IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 1, FEBRUARY 2004
123
Fuzzy Repertory Table: A Method for Acquiring
Knowledge About Input Variables to Machine
Learning Algorithm
Jose J. Castro-Schez, Juan Luis Castro, and José Manuel Zurita
Abstract—In this paper, we develop a technique for acquiring
the finite set of attributes or variables which the expert uses in a
classification problem for characterising and discriminating a set
of elements. This set will constitute the schema of a training data
set to which an inductive learning algorithm will be applied. The
technique developed uses ideas taken from psychology, in particular from Kelly’s Personal Construct Theory. While we agree that
Kelly’s repertory grid technique is an efficient way to do this, it has
several disadvantages which we shall try to solve by using a fuzzy
repertory table. With the suggested technique, we aim to obtain the
set of attributes and values which the expert can use to “measure”
the object type (class) on the classification problem in some way.
We will also acquire some general rules to identify the expert’s evident knowledge; these rules will comprise concepts belonging to
their conceptual structure.
Index Terms—Expert systems, fuzzy repertory grid, knowledge
acquisition, knowledge-based systems.
I. INTRODUCTION
W
HEN inductive machine learning techniques are applied
to build computational systems, it has been hypothesized
that there is a set of examples (training set) which is valued
according to a set of attributes or features key given. The inductive learning algorithms’ challenge is to select the most significant ones and constructing a prediction function that classifies samples according to these attributes. There are, however, situations where this hypothesis is not true, for example
in the knowledge engineering field when the knowledge engineer is acquiring knowledge for developing an expert system or
knowledge-based system and he meets with classification tasks
in which the expert cannot properly convey his knowledge, the
expert is not able to put into words perceptions that he has but
has never verbalised. In such situation, and according to ideas
from balanced cooperative modeling [24], we suggest using inductive machine learning to acquire this knowledge. In order to
carry out inductive learning in this context, the attributes used
by the expert in these tasks must be found and a set of examples
requested from the expert which are valued according to these
attributes.
Manuscript received October 23, 2001; revised September 20, 2002 and June
16, 2003.
J. J. Castro-Schez is with the Department of Computer Science, University
of Castilla-La Mancha, Escuela Superior de Informática, Ciudad Real 13071
Spain (e-mail: [email protected]).
J. L. Castro and J. M. Zurita are with the Departamento de Ciencias de la Computación e I.A., E.T.S.I. Informática, Universidad de Granada, Granada 18071,
Spain (e-mail: [email protected]; [email protected]).
Digital Object Identifier 10.1109/TFUZZ.2003.822684
Moreover, in the knowledge engineering field, we are interested in using the results obtained by the inductive machine
learning in later stages of the knowledge acquisition process.
For employing them beneficially, it is very important that the expert understands them. In order to make this easier, we suggest
that the results obtained by the inductive learning algorithms
will be expressed as they would by the expert and this involves
acquiring values that can take each attribute used by the expert,
that is to say the definition domains of the attributes.
Another case, in which can be useful the identification of a set
of relevant input variables for classification problems is when
we deal with training data that contain many features that are irrelevant to the target concept being learned. There are studies [2]
that show that the performance of inductive learning algorithms
is seriously reduced by the presence of irrelevant features. In accordance with the results of this study, one could not be sure of
these algorithms to filter out irrelevant features, some technique
should be employed to eliminate irrelevant features and focus
the learning algorithm on the relevant ones.
In this way, we can say that in some contexts, from now on we
focus our study in the knowledge engineering field, the selection of attributes and the acquisition of their definition domains
is one of the key activities which we face during the classification problem. In order to do so, many manual and automatic
methodologies could be applied, such as card or concept sorting
[23], laddered grid [26], interviewing [13], [19], [20], [30], etc.
We consider the repertory grid technique to be a good method
for this, because of it has a solid foundation in human psychological theory [22], this has already been used to acquire knowledge [4], it is easy to formalise, use and understand since it is
based on the comparison for recognising and understanding of
differences.
