Cost-sensitive methods of constructing hierarchical classifiers

Abstract

The cost of a future exploitation of a decision support system plays a key role. The paper deals with the problem of feature value acquisition cost for such systems. We present a modification of a cost-sensitive learning method for decision-tree induction with fixed attribute acquisition cost limit. Properties of the concept are established during computer experiments conducted on chosen benchmark databases from the UCI Machine Learning Repository and a real medical decision task. The results of experiments confirm that, for some decision problems, our proposition allows us to obtain a classifier with the same quality as a classifier obtained without cost limit but its exploitation is cheaper.

Full text

Cost-sensitive methods of constructing
hierarchical classifiers
Wojciech Penar
1
and Michal Wozniak
2
(1) Institute of Computer Engineering, Control and Robotics, Wroclaw University of
Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland
(2) Chair of Systems and Computer Networks, Wroclaw University of Technology, Wybrzeze
Wyspianskiego 27, 50-370 Wroclaw, Poland
Email: [wojciech.penar,michal.wozniak]@pwr.wroc.pl
Abstract: The cost of a future exploitation of a decision support system plays a key role. The paper deals with
the problem of feature value acquisition cost for such systems. We present a modification of a cost-sensitive
learning method for decision-tree induction with fixed attribute acquisition cost limit. Properties of the concept
are established during computer experiments conducted on chosen benchmark databases from the UCI Machine
Learning Repository and a real medical decision task. The results of experiments confirm that, for some
decision problems, our proposition allows us to obtain a classifier with the same quality as a classifier obtained
without cost limit but its exploitation is cheaper.
Keywords: decision tree, cost-sensitive classifier, cost limit
1. Introduction
During the design of computer decision support
systems the cost of their design and exploitation
plays a key role. The cost of exploitation can be
considered as the expense of incorrect diagnosis
or the expense of feature value acquisition. The
first problem is a typical problem for decision
theory where we want to find the classifier for
which the cost of misclassification is the lowest
(Duda et al., 2001). Of course we have to define
so-called lost function in advance which gives
the cost values of misclassification. In our paper
we concentrate our attention on the case where
the cost depends on the real expense of feature
value acquisition for decision making (Turney,
1995; Greiner et al., 2002). Of course it can be
measured by monetary units or time units. A
typical example of cost-sensitive classification is
medical diagnosis where physicians would like
to balance the costs of various tests with the
expected benefits or doctors have to make a
diagnosis quickly on the basis of low cost
(quickly measured) features because therapeutic
action has to be taken without delay. Let us note
that for many decision tasks there are no
problems in making high-quality medical deci-
sions on the basis of expensive medical tests.
One of the main tasks of designing computer-
aided diagnosis is finding the balance between
cost of exploitation and qualities of diagnosis.
As we stated the problem of cost-sensitive deci-
sion making arises frequently in medicine
(Nu´ nez, 1988), industrial production processes
(Verdenius, 1991), robotics (Tan & Schlimmer,
1989), technological diagnosis (Lirov & Yue,
1991) and many other fields, e.g. electronic
equipment testing and real-time computer sys-
tems. Additionally for many typical diagnosis
DOI: 10.1111/j.1468-0394.2010.00515.x
Article _____________________________
146 Expert Systems, July 2010, Vol. 27, No. 3 c2010 The Authors. Journal Compilation c2010 Blackwell Publishing Ltd

