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Algorithm 1
BVQ-FR algorithm
1: Train the LVQ
N
,
l
i
∈
{
(
m
1
,
l
1
),...,(
m
Q
,
l
Q
)
}
,
m
i
∈
R
y
by using the BVQ algorithm;
2: Set the elements of the matrix
BV QFM
to 0;
3:
w
tot
0;
4: For each training sample
t
k
1: Find the two code vectors
m
i
,
m
j
nearest to
t
k
;
2: If
l
i
=
from the border
S
ij
then
1: Calculate the unit normal vector to the decision boundary as:
N
ij
=
l
j
and
t
k
falls at a distance less than
ʔ
(
m
i
−
m
j
)
=
;
m
i
−
m
j
BV QFM
=
BV QFM
+
N
ij
∗
N
ij
;
3:
w
tot
=
2:
w
tot
+
1;
BV QFM
=
BV QFM
5:
w
tot
;
6: Calculate eigenvectors
u
1
,
u
2
,...,
u
N
and related eigenvalues
ʻ
1
,ʻ
2
,
...,ʻ
N
of
BV QFM
;
N
7: Set
W
=
1
ʻ
|
u
i
|
;
i
z
=
8: Sort features with respect to
W
components.
The core of theBVQ-FRalgorithm is at point 4. There, finding the two nearest code
vectors to each training sample allows us to define the effective decision boundary of
the LVQ. As a matter of fact, testing whether labels are different guarantees that the
piece of Voronoi boundary
S
ij
is actually a part of the decision boundary. Secondly,
incrementing the
BV QFM
each time a pair of code vectors is selected, allows to
weight the normal vector
N
ij
by the number of samples falling at a distance less than
ʔ
from
S
ij
. It can be proved that this is equivalent to a Parzen estimate of the integral
S
i
j
p
, while the final value of
w
tot
represents the Parzen estimate of
S
p
(
x
)
(
x
)
in
Eq. (
4.3
)[
10
].
It should be noted that the algorithm BVQ-FR can be transformed by replacing
BVQ with other FE algorithm that produces a transformation matrix EDBFM-like.
For example there can be used OLDA, SVM and ADBFE algorithms. In the next
section an experimental comparison between these alternatives will be made.
4.5 Experiments
4.5.1 Experimental Setting
This section is devoted to experimental evaluation of the EDBFM-based feature
weighting method. In particular in the present subsection we propose a synthetic
experiment which allows us to illustrate the properties of themethod.We also describe
both the experimental procedure and the evaluation criteria that will be used for
all subsequent experiments. In the next subsection various implementations of the
methodwill be tested over real-world datasets and comparedwithwell-known feature
weighting algorithms.
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