From: FLAIRS-01 Proceedings. Copyright © 2001, AAAI (www.aaai.org). All rights reserved.
Decision Tree Rule Reduction Using Linear Classifiers in Multilayer Perceptron
DaeEun Kim
Sea Woo Kim
Division of Informatics
University of Edinburgh
5 Forrest Hill
Edinburgh, EHI 2QL, United Kingdom
daeeun@dai.ed.ac.uk
Dept. of Information and
Communication Engineering, KAIST
Cheongryang l-dong
Seoul, 130-011, Korea
seawoo@ngis.kaist.ac.kr
Abstract
relationship betweendata points, using a statistical technique. It generates manydata points on the response surIt has beenshownthat a neuralnetworkis better thana
face
of the fitted curve, and then induces rules with a
direct applicationof inductiontrees in modelingcomdecision
tree. This method was introduced as an alterplex relations of input attributes in sampledata. We
native measure regarding the problem of direct applicaproposethat conciserules be extractedto supportdata
tion of the induction tree to raw data (Irani &Qian 1990;
withinput variablerelations overcontinuous-valued
atKim1991). However,it still has the problem of requiring
tributes. Thoserelations as a set of linear classifiers
manyinduction rules to reflect the responsesurface.
can be obtained from neural networkmodelingbased
onback-propagation.
Alinear classifier is derivedfrom
In this paper we use a hybrid technique to combineneua linear combinationof input attributes and neuron
ral networksand decision trees for data classification (Kim
weightsin the first hiddenlayer of neuralnetworks.It
& Lee 2000). It has been shownthat neural networks are
is shown
in this paperthat whenweuse a decisiontree
better than direct application of induction trees in modoverlinear classifiers extractedfroma multilayerpereling nonlinear characteristics of sample data (Dietterich,
ceptron, the numberof rules can be reduced.Wehave
Hild, & Bakiri 1990; Quinlan 1994; Setiono & Lie 1996;
tested this methodoverseveral data sets to compare
it
Fisher & McKusick 1989; Shavlik, Mooney, & Towell
withdecisiontree results.
1991). Neural networks have the advantage of being able to
deal with noisy, inconsistent and incompletedata. A method
Introduction
to extract symbolicrules from neural networkshas beenproposed to increase the performanceof the decision process
The discovery of decision rules and recognition of patterns
(Andrews, Diederich, &Tickle 1996; Taha & Ghosh1999;
from data examples is one of the most challenging probFu
1994; Towcll &Shavlik Oct 1993; Setiono & Lie 1996).
lems in machinelearning. If data points contain numerical attributes,
induction
treemethods
needthecontinuous- The KTalgorithm developed by Fu (Fu 1991) extracts rules
valued
attributes
tobemadediscrete
withthreshold
values. from subsets of connected weights with high activation in a
trained network. The Mof N algorithm clusters weights of
Induction
treealgorithms
suchasC4.5build
decision
trees
the trained networkand removesinsignificant clusters with
byrecursively
partitioning
theinput
attribute
space
(Quinlow active weights. Then the rules are extracted from the
lan1996).
Thetreetraversal
fromtherootnodetoeachleaf
weights (Towell & Shavlik Oct 1993).
leads
tooneconjunctive
rule.
Eachinternal
nodeinthedecisiontree
hasa splitting
criterion
orthreshold
forcontinuous- A simple rule extraction algorithmthat uses discrete activalued
attributes
topartition
somepartoftheinput
space,
vations over continuous hidden units is presented in by Seandeachleafrepresents
a class
related
totheconditions
of
tiono and Taha (Setiono &Lie 1996; Taha & Ghosh1999).
eachinternal
node.
They used in sequence a weight-decay back-propagation
Approaches
basedon decision
treesinvolve
making
the
over a three-layer feed-forward network, a pruning process
continuous-valued
attributes
discrete
ininput
space,
creatto removeirrelevant connectionweights, a clustering of hidingmanyrectangular
divisions.