systems we cannot exceed the cost limit which
means that the maximum diagnosis cost (or
time) is fixed. Our paper addresses the most
popular inductive learning method (Mitchell,
1997) based on top-down decision-tree induction.
We propose cost-sensitive modifications of the
decision-tree induction algorithm with respect to
on the one hand the cost of feature value acquisi-
tion and on the other the maximum cost limit.
The content of the work is as follows. Section 2
provides the idea of the decision-tree induction
algorithm and related work which takes into
account cost during the learning process. Section
3 describes our modification of the cost-sensitive
approach to the decision-tree induction algo-
rithm. In the following section the results of
experimental investigations of the algorithms are
presented. The last part concludes the paper.
2. Cost-sensitive decision-tree learning
Let us discuss the methods used in this paper.
First we present the idea of the top-down
decision-tree induction algorithm (Quinlan,
1986), and related modifications of the algo-
rithm which respect the feature value acquisition
cost will be proposed. The basic idea involved in
any multi-stage approach is to break up a
complex decision into several simpler classifica-
tions (Safavian & Landgrebe, 1991). The deci-
sion-tree classifiers are one possible approach to
multi-stage pattern recognition. The synthesis of
a hierarchical classifier is a complex problem
(Burduk, 2010). It involves specification of the
following components (Mui & Fu, 1980):
(1) design of a decision-tree structure;
(2) feature selection used at each non-terminal
node of the decision tree;
(3) choice of decision rules for performing the
classification.
The approach under consideration is based
on the top-down induction algorithm and fo-
cuses its attention on decision-tree construction.
In each node only one test on an individual
feature is performed. Decision-tree induction
algorithms have been used for several years
(Mitchell, 1997; Alpaydin, 2010). In general,
they propose an approximate discrete function
method which is adopted to the classification
task. It is one of the most important methods for
classification which achieves very good classifi-
cation quality in many practical decision sup-
port systems. Many decision-tree algorithms
have been developed. The most famous are
CART (Breiman et al., 1984), ID3 (Quinlan,
1986) and its modification C4.5 (Quinlan, 1993).
2.1. Idea of the ID3 learning algorithm
As we mentioned above ID3 is a typical deci-
sion-tree algorithm. It introduces information
entropy as the splitting attribute choosing
measure. It trains a tree from root to leaf, a
top-down sequence which is shown in the pseu-
docode presented below.
function ID3(examples, target_concept, attributes)
parameters: examples learning set, target_concept, attributes list of
attributes
[1] Create a Root node for tree
[2] IF all examples belong to the same class
[3] THEN return the single node tree Root with this class label and return
[4] IF set of attributes is empty
[5] THEN return the single node tree Root with label ¼most common
value of label in examples and return
[6] Choose ‘the best’ attribute A from the set of attributes
[7] FOR EACH possible value vi of attribute
[8] Add new tree branch below Root, corresponding to the test A ¼vi.
c2010 The Authors. Journal Compilation c2010 Blackwell Publishing Ltd Expert Systems, July 2010, Vol. 27, No. 3 147

The central choice in the ID3 algorithm is
selecting the best attribute to test at each node in
the tree. The proposed algorithm uses the in-
formation gain that measures how well the given
attribute separates the training examples ac-
cording to the target classification. This measure
is based on the Shannon entropy of set S:
entropyðSÞ¼ X
M
i¼0
pilog2pið1Þ
where p
i
is the proportion of Sbelonging
to class i. The information gain of an attribute
Arelative to the collection of examples Sis
defined as
gainðS;AÞ¼entropyðSÞ
X
c2valuesðAÞ
jSvj
jSjentropyðSvÞð2Þ
where values(A) is the set of all possible values
for attribute Aand S
v
is the subset of Sfor
which A¼v.
2.2. Cost-sensitive modification of information
gain function
As we mentioned earlier the C4.5 algorithm is an
extended version of ID3. It improves appropri-
ate attribute selection measures, avoids data
overfitting, reduces error pruning, handles attri-
butes with different weights, improves comput-
ing efficiency, handles missing value data and
continuous attributes, and performs other func-
tions. Instead of the information gain in ID3,
C4.5 uses an information gain ratio (Quinlan,
1993). One of the main advantages of this
decision tree is that we can easily convert the
obtained tree into a set of rules (each path
from that form is a decision rule). This form of
knowledge is the most popular form of knowl-
edge representation in most expert systems
(Liebowitz, 1998). We can find some modifica-
tions of the function which evaluates attributes.
Some well-known propositions are presented
in Table 1. Let us present some propositions
which take the feature acquisition cost into
consideration.
3. Propositions of the new method
We propose the following modification of the
decision-tree algorithm to respect the cost limit.
We use the proposition known as EG2 (Nu´ nez,
1991) with different values instead of the infor-
mation gain function to determine which attri-
bute has to be chosen for node creation (Penar
& Wozniak, 2007). First, we add the following
input parameters:
COST-LIMIT, which means the limit of cost
connected with classifier exploitation (a gen-
eral initial value of COST-LIMIT is fixed by
experts)
MAX-COST-LIMIT which means the max-
imum cost limit connected with classifier
exploitation
STEP is a grain of the increasing cost
EXPECTED-QUALITY
We then propose to modify lines [4] to [6] of the
ID3 algorithm and similarly for any other algo-
rithm the code that uses functions which evalu-
ate the information gain of the chosen attribute,
e.g. for C4.5 the function that is modified is the
so-called information ratio.
[9] Let Evi be the subset of set of examples that has value vi for A.
[10] IF Evi is empty
[11] THEN below this new branch add a leaf node with label ¼most
common value of label in the set of examples
[12] ELSE below this new branch add new subtree and call
ID3(Evi, target_concept, attributes fAg).
[13] END
[14] RETURN Root
148 Expert Systems, July 2010, Vol. 27, No. 3 c2010 The Authors. Journal Compilation c2010 Blackwell Publishing Ltd

