Information Technology Reference
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In the work of Santos [203] an empirically tuned formula for the choice
of an
h
value affording a convenient fat estimation of the error PDF was
proposed. The formula is inspired by another one proposed in [31] for the
choice of an optimal
h
value in the IMSE sense
2
:
h
opt
=
s
1
d
+4
4
(
d
+2)
n
,
(6.7)
where
s
is the sample standard deviation of the data and
d
the dimensionality
of the PDF. For the estimation of the error PDF of
c
classes one should
substitute
d
for
c
in formula (6.7). For the fat estimation of the error PDF
the cited work [203] proposed, based on a large number of experiments with
different datasets, the following formula for
h
giving higher values than (6.7)
but with a similar behavior:
h
fat
=25
c
n
.
(6.8)
Note that
nh
fat
→∞
as required (see Appendix E), and increases with
c
.
A comparison between the values of
h
fat
and
h
opt
for different values of
c
is
shown in Fig. 6.6. Note that
h
fat
>h
opt
specially for small values of
n
.
Table 6.2 summarizes results of experiments reported in the cited work
[203], where details on the datasets are provided. The "Best
h
"column
presents the values of
h
achieving the minimum number of classification er-
rors in 20 experiments; the "Suggested
h
" column presents the averages of
the
h
values achieving the 10 smallest classification errors. For comparison,
Table 6.2 also shows the values supplied by formulas (6.7) and (6.8).
Tabl e 6 . 2 Values of
h
for the experiments reported in [203] and those obtained
with formulas (6.7) and (6.8).
Datasets
cn
Best
h
Suggested
hh
fat
h
opt
Ionosphere
2
351
4.6
3.9
2.67
0.85
Sonar
2
208
3.4
3.7
3.47
0.92
Wdbc
2
569
1.4
1.7
2.10
0.78
XOR-n
2
200
4.0
4.7
3.54
0.93
Iris
3
150
4.0
3.6
5.00
1.05
Wine
3
178
4.6
4.2
4.59
1.02
PB12
4
608
1.8
2.2
2.87
0.93
XOR-n
4
200
5.2
5.0
5.00
1.07
Olive
9
572
4.6
5.4
4.43
1.20
2
Note that for the univariate case (
d
=1
) formula (6.7) reproduces formula (E.19),
when using the sample estimate of the standard deviation.