Asa result,
theymayhave
den unit activations, and extraction of rules from discrete
theinability
todetect
datatrends
ordesirable
classifica- unit activations. They derived symbolic rules from neural
tionsurfaces.
Eveninthecaseofmultivariate
methods
of
networks that include oblique decision hyperplanes instead
discretion
which
search
inparallel
forthreshold
values
for
of general input attribute relations (Setiono &Liu 1997).
morethanonecontinuous
attribute
(Fayyad
& Irani1993;
Also the direct conversion from neural networks to rules
has an exponential complexity whenusing search-based alKweldo
& Kretowski
1999),
thedecision
rulesmaynotreflect
datatrends
orthedecision
treemaybuild
manyrules
gorithm over incoming weights for each unit (Fu 1994;
withthesupport
of a smallnumber
of examples
or ignore
Towell & Shavlik Oct 1993). Most of the rule extraction
somedatapoints
bydismissing
themasnoisy.
algorithms are used to derive rules from neuron weights
A possible process is suggested to grasp the trend of the
and neuronactivations in the hidden layer as a search-based
method. An instance-based rule extraction methodis sugdata. It first tries to fit it with a given data set for the
Copyright
©2001,
AAAI,
All rightsreserved.
48O
FLAIRS-2001
gested to reduce computationtime by escaping search-based
methods(Kim &Lee 2000). After training two hidden layer
neural networks, the first hidden layer weight parametersare
treated as linear classifiers. Theselinear differentiated fimctions are chosenby decision tree methodsto determinedecision boundariesafter re-organizing the training set in terms
of the newlinear classifier attributes¯
Our approach is to train a neural network with sigmoid
functions and to use decision classifiers based on weight
parameters of neural networks. Then an induction tree selects the desirable input variable relations for data classification. Decision tree applications have the ability to determineproper subintervals over continuousattributes by a discretion process. This discretion process will cover oblique
hyperplanes mentionedin Setiono’s papers. In this paper,
we havetested linear classifiers with variable thresholds and
fixed thresholds. The methodsare tested on various types of
data and compared with the method based on the decision
tree alone.
Problem Statement
Induction trees are useful for a large numberof examples,
and they enable us to obtain proper rules from examples
rapidly (Quinlan 1996). However,they have the difficulty
in inferring relations betweendata points and cannot handle
noisy data.
.... f ..............................
"._~..
° i
1
(a)
(b)
(c)
Figure 1: Example(a) data set and decision boundary(O
class 1, X : class 0) (b)-(c) neural networkfitting (d) data
with 900 points (Kim & Lee 2000)
Wecan see a simple exampleof undesirable rule extraction discoveredin the inductiontree application¯ Figure1 (a)
displays a set of 29 original sample data with two classes¯
It appears that the set has four sections that have the boundaries of direction from upper-left to lower-right. A set of
the dotted boundarylines is the result of multivariate classification by the induction tree. It has six rules to classify
data points. Even in C4.5 (Quinlan 1996) run, it has four
rules with 6.9 % error, makingdivisions with attribute y.
Therules do not catch data clustering completelyin this example. Figure l(b)-(c) showsneural networkfitting with
back-propagationmethod¯In Figure l(b)-(c) neural network
nodes have slopes alpha = 1.5, 4.0 for sigmoids, respectively. After curve fitting, 900 points were generated uniformly on the response surface for the mappingfrom input
space to class, and the response values of the neural networkwere calculated as shownin Figure l(d). The result
C4.5 to those 900 points followed the classification curves,
but produced55 rules¯ The production of manyrules results
from the fact that decision tree makespiecewise rectangular divisions for each rule. This happensin spite of the fact
that the responsesurface for data clustering has a correlation
betweenthe input variables.
As shownabove, the decision tree has a problemof overgeneralization for a small number of data and an overspecialization problem for a large numberof data. A possible suggestion is to consider or derive relations between
input variablt s as another attribute for rule extraction¯ However, it is difficult to find input variablerelations for classification directly in supervised learning, while unsupervised
methodscan use statistical methodssuch as principal component analysis (Haykin 1999).
