Hubness-Aware Classification, Instance Selection and Feature Construction: Survey and Extensions to Time-Series - Feature Selection for Data and Pattern Recognition - page 235

Table 11.2 GN 1 ( x ) , BN 1 ( x ) , N 1 ( x ) , N 1 , C 1 ( x ) and N 1 , C 2 ( x ) for the instances shown in Fig. 11.6

Instance GN 1 ( x ) BN 1 ( x ) N 1 ( x ) N 1 , C 1 ( x ) N 1 , C 2 ( x )

1 1 0 1 1 0

2 2 0 2 2 0

3 2 0 2 2 0

4 0 0 0 0 0

5 0 0 0 0 0

6 0 2 2 0 2

7 1 0 1 0 1

8 0 0 0 0 0

9 1 1 2 1 1

10 0 0 0 0 0

Mean μ GN 1 ( x ) = 0 . 7 μ BN 1 ( x ) = 0 . 3 μ N 1 ( x ) = 1

Std. ˃ ) = 0 . 823 ˃ ) = 0 . 675 ˃ ) = 0 . 943 GN 1 ( x BN 1 ( x N 1 ( x

and

BN 1 ( x 9 ) − μ BN 1 ( x )

˃ BN 1 ( x )

1 − 0 . 3

e − e − h b ( x 9 ) = e − w 9 = = = 0 . 3545 .

0 . 675

As w 9 > w 6 , instance 11 will be classified as rectangle according to instance 9.

From the example we can see that in hw- k NN all neighbors vote by their own
label. As this may be disadvantageous in some cases [ 49 ], in the algorithms consid-
ered below, the neighbors do not always vote by their own labels, which is a major
difference to hw- k NN.

11.5.2 h-FNN: Hubness-Based Fuzzy Nearest Neighbor

Consider the relative class hubness u C ( x i ) of each nearest neighbor x i :

N k , C ( x i )

u C ( x i ) = . (11.10)

N k ( x i )

The above u C ( can be interpreted as the fuzziness of the event that x i occurred

as one of the neighbors, C denotes one of the classes: C

x i )

. Integrating fuzziness
as a measure of uncertainty is usual in k -nearest neighbor methods and h-FNN [ 54 ]
uses the relative class hubness when assigning class-conditional vote weights. The
approach is based on the fuzzy k -nearest neighbor voting framework [ 27 ]. Therefore,
the probability of each class C for the instance x ∗ to be classified is estimated as:

∈ C

x i ∈ N k ( x ∗ )

u C ( x i )

x ∗ ) = u C ( x i ∈ N k ( x ∗ ) C ∈ C

x i ) .

(11.11)

u C (