Kelly’s repertory grid technique is a method which was
developed with the cushion of the Personal Construct (PC)
Theory [22]. This theory reports how people make sense of the
world and the people in it. It was axiomatized as a fundamental
postulate together with eleven corollaries [22], two of which,
dichotomy and construction corollaries, give theoretical reasons
for the technique developed in this paper. The dichotomy corollary of the PC theory states “A person’s construction system
is composed of a finite number of dichotomous constructs.”
Kelly assumes that people develop a set of personal constructs
(person’s construction system) which is used to evaluate the
world around them and anticipate events and experiences.
The construct notion is like a verbal label, which represents
a bipolar distinction with opposite poles, we usually think in
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 1, FEBRUARY 2004
Fig. 1. Partial example of a repertory grid for eliciting role models.
Fig. 2. Partial example of an extended-classic repertory grid.
terms of contrast. For example, a buyer might use the construct
cheap-expensive, among others, to judge the seller’s offers.
The construction corollary states “A person anticipates events
by construing replications,” These constructs are a useful tool
to anticipate and control events. Therefore, if we want to know
how people perform, it is very useful to clarify their personal
constructs. In order to do so, Kelly developed the repertory
grid technique.
The repertory grid is individual test to assess his/her personal
construct system, it is essentially a complex sorting test in which
a list of elements are judged successively on the basis of a set
of bipolar constructs [1]. The person first defines a set of elements (concrete or abstract entities taken from the area to which
the test is to be applied to), then he introduces a set of constructs (the distinctions that he/she makes among the elements).
In order to acquire this last set, a small group (usually three similar elements) is selected and presented to him and he is asked to
name a construct that can divide this group into two subgroups.
For each construct obtained in this way, the person must decide
which construct pole to apply to each element. By repeating this
process we shall obtain the required set of personal constructs
that form his personal construct system.
Elements and constructs (dimensions of similarity and differences between elements) are central to knowledge representation in repertory grids. The most basic form of a repertory grid
is a rectangular matrix with elements as columns and constructs
as rows (Fig. 1). Each row-column intersection in the grid contains a rating to show how a person applied a given construct to
a particular element. Thus, if the element is closest to the left
pole of the construct, he places a tick; otherwise, a cross.
Within classification problems, the elements of a repertory
grid will be those classes which we want to learn to classify and
the constructs will be the input variables, whose different values
can distinguish a class from another one. Since we are interested
in to obtain the input variables plus their definition domains (the
rating used by the expert to value each input variable) that the
expert uses in a classification task and in the same way in which
he uses them, for facilitating their integration into the knowledge
acquisition process, we will need to make several changes to the
classic repertory grid.
In this paper, we shall modify this technique in several ways.
We shall present the fuzzy repertory table (FRT), which will be
useful when it comes to acquiring the input variables and their
domains, which are used by the expert in classification tasks.
Furthermore, we shall obtain a set of very general rules which
model the way the expert classifies. These rules may be used
to plan future interviews. Moreover, the information obtained is
highly individual, and it can be compared to those seen in other
expert’s domain for obtain more knowledge. The proposed technique allows the domain expert to build simple knowledge bases
directly, only with a very little guidance from programmers or
specially trained knowledge engineers.
The rest of this paper is organized in the following way. Section II presents the main difficulties underlying the repertory
grid for our aim; Section III introduces a data structure which
allows a number of different data types to be represented uniformly; Section IV specifies the proposed method; Section V
illustrates the operation of our method with an example; and
finally, Section VI summarizes our main contributions and outlines the way we could use the results of our method to obtain a
better knowledge base and some directions for future research.
II. LIMITATIONS CURRENT REPERTORY GRIDS
There are as many variants on repertory grid technique as
possible values which may be introduced into the grid [1], [25].
No variant, however, allows experts to express themselves with
the total freedom as they usually do. In this regard, the following
hold.
• Classic or Kelly’s repertory grids, which work with
bipolar distinctions (Fig. 1): The classic repertory grid
requires the expert to assign one pole of the construct or
the other that has been established to each element. This
will often not be possible because in the expert’s mind
there are intermediate states between the left and right
construct poles. The construct “transport service quality”
with the poles “Good service” and “Bad service” is an
example of a construct which allows intermediate states.
