9
. The self-information associated with
the occurrence of these instances as nearest neighbors:
x
i
)
values calculated with
k
=
2
10
=
1
0
P
(
x
6
∈
N
1
)
=
0
.
2
,
I
x
6
=
log
2
2
=
log
2
5
,
.
2
10
=
1
0
P
(
x
9
∈
N
1
)
=
0
.
2
,
I
x
9
=
log
2
2
=
log
2
5
.
.
The relevance factors are:
log
2
5
ʱ(
x
6
)
=
ʱ(
x
9
)
=
0
,ʲ(
x
6
)
=
ʲ(
x
9
)
=
log
2
10
.
The fuzzy votes according to (
11.19
):
y
∗
=
y
∗
=
P
k
(
C
1
|
x
6
)
=
u
C
1
(
x
6
)
=
0
,
P
k
(
C
2
|
x
6
)
=
u
C
2
(
x
6
)
=
1
,
y
∗
=
y
∗
=
P
k
(
C
1
|
x
9
)
=
u
C
1
(
x
9
)
=
0
.
5
,
P
k
(
C
2
|
x
9
)
=
u
C
2
(
x
9
)
=
0
.
5
.
The sum of fuzzy votes (without taking the distance weighting factor into account):
log
2
5
log
2
5
u
C
1
(
x
11
)
=
log
2
10
·
0
+
log
2
10
·
0
.
5
,
log
2
5
log
2
5
u
C
2
(
x
11
)
=
log
2
10
·
1
+
log
2
10
·
0
.
5
.
As
u
C
2
(
x
11
)>
u
C
1
(
x
11
)
, instance 11 will be classified as rectangle (
C
2
).
11.5.5 Experimental Evaluation of Hubness-Aware Classifiers
Time series datasets exhibit a certain degree of hubness, as shown in Table
11.3
.This
is in agreement with previous observations [
43
].
Most datasets from the UCR repository [
28
] are balanced, with close-to-uniform
class distributions. This can be seen by analyzing the relative imbalance factor (RImb)
of the label distribution which we define as the normalized standard deviation of the
class probabilities from the absolutely homogenous mean value of 1
/
m
, where
m
denotes the number of classes, i.e.,
m
=|
C
|
:
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