Let CS-ID3 be the name of such modified top-
down induction algorithm. Taking advantage of
this function we propose the following modifi-
cation:
4. Experimental investigation
The aim of the experiment was to compare
errors and size of the decision-tree classifiers
obtained via the C4.5 procedure which consider
the classification (attribute acquisition) cost lim-
it. We carried out two groups of experiments.
The first was performed on benchmark learning
sets; however, the second one evaluated a cost-
sensitive decision tree for the medical diagnosis
problem. The conditions of the experiments
were as follows.
All experiments were made for different limits
of cost and ovalues for EG2 modification.
For the experiments we chose a pruned
decision tree obtained via C4.5 (Quinlan,
1993).
Table 1: Cost-sensitive modifications of the information gain function
Name Description Comment
EG2 (Nu´ nez, 1991)
ICFðS;AÞ¼ 2gainðS;AÞ
½costðAÞ1o
ois the strength of the bias toward the
lowest cost attributes. In the case of
o¼0 the feature acquisition cost is
ignored and ICF has the same features as
the gain function; if o¼1 the mentioned
cost plays the most important role
CS-ID3 (Tan & Schlimmer,
1989, 1990; Tan, 1993) CSðS;AÞ¼gainðS;AÞ
costðAÞ2
IDX (Norton, 1989) IDXðS;AÞ¼gainðS;AÞ
costðAÞ
[4] IF set of attributes is empty or cost of previously chosen attributes
and cost of any remaining attributes exceed the COST_LIMIT
[5] THEN return the single node tree Root with label ¼most common
value of label in examples and return
[6] Choose the best attribute A from the set of attributes for which
summarized cost of attributes does not exceed the cost limit
[1] Call function ID3(examples, target_concept, attributes, COST-LIMIT)
[2] Evaluate obtained tree
[3] IF quality of classifier <EXPECTED-QUALITY
[4] THEN COST-LIMIT: ¼COST-LIMIT þSTEP
[5] ELSE return tree
[6] FI
[7] IF COST-LIMIT exceeds MAX-COST-LIMIT
[8] THEN return tree and information that EXPECTED-QUALITY was not
achieved
[9] FI
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The rule post pruning method was used for
pruning.
All experiments were carried out using a
modified Quinlan implementation of C4.5
and our own software created in the Matlab
environment using PRTOOLS (Duin et al.,
2004) and the classification toolbox (Stork &
Yom-Tov, 2004). For this purpose we mod-
ified the C4.5 algorithm source code.
Probabilities of errors of the classifiers were
estimated using the 10-fold cross-validation
method.
4.1. Experiments on benchmark databases
For all experiments five databases from the UCI
Machine Learning Repository (Newman et al.,
1998) were chosen. They concern different fields
of medicine and differ from each other in respect
of numbers of examples, amounts and types of
attributes, and numbers of classes. The details of
the tested databases are presented in Table 2.
The results of the experimental investigations
are presented in Figures 1–5.
4.2. Type of hypertension diagnosis
For hypertension therapy it is very important to
recognize the state of the patient and the correct
treatment. The physician is responsible for de-
ciding if the hypertension is of an essential or a
secondary type (so-called first-level diagnosis).
Senior physicians from the Hoˆ pital Broussais
Hypertension Clinic and Wroclaw Medical
Academy suggest 30% as an acceptable error
rate for first-level diagnosis. The presented pro-
ject was developed together with the Service
d’Informatique Me
´dicale from the University
Paris VI. All data were obtained from the
medical database ARTEMIS, which contains
the data of patients with hypertension who have
been treated at the Hoˆ pital Broussais in Paris.
The mathematical model was simplified. How-
ever, our experts from the Hoˆ pital Broussais
and Wroclaw Medical Academy regarded this
problem of diagnosis very useful. It led to the
following classification of hypertension type:
(1) essential hypertension (abbreviation, es-
sential);
(2) fibroplastic renal artery stenosis (abbrevia-
tion, fibro);
(3) atheromatous renal artery stenosis (abbre-
viation, athero);
(4) Conns syndrome (abbreviation, conn);
(5) renal cystic disease (abbreviation, poly);
(6) pheochromocystoma (abbreviation, pheo).
Although the set of symptoms necessary to
correctly assess existing hypertension is pretty
wide, in practice for the diagnosis the results of
19 examinations (which came from general in-
formation about the patient, blood pressure
measurements and basic biochemical data) are