• “Extended-Classic” repertory grids, i.e., those with permitted values which are chosen from a predefined rating
scale (Fig. 2): Instead of regarding each construct as a
bipolar distinction, the extended-classic repertory grid regards it as a scale. The minimum value of the scale is
at the left-hand pole of the construct, and the maximum
value is at the right-pole, while the remaining values fall
between the poles. This kind of grid requires the expert
CASTRO-SCHEZ et al.: FUZZY REPERTORY TABLE
125
Fig. 3. Hwang’s fuzzy table.
to rate each element on the construct that has been established, i.e., they will make crisp distinctions. However, it
will often not be possible since there is vagueness in the
expert’s mind.
Some researchers treat the vagueness present in the expert’s
mind in various ways.
• Bathia’s approach [3] allows a range of ratings to be expressed rather than a crisp value. Bathia et al. present the
idea of an interval to represent the ratings of a construct
with respect to an element. The interviewee provides an
interval to describe the ratings, without a firm commitment
to any particular integer even within the interval.
• Gaines’ approach [18]. A good solution to the problem
mentioned above is to use the concept of a fuzzy set to represent each construct pole. In this kind of repertory grid,
assigning a rating to an element for a given construct is
considered to be the same as providing a representation
for that element’s degree of membership in the fuzzy set
which defines each construct pole.
However, these repertory grids do not hold information linguistic, which is frequently used to express expertise [32]. In
order to do so, Hwang [21] developed the fuzzy table.
• Hwang’s fuzzy table adapts the grid so that it is suitable
for treating fuzzy logic instead of using fuzzy logic with
grids. In the fuzzy table, constructs are fuzzy variables
that take values from a finite set of fuzzy linguistic values.
Each table entry consists of the most desirable fuzzy value
and the degree of certainty for that choice (Fig. 3). The
most desirable fuzzy value is a combination of linguistic
hedges (very, normal, more or less) and fuzzy linguistic
values which are represented by means of a seven-scale
rating mechanism. The ratings are integers from the scale
3 to 3. For example, if we consider the ratings of
variable Height in Fig. 3 then 3 means VERY TALL, 2
means TALL, 1 means MORE OR LESS TALL, 0 means
MIDDLE, 1 means MORE OR LESS SHORT, 2 means
SHORT and 3 means VERY SHORT. “S” and “N” are
the degree of certainty for each rating, and they mean
VERY SURE and NOT VERY SURE, respectively.
Although, it is clear that there is uncertainty and vagueness in
the expert’s mind, it is also obvious that not everything in their
mind is vague or uncertain. For this reason, a repertory grid that
only considers fuzzy values does not appear to be appropriate
for our purpose of allowing the expert to express him/herself in
total freedom.
• Shaw’s repertory grid [27], “WebGrid II”, is prepared for
categorical and numerical data types in addition to rating
scales, and also allows symbolic values.
On the other hand, there are many approaches to grid analysis
such as nonparametric analysis, principal component analysis,
and hierarchical cluster analysis. These analysis techniques are
based on the distance measures between vectors (either rows or
columns in the grid) to obtain similarities and differences [1].
There are studies that show that the distance-based approaches
may only find symmetric relations, i.e., they only find those relations with the following structure: construct/element A is similar
to construct/element B. Such methods do not find a one-sided
implication relationship, it is not possible to infer that one construct entails another one or an element. In response to this limitation, some researchers have focused on clarifying the logical
rationale underlying constructivist knowledge acquisition and
they develop procedures for the logical analysis of construct entailment [7], [14]–[16] that are used in automatic knowledge acquisition tools [17].
We shall develop a repertory grid which will be suitable for
uniformly manipulating the most frequent features or input variables types that are used by the expert in classification problems.
We will take into account the following types which we consider
to be the most common:
—
ordered-discrete or ordinal (e.g., marks in an examination);
—
unordered-discrete or nominal (e.g., hair color);
—
Boolean (e.g., yes/no answers in a test);
—
ranking (e.g., quality of the service);
—
continuous or numerical (e.g., age).