Method
The goal for our approachis to generate rules following the
shape and characteristics of response surfaces. Usually induction trees cannot trace the trend of data, and they determinedata clustering only in terms of input variables, unless
we apply other relation factors or attributes. In order to improveclassification rules from a large training data set, we
allow input variable relations for multi-attributes in a set of
rules¯
NeuralNetworkandLinearClassifiers
Weuse a two-phase method for rule extraction over
continuous-valuedattributes. Given a large training set of
data points, the first phase, as a feature extraction phase, is
to train feed-forward neural networks with back-propagation
and collect the weight set over input variables in the first
hidden layer. A feature useful in inferring multi-attribute
relations of data is foundin the first hiddenlayer of neural
networks. The extracted rules involving networkweight values will reflect features of data examplesand provide good
classification boundaries. Also they maybe more compact
and comprehensible,comparedto induction tree rules.
In the secondphase, as a feature combinationphase, each
extracted feature for a linear classification boundaryis combined together using Booleanlogic gates. In this paper, we
use an induction tree to combineeach linear classifier.
The highly nonlinear property of neural networks makes
it difficult to describe howthey reach predictions. Although
their predictive accuracy is satisfactory for manyapplications, they have long been considered as a complexmodelin
terms of analysis. By using expert rules derived from neural networks, the neural networkrepresentation can be more
understandable¯
It has been shownthat a particular set of functions can
NEURAL
NETWORK
/ FUZZY 481
be obtained with arbitrary accuracy by at most two hidden layers given enough nodes per layer (Cybenko 1988).
Also one hidden layer is sufficient to represent any Boolean
function (Hertz, Palmer, & Krogh 1991). Our neural network structure has two hidden layers, where the first hidden layer makesa local feature selection with linear classifiers and the second layer receives Boolean logic values from the first layer and maps any Boolean function. The second hidden layer and output layer can be
thought of as a sum of the product of Boolean logic
gates. The n-th output of neural networks for a set of
data is Fn
= f(EN2w ,d()Sj
2 ---N1 wjkf(
1 --No w°ad))
After training data patterns with a neural networkby backpropagation,wecan havelinear classifiers in the first hidden
layer.
For a nodein the first hiddenlayer, the activation is defined as Hj = f(Y~i N° aiWij) for the j-th node where
No is the numberof input attributes, ai is an input, and
f(x) = 1.0/(1.0 -’ ~z) is a s igmoid function. When we
train neural networks with the back-propagationmethod, a,
the slope of the sigmoid function is increased as iteration
continues. If we have a high value of a, the activation of
each neuronis close to the property of digital logic gates,
whichhas a binary value of 0 or 1.
Exceptfor the first hiddenlayer, we can replace each neuron by logic gates if we assume we have a high slope for
the sigmoidfunction. Input to each neuronin the first hidden layer is represented as a linear combinationof input attributes and weights, )--~N aiWij. This forms linear classifiers for data classification as a feature extraction over data
distribution.
WhenFigure l(a) data is trained, we can introduce new
attributes aX + bY + e, where a, b, c are constants. We
use two hidden layers with 4 nodes and 3 nodes, respectively, where every neuron node has a high sigmoid slope
to guarantee desirable linear classifiers as shownin Figure l(c). Wetransformed 900 data points in Figure l(d)
into four linear classifier data points, and then we added
the classifier attributes {L1, L2, L3, L4}to the original attributes x, y. Induction tree algorithm used those six attributes {x, y, LI, L2, Ls, L4} for its input attributes.
Then we could obtain only four rules with C4.5, while a
simple application of C4.5 for those data genetated 55 rules.
