Computational Intelligence - August 2015 - 59

Table 1 The average testing error rate and standard derivation of ENN ^ k = 3 h compared with KNN ^ k = 3 h , naive Bayes, LDA, and
neural network. All results are shown in percentage. Each result is averaged over 100 random runs: in each run, we randomly select
half of the data as the training data and the remaining half as the test data. For each dataset, we highlight the best result with bold
value among all these five methods. To specifically compare the results of ENN and KNN, we also underline the value if ENN
performs significantly better than KNN under one-tailed t-test ^ p = 0.01 h .
DaTaseTs

eNN

KNN

Naive bayes

LDA

Neural NeTworK

IONOsphErE

17.35 ! 2.69

18.55 ! 2.94

19.83 ! 2.86

20.68 ! 3.00

18.48 ! 2.90

VOwEl

8.50 ! 1.92

11.73 ! 1.80

43.90 ! 2.98

40.94 ! 1.97

45.17 ! 3.15

sONar

22.67 ! 3.97

24.49 ! 4.06

29.22 ! 4.16

33.75 ! 5.11

27.24 ! 4.37

wINE

4.49 ! 2.16

7.08 ! 2.20

5.07 ! 1.71

2.58 ! 2.13

7.21 ! 2.88

BrEast-caNcEr

4.04 ! 0.87

4.44 ! 1.07

5.76 ! 1.04

5.88 ! 1.18

4.57 ! 1.17

haBErmaN

31.32 ! 6.53

32.13 ! 5.79

36.35 ! 10.85

34.63 ! 9.92

37.40 ! 10.58

BrEast tIssuE

36.71 ! 6.37

42.40 ! 6.19

44.02 ! 6.18

41.24 ! 6.60

67.62 ! 5.22

mOVEmENt lIBras

26.33 ! 2.88

32.16 ! 2.97

45.41 ! 3.39

39.90 ! 3.31

40.87 ! 4.34

mammOgraphIc massEs

21.16 ! 1.43

22.27 ! 1.55

18.96 ! 1.57

19.17 ! 1.72

49.40 ! 0.29

sEgmENtatION

24.71 ! 3.07

27.85 ! 3.04

12.64 ! 2.93

12.79 ! 2.88

23.06 ! 5.95

IlpD

40.04 ! 3.58

40.91 ! 3.68

26.87 ! 2.69

29.64 ! 3.39

32.09 ! 3.53

pIma INDIaNs DIaBEtEs

31.22 ! 2.15

33.08 ! 2.37

29.44 ! 2.19

28.29 ! 2.01

25.38 ! 2.77

KNOwlEDgE

23.93 ! 4.69

27.11 ! 4.45

12.66 ! 2.45

6.97 ! 2.53

14.42 ! 3.86

VErtEBral

35.13 ! 4.83

37.64 ! 5.06

47.93 ! 3.41

36.88 ! 4.83

45.11 ! 3.12

BaNK NOtE

0.09 ! 0.18

0.12 ! 0.23

15.27 ! 1.25

2.60 ! 0.60

0.18 ! 0.37

magIc

20.10 ! 0.33

20.42 ! 0.36

25.69 ! 0.61

23.30 ! 0.34

29.62 ! 0.38

pEN DIgIts

0.74 ! 0.15

0.94 ! 0.17

15.38 ! 0.41

11.22 ! 0.52

11.65 ! 0.70

Faults

0.91 ! 0.52

1.65 ! 0.86

0.00 ! 0.00

0.00 ! 0.00

0.00 ! 0.00

lEttEr

5.60 ! 0.25

7.44 ! 0.25

40.09 ! 0.47

29.80 ! 0.37

28.33 ! 0.52

spam

10.08 ! 0.59

11.52 ! 0.63

10.31 ! 0.78

9.64 ! 0.61

15.32 ! 1.02

p = 0.01h . In the neural network
implementation, we use the classic multilayer perceptron (MLP) structure with
10 hidden neurons, and with 800 backpropagation iterations for training at the
learning rate of 0.01. The results are
averaged over 100 random runs, and in
every run, we randomly select half of
the data as the training data and the
remaining half as the test data. Let's consider the Spam database as an example.
This database includes 4601 e-mail messages, in which 2788 are legitimate messages and 1813 are spam messages. Each
message is represented by 57 attributes,
of which 48 are the frequency of a particular word (FW), 6 are based on the
frequency of a particular character (FC),
and 3 are continuous attributes that
reflect the use of capital letters (SCL) in
the e-mails. Fig. 5 shows detailed classification error rates (in percentage) with
different parameters of k , which clearly
demonstrates that ENN method can
achieve consistently lower error rates
than those of KNN rule. For a detailed
performance comparison between ENN
and KNN for all these twenty datasets,

please refer to the Supplementary Material Section 3 for further information.
We would like to note that for all
these experiments, there appears to be
no significant difference between ENN
and KNN when k = 1. The reason for
this might be that under such a small
value of k = 1, the classification performance will be determined by the single
closest neighbor. Therefore, when k = 1,
both methods do not consider the data
distribution anyway. That might explain
why under k = 1, both methods show a
very close performance.

Overall Classification
Error (in Percentage)

We also evaluate the classification
performance of our proposed ENN
classifier on the entire MNIST handwritten digits dataset [34], which is a
widely used benchmark in the community. In this experiment, we use 60,000
images as training data and 10,000
images as test data. Fig. 4 shows the
comparison of classification error rates
(in percentage) between classic KNN
rule and our ENN rule, where the minimum error of 2.61% is obtained at
k = 7 for our ENN method. The overall
classification error rates using the ENN
rule are notably less than those with the
KNN rule (Fig. 4).
We further apply our ENN classifier
to twenty real world datasets from UCI
Machine Learning Repository [35].
Table I presents the classification error
rates (in percentage) for these twenty
UCI datasets in comparison to the classic KNN, naive Bayes, linear discriminant analysis (LDA), and neural network.
It shows that ENN always performs better than KNN, and in 17 out of these 20
datasets, the performance improvement
is significant (one-tailed t -test,

13.5
13
12.5
12
11.5
11
10.5
10
9.5

KNN
ENN

5

10

k

15

20

Figure 5 Overall classification error rate (in
percentage) of ENN and KNN for spam
e-mail classification.

August 2015 | IEEE ComputAtIonAl IntEllIgEnCE mAgAzInE

59



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