Furthermore, since we want to give the expert freedom when
s/he expresses his/her knowledge, we shall have to treat the
vagueness which is inherent to the expert’s mind. In order to
do so, we allow the interval (as in Bathia’s approach) and linguistic fuzzy values (as in Hwang’s approach) to be assigned
to continuous and ranking variables. We also allow the expert
to assign numerical and symbolic values like any, several or no
values to one construct (as in Shaw’s approach).
We shall also suggest a gap measurement-based repertory
grid analysis technique. This analysis can generate knowledge
from which one-sided implication relationships will be derived.
Its structure shall be construct A entails element B. These entailments will be used in the FRT construction process.
III. PRESENTATION OF THE SUGGESTED REPRESENTATION
SCHEMA
We need a representation which allows all the previously
mentioned values to be described. This representation must
have the following properties.
•
•
•
•
Flexible: It can hold any value used by the expert.
Linguistic: It will be understood by the expert.
Mixed: It allows vague and crisp values.
Uniform: It can be used by inductive learning algorithm.
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 1, FEBRUARY 2004
Fig. 4. Definition of the proposed function to represent any value.
Fig. 5.
Fuzzy repertory table.
We also need another representation which holds knowledge
that computes the mapping from inputs to output. Since we want
to use this knowledge during repertory grid construction and
in later stages in the knowledge acquisition process, the expert
must understand the selected representation. In order to make
this easier, we recommend that the chosen representation be
IF–THEN rules formed by the concepts belonging to the expert’s
conceptual structure and that these be expressed in the way the
expert uses them.
The choice of these representations is an important decision
because they may influence the selection of attributes and the
acquisition of their definition domains. Moreover, also they may
have an effect on the knowledge acquisition process.
We suggest the popular trapezoidal function to represent the
crisp and vague values mentioned before. Crisp sets will be considered as especial cases of fuzzy sets. This function is defined
as shown in Fig. 4.
The FRT form is a rectangular matrix with elements as
columns and constructs as rows. Each row-column intersection
in the grid contains a rating which comprises a set of trapezoidal
functions (that may be the empty set) showing how a person
applied a given construct to a particular element (Fig. 5).
As it has been mentioned before, the repertory grid technique
is based on the search for elements that are similar. In order
to do so, it uses an analysis technique. Since all the values in
the fuzzy repertory table are represented as if they were fuzzy
values, i.e., by means of a trapezoidal function, we need an analysis technique that uses a measurement for comparing fuzzy
sets. Numerous measures of comparison between fuzzy sets can
be found in the literature [10], [11], [33]. We shall require from
the measure of comparison selected, , the following properties.
• P.1:
(Non-negative)
• P.2:
if and only if
.
In particular, it is verified when we compare the same
(extension of minimality propfuzzy set,
, then
.
erty) or a subset, if
. The comparison between two
• P.3:
fuzzy sets must be reciprocal (Symmetry property).
if and only if is closer to
• P.4:
than to . Sensitivity among different pairs of fuzzy
disjoint sets. The measure must differentiate between different pairs of disjoint sets (that is,
) according to their position in the domain of discourse
on which they are defined, providing a lower value for
those that are closest.
For any fuzzy sets , and belonging to the class of all
fuzzy sets, , defined over the domain, X.
As well as, we are interested in the following additional subjective property.
• P.5: Understandable by the user of the fuzzy repertory
table. The FRT development process requires interaction
with its user, the user frequently examines the result obtained by the analysis technique. The ability to understand
the comparisons helps to focus the analysis and makes interaction with the FRT easier. Because it, we must use a
measure of comparison that will be easily interpretable
and intuitive for the user.
The measures of comparison between fuzzy sets can be
grouped together in two broad classes: geometrical measures,
based on the classical concept of metric space and its associated
distance function and set theoretical measures, based on the
use of operators on sets. When we have need of make the
result of a comparison understandable by the user (P5), the
measures belonging to the second class deemed are not the
most appropriate, since their interpretation is complicated for
persons that do not know anything about fuzzy sets. Moreover,
they do not verify the P4 property and some of them not either
the P2 property. On the other hand, all measures of the first
class considered verify these properties, but do not verify the P2
property. Thus, a specific and particular measure is presented
which verifies the properties mentioned earlier.