The rules are given as follows:
rule 1
rule2
rule 3
rule4
if (1.44x + 1.73y <= 5.98), thenclass
if(1.44x+ 1.73y > 5.98)
and (1.18x + 2.81y <= 12.37) then class
: if(1.44x + 1.73y > 5.98)
and (1.18x + 2.81y > 12.37)
and (0.53x + 2.94y < 14.11), then class
: if(1.44x + 1.73y > 5.98)
and (1.18x + 2.81y > 12.37)
and (0.53x + 2.94y > 14.11), then class
:
:
These linear classifiers exactly match with the boundaries
shownin Figure 1 (c), and they are moredominantfor classi482
FLAIRS-2001
fication in terms of entropy minimizationthan a set of original input attributes itself. Evenif weinclude input attributes,
the entropy measurementleads to a rule set with boundary
equations. Theserules are moremeaningfulthan those of direct C4.5 application to raw data since their divisions show
the trend of data clustering and howeach attribute is correlated.
Linear Classifiers
for Decision Trees
Induction trees can split any continuous value by selecting
thresholds for given attributes, while it cannot derive relations of input attributes directly. Thus, before induction
trees are applied to a given training set, weput newrelation
attributes consisting of linear classifiers in the training set,
whichare generated fromthe weight set in a neural network.
Wecan represent training data as a set of attribute column
vectors. Whenwe have linear classifiers extracted from a
neural network,each linear classifier can be a columnvector
in a training set, wherethe vector size is equal to the number
of the original training data. Eachlinear classifier becomesa
newattribute in the training set. If we represent original input attribute vectors and neural networklinear classifiers as
U-vectorsand L-vectors, respectively, C-vectors, L-vectors,
and {U + L}-vectors will form a different set of training
data; each set of vector is transformedfrom the same data.
Thosethree vector sets were tested with several data sets in
the UCI depository (Blake, Keogh, & Merz 1998) to compare the performance (Kim &Lee 2000).
It is believed that a compactset of attributes to represent
the data set shows a better performance. Addingoriginal
input attributes does not improvethe result, but it makesits
performanceworse in most cases. C4.5 has a difficulty in
selecting properly the mostsignificant attributes for a given
set of data, because it chooses attributes with local entropy
measurementand the methodis not a global optimization of
entropy. Also, especially whenonly linear classifiers from
neural network,L-vectors, are used, it is quite effective in
reducing the numberof rules (Kim& Lee 2000).
Generally manydecision tree algorithms have muchdifficulty in feature extraction. Whenwe add manyunrelated
features(attributes)to a training set for decisiontrees, it has
tendency to worsenperformance.This is because the induction tree is basedon a locally optimalentropy search. In this
paper, a compactL-linear classifier methodwas tested. We
used L-linear classifiers with fixed thresholds and variable
thresholds.
In the linear classifier methodwith fixed thresholds, all
instances in the training data are transformed into Boolean
logic values through dichotomyof node activations in the
first hidden layer; then Booleandata are applied to the induction algorithmC4.5. In the linear classifiers with variable
thresholds, a set of linear classifiers are taken as continuousvalued attributes. TheC4.5 application over instances of linear classifiers will try to find the best splitting thresholdsfor
discretion over each linear classifier attribute. In this case,
each linear classifier attribute mayhave multivariate thresholds to handle marginalboundariesof linear classifiers.