used (Table 3) and the results of the classifier for
diagnosis of hypertension type are presented in
Figure 6.
4.3. Evaluation of the experimental results
In general the results of the simulations did not
surprise us. The quality of the classifiers strongly
depends on cost limit. It increased as the cost
limit rose. We noted that the quality of the
classifiers was not improved when some values
of cost limit were exceeded.
For some databases like the thyroid (Figure
1) and liver (Figure 2) we observed that
there were values of cost limit for which the
classifiers obtained were good enough. Increas-
ing them provided slightly better classifiers but
Table 2: Description of databases used in experiments
Database Number of examples Number of attributes Number of classes
1 Pima Indians diabetes 768 8 2
2 Heart disease 303 13 2
3 Hepatitis 155 19 2
4 Liver disorders 345 5 2
5 Thyroid disease 7200 20 3
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0
1
2
3
4
5
6
7
8
0 20 40 60 80 100 120 140 160
Error rate [%]
Maximum classification cost
Classification error for Thyroid Disease
0
5
10
15
20
25
30
35
40
0 20 40 60 80 100 120 140 160
Tree size
Maximum classification cost
Tree size for Thyroid Disease
Figure 1: Classification error and decision-tree size versus maximum classification cost for thyroid
database.
32
34
36
38
40
42
44
46
0 20 40 60 80 100
Error rate [%]
Maximum classification cost
Classification error for Liver Disorders
0
5
10
15
20
25
30
35
40
45
0 20 40 60 80 100
Tree size
Maximum classification cost
Tree size for Liver Disorders
Figure 2: Classification error and decision-tree size versus maximum classification cost for liver
database.
15
20
25
30
35
40
45
0 100 200 300 400 500 600
Error rate [%]
Maximum classification cost
Classification error for Heart Disease
0
10
20
30
40
50
60
0 100 200 300 400 500 600
Tree size
Maximum classification cost
Tree size for Heart Disease
Figure 3: Classification error and decision-tree size versus maximum classification cost for heart
disease database.
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tree size grew rapidly (especially for the thyroid
database), which might cause the classifiers to be
overtrained.
But we have to mention an interesting phe-
nomenon that we observed in some experiments.
For example, for the hepatitis (Figure 4) or
Pima Indian diabetes (Figure 5) databases we
noted that an increasing cost limit did not result
in classification quality improvement. A cheap
classifier (for a cost limit of about 20) had good
quality and was based on a small decision-tree
structure. The small decision-tree size protected
the classifier from overfitting. The observed
effect of decreasing tree size and error rate with
increasing cost limit were caused by the avail-
ability of new, more expensive features, which
were better then the cheaper ones used pre-
viously. The cheap features were responsible
for building an overgrown tree. When the cost
limit increased, the cheap attributes became
available again and they caused tree overgrowth
and a lower quality of classifier.
The same observation was made in the second
experiment (Figure 6). A high quality and very
cheap classifier is obtained for a cost limit equal
to about 12–15 (the maximum cost limit is 44)
and it depends on the ovalue. The second
advantage of this classifier is that tree size is
rather small which could protect against over-
fitting. A similar problem of computer-aided
diagnosis of hypertension type was described
by Blinowska et al. (1991) but they used another
mathematical model and implemented Bayes’s
decision rule. They obtained a better classifier
17
17.5
18
18.5
19
19.5
20
20.5
0 10 20 30 40 50
Error rate [%]
Maximum classification cost
Classification error for Hepatitis
0
2
4
6
8
10
12
14
16
0 10 20 30 40 50
Tree size
Maximum classification cost
Tree size for Hepatitis
Figure 4: Classification error and decision-tree size versus maximum classification cost for hepatitis
database.
22
24
26
28
30
32
34
36
0 20 40 60 80 100
Error rate [%]
Maximum classification cost
Classification error for Pima Indian Diabetes
0
10
20
30
40
50
60
0 20 40 60 80 100
Tree size
Maximum classification cost
Tree size for Pima Indian Diabetes
Figure 5: Classification error and decision-tree size versus maximum classification cost for Pima
Indian diabetes database.
152 Expert Systems, July 2010, Vol. 27, No. 3 c2010 The Authors. Journal Compilation c2010 Blackwell Publishing Ltd