We ask for a measurement which holds information about the
gap separation between two fuzzy sets. The separation between
CASTRO-SCHEZ et al.: FUZZY REPERTORY TABLE
Fig. 6.
127
Separation between two fuzzy sets A and B .
Fig. 7. Area of the grey zone is the separation measurement between A and B values. (a) Separation between two crisp values S (A; B ) = a1 + a2 =
(b) Separation between two nearby values S (A; B ) = (b
c)=4.
0
two fuzzy sets has a close similarity to the fuzzy set between
(Fig. 6).
Let us make some prior definitions which will be needed to
formally define the separation measure. These definitions are
carried out over a fuzzy set , that is defined by means of a
trapezoidal function with parameters , , , and .
, which is the set conDefinition 1: The fuzzy set
taining values greater than , is defined in (1).
area. The measure
is defined in (4).
is a real function,
, which is the set con, is defined in (2)
(2)
Definition 3: The fuzzy set between the fuzzy sets and ,
i.e.
, is defined as the set of values that are less
than the major value and greater than the minor value, and is
defined in (3).
(3)
In (3), the symbol is the minimum t-norm and is the maximum t-conorm (Fig. 6).
The suggested measure of comparison, , between fuzzy sets
and
is based on the just recently
defined fuzzy set between and it lies in the calculation of its
0
c
.
which
(4)
(1)
Definition 2: The fuzzy set
taining values that are less than
b
In (4),
and
are calculated in (5)
(5)
and
are based on the definitions of the
The functions
membership functions of the fuzzy sets and , and they calculate the area corresponding to the fuzzy sets
and
, respectively. By the defand
sets, we can state that
iminitions of
and the contrapositive statement. To proof this
plies that
statement it is enough to prove that is not possible that a element from the X domain, belongs simultaneously to the sets
and
.
Some examples are shown in Fig. 7.
The suggested measure of comparison verifies the aforementioned properties.
• P.1:
, the area never is negative.
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 1, FEBRUARY 2004
• P.2:
When
if and only if
.
then the set
is the empty set, there are not ele.
ments over the domain X belong to the set
Therefore, the area of the empty set is null, that is
.
. The symmetry property is ver• P.3:
ified by the measure definition. The separation measure
and ,
is calculated as the area of
between
the fuzzy set
, that is
. On the other hand, the separation between and ,
is calculated as the area of
,
the set
. Because the sum operator
is commuthat is
tative we can state that symmetry property is verified.
if and only if
is closer to
• P.4:
than to . It can be easily observed in the graphical
.
definition of the fuzzy set
• P.5: The proposed measure is understandable, this affirmation is justified in the fact that it has a graphical representation that is intuitive.
In this regard, the proposed measurement is the one most suitable for our purpose of obtaining a set of attributes which clearly
distinguishes between the elements introduced by the domain
expert in the FRT.
The suggested measurement of separation cannot be considered as a measure of distance [31] because it does not verify
. This
the triangular property
is not, however, a problem according to Tversky and Gati [29],
who have demonstrated that in psychology this property is not
always true and that sometimes it is necessary to find a procedure that only partially occurs. According to the Bouchon–Meunier’s classification of measures [6], the suggested above could
be classified as a measure of dissimilarity extended with some
additional properties (P.2, P.4, and P.5).
Next, we need to include the suggested measure as part of
the FRT. For doing it, we make some previous definitions and
conventions.
Since the fuzzy repertory table allows the special and symbolic values All and None to be included to indicate that an attribute can take any or no value from its definition domain respectively, we need to state how the proposed separation measurement behaves with these values. Thus, the separation between any value and the value All will always be 0; the value
None separation is not considered.
On the other hand, the expert can include a set of values
in the fuzzy repertory table as a value of an
attribute for an element. For this, the separation measurement
must be defined between sets of values.
Definition 4: Separation between two sets of values
and
is defined
as the least separation between any two values from these sets,
that is to say
(6)
The separation calculation is carried out in the same way
without depending on the type of the attribute. It is, however,
necessary for us to make some comments when the calculation is carried out among nominal and nonequidistant ordered
values. Since, these values can not be represented by mean of
the trapezoidal function.