data
wine
iris
blagast-w
ios~sphere
pinla
glass
bupa
C4.5
orain (%) [ test (%)
1.2 4- 0.1
7.9 4- 1.3
1.9 4- 0.1
5.4"4-0.7
I.I 4-0.1
4.74- 0.5
1.6 4- 0.2
10.44- I.I
15.14- 0.8 26.4 4- 0.9
6.7 4- 0.4 32.0 4- 1.5
12.94- 1.5 34.54- 2.0
Linear
Classifier
[
train (%) [ test (%)
0.0 4- 0.0
3.6 4- 0.9
0.7 4- 0.2
4.7 4- 1.5
0.9 4- 0.2
4.4 4- 0.3
1.2 4- 0.2
8.8 4- 1.5
15.84- 0.4 27.04- 0.9
6.6 4- 0.8
33.6
4-3.I
10.24- 1.1
33.74- 2.1
nodes
4-3-3
4-5-3
4-7-3
4-10-3
4-3-3-3
4-5-4-3
4-7-4-3
neuralnetwork
wain(%)
test (%)
O.84- 0.1
4.3 4- 1.5
0.5 4- 0.1
4.74- I.I
5.2 4- 1.1
0.5 4- 0.2
0.4-I- 0.2 5.1 4- 1.2
4.5-II.I
0.7
-40.1
0.64.0.1
4.1-1.-1.3
0.54-0.1 4.74- 0.9
veriable
T’[ L.I’
# rules
3.8-1-0.3
3.9 4- 0.4
3.94-0.2
3.94-0.1
3.7 q- 0.2
4.04* 0.4
3.9 4- 0.2
’[.L}
fixedT
# rules
3.44- 0.2
4.0 4- 0.5
4.1.4- 0.4
4.4 4- 0.6
3.8 4- 0.2
4.1 4-0.2
4.0 4- 0.4
Ca)
Table1: Dataclassification errors in C4.5andlinear classifier methodwith variablethresholds(error rates in linear
classifier methodshowthe best result amongseveral neural
networkexperiments)
0 1
-- g .m .
.
t z, J.
variable
thresholds
nodes train (%) test (%)
4-3-3
0.74.0.1
5.14. 1.3
4-5-3
0.7 4* 0.2 6.O4- 1.7
4-7-3 0.7 4- 0.2 4.64*1.1
4-10-3 0.8-4-0.1 5.2 4- 1.6
4-3-3-3 0.9 "4"0.1 5.9 4- 1.4
4-5-4-30.7 4- 0.2 4.7 4- 1.5
4-7-4-3 0.8 4- 0.2 4.7 4- 1.0
fixedthresholdst
t, }
train (%) (%)
1.7 4-0.3
4.84- 1.4
1.8 4-0.3 5.1 4- 1.2
1.5 4-0.3
5.3 4- 1.1
1.4 4- 0.2 5.2 4- 1.4
2.6 4- 0.7 6.14- 1.7
2.0 4- 0.4 5.3 4- 1.2
2.2 4- 0.4 5.14- 1.0
Table2: iris dataclassification result (a) neuralnetworkerror rate and the numberof rules with the linear classifier
method(b) error rate in linear classifier methodwith variable thresholdsandfixed thresholds
¯ --
(a)
(b)
Figure 2: ComparisonbetweenC4.5 and linear classifier
method(a) averageerror rate in test data with C4.5, neural
network,andlinear classifier (b) the numberof rules with
C4.5 andlinear classifier method
Experiments
Ourmethodhas been tested on several sets of data in the
UCI depository (Blake, Keogh, & Merz1998). Figure
showsaverageclassification errorrates for C4.5, neuralnetworksandthe linear classifier method.Table 1 showserror
rates to comparethe pure C4.5 methodandour linear classifier method.Theerror rates wereestimatedby runningthe
complete10-fold cross-validation ten times, andthe average
andthe standarddeviation for ten runs weregivenin the table. Several neural networkswere tested for each data set.
Table 2-3 shows examplesof different neural networksand
their linearclassifier results.
Ourmethodsusing linear classifiers are better than C4.5
in somesets and worsein other data sets such as glass and
pima whichare hard to predict even in neural network.The
result supportsthe fact that the methodsgreatly dependon
neuralnetworktraining. If neuralnetworkfitting is not correct, thenthe fitting errors maymisleadthe result of linear
classifier methods.Normally,the C4.5 applicationshowsthe
errorrate is very highfor trainingdata in Table1. Theneural networkcan improvetraining performanceby increasing
the numberof nodes in the hiddenlayers as shownin Table
2-3. However,it does not guaranteeto improvetest set performance.In manycases, reducingerrors in a training set
tendsto increasethe errorrate in a test set by overfitting.