than ours: its frequency of correct classification
of the secondary type of hypertension is about
85% (ours was about 70%). The advantage of
our proposition is that it is simpler and extre-
mely cheap compared with the model in the
above-mentioned paper.
5. Conclusion
The idea of a cost-sensitive learning method for
a decision tree with fixed cost limit was pre-
sented in this paper. The properties of the
proposed concept were established during com-
puter experiments conducted on four bench-
mark databases from the medical area. The
results did not surprise us but we noted some
interesting properties of the method under con-
sideration. We hope that the idea presented will
be helpful for constructing real decision systems,
especially for decision-aided systems where the
cost of classification plays an important role.
Acknowledgements
This work is supported by the Polish State
Committee for Scientific Research under a grant
which was realized in the years 2006–2009.
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The authors
Wojciech Penar
Wojciech Penar is a PhD student in the Institute
of Computer Engineering, Control and Ro-
botics, Faculty of Electronics, Wroclaw Univer-
sity of Technology, Poland. He received an MS
degree in computer science from the Wroclaw
University of Technology in 2004. His research
focuses on distributed systems, communication
networks and machine learning.
Michal Wozniak
Michal Wozniak is Professor of Computer
Science in the Department of Systems and
Computer Networks, Faculty of Electronics,
Wroclaw University of Technology, Poland. He
received an MS degree in biomedical engineer-
ing in 1992 from the Wroclaw University of
Technology, and PhD and DSc (habilitation)
degrees in computer science in 1996 and 2007
respectively from the same university. His re-
154 Expert Systems, July 2010, Vol. 27, No. 3 c2010 The Authors. Journal Compilation c2010 Blackwell Publishing Ltd

search focuses on multiple classifier systems,
machine learning, data and web mining, Bayes
compound theory, distributed algorithms, com-
puter and networks security and teleinformatics.
Professor Wozniak has published over 120 pa-
pers and two books and has edited three books.
He is editor in chief of International Journal of
Computer Networks and Communications and
associate editor of several international journals
including Pattern Analysis and Applications,Ex-
pert Systems,Information Fusion,Logic Journal
of the IGPL and International Journal of Com-
munication Networks and Distributed Systems.
He serves on the programme committees of
numerous international conferences. His works
have been transitioned into commercial applica-
tions. Professor Wozniak has been involved in
many research projects related to machine learn-
ing, computer networks and telemedicine.
Moreover, he has been a consultant on several
commercial projects for well-known Polish
companies and for the Polish public administra-
tion. Professor Wozniak is a member of
the IEEE (Computational Intelligence Society
and Systems, Man and Cybernetics Society)
and IBS (International Biometric Society).
For a more detailed profile see http://www.kssk.
pwr.wroc.pl/pracownicy/michal.wozniak-en.
c2010 The Authors. Journal Compilation c2010 Blackwell Publishing Ltd Expert Systems, July 2010, Vol. 27, No. 3 155