When the attribute is nominal, two things may happen.
• Its values can be compared. We have suggested that any
value that the attribute can take will be transformed into
the elements of a new fuzzy repertory table. A group of
attributes is then searched for which allows these new elements to be distinguished between. The separation of these
elements in the new table will be used to establish the separation of those values in the original table.
• Its values cannot be compared. In this case, we use the following heuristic to calculate the separation between values
according to the attribute: If both values are the same, the
separation between them is zero. If on the contrary they
are different, then the separation is the greatest and this is
1 in normalized spaces.
The choice between one way of acting and the other will depend on the problem, the attribute and the expert’s knowledge.
The separation among attributes with nonequidistant values
will be stored in an auxiliary table, when we will need calculate
the separation of these attributes in the FRT, we will look up the
auxiliary table.
Up till now, we have defined a measure that allows values to
be compared. It is now necessary for us to establish a method
to calculate the closeness between elements in order to be able
to make comparisons between them. Since the elements are
valued according to a set of attributes, we must know how each
particular attribute separates to the elements, we note it as
, and it is
where is the value that the element takes in the attribute and is the value of the element
in the same attribute.
and
Definition 5: The separation between two elements
,
, is defined as the sum of the separation between
these two elements according to each of the attributes that defines them. This is calculated in (7)
(7)
with being the number of total attributes, the number of attributes with values different from None and
is the normalized separation which is defined in
(8)
with
being the separation between elements
and
according to the construct , as it has been previously
being the separation between
mentioned, and
the maximum (rightmost value) and minimum (leftmost value)
values of the attribute . The leftmost value will be the one that
has the lowest and parameters. The rightmost value will be
the value with the parameters and greatest.
CASTRO-SCHEZ et al.: FUZZY REPERTORY TABLE
Fig. 8.
129
Links and connections between the algorithms.
The calculation of the closeness between any pair of elements
of the input group will enable detection of those clearly distinguished elements and those for which it will be necessary to introduce new attributes to discriminate between them. We shall
define the distinguished condition here.
and
are disDefinition 6: We say that two elements
, with
being a
tinguished if and only if
parameter named distinction distance that takes its values in the
interval [0,1].
takes will depend on the
The value that the parameter
problem we are faced with and will be defined by us. The
greater the value that the parameter takes, the greater the
separation between the domain elements and consequently
greater is the quality of the knowledge acquired. Normally,
. We recommend the
it takes its values from
following values: when it works with only boolean values,
; Ordered-discrete or ordinal and Ranking values
;
Continuous values (crisp or fuzzy)
.
Once we have defined the types of attributes that the expert
will be able to use and a measure to compare fuzzy values and
it has been extended for comparing elements, we proceed to develop the algorithms to build the fuzzy repertory table, and also
the procedures to analyze and search evident rules of knowledge.
IV. ALGORITHMS FOR WORKING WITH THE SUGGESTED
FUZZY REPERTORY TABLE
In this section, we present the algorithms necessary to build
the suggested fuzzy repertory table (Algorithm 1), cluster the
elements and search for the elements that constitute the group of
maximal equivalence (Algorithm 2), obtain the attributes
and their definition domains
introduced by the expert
(Algorithm 3) and obtain a set of rules that model the way the
expert reasons (Algorithm 4).
In Fig. 8, we show a global scheme that shows the links and
connections between each algorithm.
Algorithm 1 builds the fuzzy repertory table. The operation
of this algorithm is based on the search for nondistinguished
elements so that the expert can introduce new attributes which
allow them to be distinguished. The aim of the algorithm is to
obtain a set of attributes or input variables , and their definition
that allow each of the elements belonging to the
domains
system output set to be discriminated against.
The algorithm uses two matrixes, and . Matrix is the
fuzzy repertory table. Matrix
is an innovation that we make
is named the distincto the repertory grid technique. Matrix
tions matrix. It is a bi-dimensional matrix with dimension
cells (Fig. 12) where is the number of FRT elements. Each
cell ( , ) will store information about the attributes that disand , and the strength to
tinguish between the elements
which they are distinguished. This strength is the normalized
separation between these two elements against that attribute, i.e.