The error rate difference betweena neural networkand
the linear classifier methodexplains that somedata points
are located on marginalboundariesof classifiers. It is due
to the fact that our neuralnetworkmodeluses sigmoidfunctions with high slopes instead of step functions. When
acti-
n~s
6-5-2
6-10-2
6-5-5-2
6-8-6-2
6-10-7-2
neural network
~n <%) I testc%)
17.94- 1.1
31.94- 1.6
10.24.1.1
32.84- 2.1
15.24- 0.9
32.84- 1.9
9.3 4- 0.7
32.14- 1.9
7.34-1.4
32.74- 2.2
variable
T~L }
#~
10.04- 1.3
15.3
4*1.7
10.6.4-1.3
14.34*2.1
16.24- 1.7
fixedT~-LJ.
#r~ es
7.2
4-0.7
16.34- 1.7
9.4 4- 0.9
19.54- 1.3
24.4
4*2.3
(a)
nodes
6-5-5-2
6-8-6-2
6-10-7-2
"[L}
L} [1
variable thresholds
fixed thresholdsf
train (%)
test (%)
train (%)
(%)
17.84- 1.5 32.84- 1.9
25.24- 1.9
34.14- 1.3
14.84- 1.4 33.74*2.1
19.94*1.5
36.24- 3.1
17.54- 1.0 33.64* 2.3
22.24- 1.1
34.34- 2.3
13.64*1.5
32.54* 1.7 15.14- 0.6
34.04- 2.4
15.24- 1.3
32.74- 2.9
17.34- 0.7 32.74- 1.6
(b)
Table 3: bupadata classification result (a) neuralnetwork
error rate andthe number
of rules with the linear classifier
method
(b) errorrate in linear classifier method
withvariable
thresholds andfixed thresholds
vation is near 0.5, the weightedsumof activations maylead
to different output classes. If the numberof nodes in the
first hiddenlayer is increased, this marginaleffect becomes
larger as observedin Table2 - see fixed thresholds.
Figure 2(b) shows that the numberof rules using our
methodis significantly smaller than that using conventional
C4.5in all the data sets. To reducethe number
of rules, linear classifiers with the Booleancircuit modelgreatly depend
on the number
of nodesin the first hiddenlayer. It decreases
the numberof rules whenthe numberof nodes decreases
in the first hiddenlayer, whilethe errorrate performance
is
similar withinsomelimit, regardlessof the number
of nodes.
Thelinear classifier methodwithvariablethresholdsalso depends on the numberof nodes. The reason whythe number
of rules is proportionalto the number
of nodesis related to
the searchspace of Booleanlogic circuits. Thelinear classifier methodwith the Booleancircuit modeloften tends to
generate rules that have a small numberof support examples, while variable threshold modelprunesthose rules by
NEURAL
NETWORK
/ FUZZY 483
adjusting splitting thresholds in the decision tree. Twohidden layer neural networks are not significantly more effective in terms of error rates and the numberof rules than one
hidden layer as shownin Table 2-3. Thus, neural networks
with one hidden layer maybe enoughfor UCIdata set.
Mostof the data sets in the UCIdepository have a small
numberof data examplesrelative to the numberof attributes.
The significant difference between a simple C4.5 application and a combinationof C4.5 application and a neural networkis not seen distinctively in UCIdata in terms of error
rate, unlike the synthetic data in Figure 1. Information of
data trend or input relations can be moredefinitely described
whengiven manydata examplesrelative to the numberof attributes.
Table 2-3 showsthat neural networkclassification is better than linear classifier applications. Even thoughlinear
classifier methods are good approximations to nonlinear
neural networkmodelingin the experiments, we still need
to reduce the gap betweenneural networktraining and linear
classifier models. There is a trade-off betweenthe number
of rules and error rate performance.Weneed to explain what
is the optimal numberof rules for a given data set for future
study.
Conclusions
This paper presents a hybrid methodfor constructing a decision tree from neural networks. Our methoduses neural
network modeling to find unseen data points and then an
induction tree is applied to data points for symbolicrules,
using features from the neural network. The combination
of neural networks and induction trees will compensatefor
the disadvantages of one approach alone. This methodhas
advantages over a simple decision tree method. First, we
can obtain goodfeatures for a classification boundaryfrom
neural networks by training input patterns. Second, because
of feature extractions about input variable relations, we can
obtain a compactset of rules to reflect input patterns.
Westill have muchwork ahead, such as reducing the
numberof rules and error rate together, and finding the
optimal numberof linear classifiers.
References
Andrews,R.; Diederich, J.; and Tickle, A. 1996. A survey
and critique of techniquesfor extracting rules from trained
artificial neural networks. Knowledge-Based
Systems 8(6).
Blake, C.; Keogh, E., and Merz, C. 1998. UCIrepository
of machinelearning databases. In Preceedingsof the Fifth
International Conference on MachineLearning.
Cybenko, G. 1988. Continuous valued neural networks
with two hidden layers are sufficient. Technical report,
Technical Report, Departmentof ComputerScience, Tufts
University, Medford, MA.
Dietterich, T.; Hild, H.; and Bakiri, G. 1990. A comparative study of ID3 and backpropagation for english textto-speech mapping. In Proceedings of the 1990 Machine
Learning Conference, 24-31. Austin, TX.
484
FLAIRS-2001
Fayyad, U., and Irani, K. 1993. Multi-interval discretization of continuous-valued
attributes for classification learning. In Proceedings of IJCAI’93, 1022-1027. Morgan
Kaufmann.
Fisher, D., and McKusick,K. 1989. An empirical comparison of ID3 and backpropagation. In Proceedings of 11th
International Joint Conferenceon AI, 788-793.
Fu, L. 1991. Rule learning by searching on adaptive nets.
In Preceedingsof the 9th NationalConferenceon Artificial
Intelligence, 590-595.
Fu, L. 1994. Neural Networks in ComputerIntelligence.
NewYork: McGraw-Hill.
Haykin, S. 1999. Neural networks : a comprehensivefoundation. UpperSaddle River, N.J.: Prentice Hall, 2nd edition.
Hertz, J.; Palmer, R.; and Krogh,A. 1991. Introduction to
the Theory of Neural Computation.RedwoodCity, Calif.:
Addision Wesley.
Irani, K., and Qian, Z. 1990. Karsm: A combinedresponse
surface / knowledgeacquisition approachfor deriving rules
for expert system. In TECHCON’90
Conference, 209-212.
Kim,D., and Lee, J. 2000. Handlingcontinuous-valuedattributes in decision tree using neural networkmodeling.In
EuropeanConference on MachineLearning, Lecture Notes
in Artificial Intelligence 1810, 211-219.Springer Verlag.
Kim, D. 1991. Knowledgeacquisition based on neural
network modeling. Technical report, Directed Study, The
University of Michigan, AnnArbor.
Kweldo,W., and Kretowski, M. 1999. An evolutionary algorithmusing multivariate discretization for decision rule
induction. In Proceedings of the Third EuropeanConference on Principles of Data Miningand KnowledgeDiscovery, 392-397. Springer.
Quinlan, J. 1994. Comparingconnectionist and symbolic
learning methods. In Computational Learning Theory and
Natural Learning Systems, 445-456. MITPress.
Quinlan, J. 1996. Improveduse of continuousattributes in
C4.5. Journal of Artificial Intelligence Approach(4):7790.
Setiono, R., and Lie, H. 1996. Symbolicrepresentation of
neural networks. Computer29(3):71-77.
Setiono, R., and Liu, H. 1997. Neurolinear: A system
for extracting oblique decision rules fromneural networks.
In European Conference on Machine Learning, 221-233.
Springer Verlag.
Shavlik, J.; Mooney,R.; and Towell, G. 1991. Symbolic
and neural learning algorithms: Anexperimental comparison. MachineLearning 6(2): 111-143.
Taha, I. A., and Ghosh, J. 1999. Symbolicinterpretation
of artificial neural networks. IEEETransactions on Knowledge and Data Engineering 11(3):448-463.
Towell, G., and Shavlik, J. Oct. 1993. Extracting refined
rules from knowledge-based neural networks. Machine
Learning 13(1):71-101.