. Initially each cell of the matrix will have the NIL value.
For example, in the distinctions matrix shown in Fig. 19, the
cell (2,5) contains the following information: the attributes that
and , all of them with the
distinguish from are ,
maximum strength, i.e., 1.
The distinctions matrix allows us to do the following.
130
• Reconstruct the substantive points that occurred in the interview by looking at it.
• Avoid that elements distinguished by means of one or several attributes will be presented as similar elements to the
expert. In traditional repertory grids (see Section II), the
introduction of attributes can do that elements clearly distinguished will be shown as similar elements, with the distinctions matrix we are sure that distinguished elements do
not be compared again.
• Extract easily a set of rules that contains those attributes
and their values that allow distinguish each class from the
other (in an approximate way!).
The information contained in the distinctions matrix will be
used when we search for the group of maximal equivalence and
the rules that relate attributes with elements. The distinctions
matrix is obtained during the Algorithm 1 run.
Algorithm 1
Input: The set of elements that we want
to learn to classify, .
and Distinctions matrix
Output: FRT
.
, set up two matrices
and
1) Let
. Matrix
with
columns and zero
with
columns and
rows. Matrix
rows.
will be initialized to ‘Nil’.
(Al2) Search for similar elements in
gorithm 2).
3) If there are no similar elements (Algorithm 2 returns the value “Stop”), then
terminate this algorithm. The job is
done.
4) Otherwise, assume that we have a set
(Algorithm
of three similar elements
of elements). Ask
2 returns a subset
the domain expert to provide a new con( must not be any of the construct
structs which always exist in ), such
clearly divides the three elethat
into at least two nonempty
ments of
subgroups. All elements in the same
subgroup have the same behavior regarding , but the elements in different subgroups have different behaviors.
by one row, which
5) Increase matrix
corresponds to the new construct . Let
be its -th row.
this new row of
consider the new
6) For each element of
construct :
takes its values in the contin• If
uous domain of the real numbers, go
to step 6.1.
takes its values in the dis• If
crete domain, go to step 6.2.
is defined for the -th ele6.1. If
ment of , then ask the domain ex-
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 1, FEBRUARY 2004
pert to assign one or several values
to the matrix entry ( , ) or the
All value if it can take any value.
is defined for the m-th ele6.2. If
ment of , ask the domain expert
the class of the construct:
is a Boolean construct, go
• If
to step 6.2.1.
is a graduated predicated
• If
(ranking), go to step 6.2.2.
is an ordinal construct, go
• If
to step 6.2.3.
is a nominal construct, go
• If
to step 6.2.4.
6.2.1
.
The matrix entry ( , ) will be 0
for the false value, 1 for the true
value and All for any value.
6.2.2
.
Ask the domain expert to assign a
value between 1 and maximum value of
scale to the matrix entry ( , ) or
the All value if can take any value
of the scale.
6.2.3
.
Ask the domain expert to enter a
value to the matrix entry ( , ),
with
being the value of
for this
element. We also ask for an order
for this value.
6.2.4
.
The same as the ordinal constructs,
without asking for the order for
this value.
is not defined with
7) If the construct
respect to the m-th element, we assign the value None to the matrix entry
( , ). Go to step 2.
and
8) For all the couples of elements
belonging to
that are distinguished using the attribute , i.e.
to make:
(with+ being an operator of concatenation).
9) End of algorithm.
Before studying the operation of Algorithm 2, let us make
some prior definitions:
Definition 7: An equivalence group is a group of elements
in which any two elements are not distinguished.
Definition 8: An equivalence group is maximal if and only if
there is no other group with more elements.
Algorithm 2 clusters nondistinguished elements into equivalence groups and returns three elements of the group of maximal
equivalence. The proposed algorithm implements a method of
hierarchical clustering that uses the average measure to perform
the cluster.
CASTRO-SCHEZ et al.: FUZZY REPERTORY TABLE
Fig. 9. Representations of values introduced by the expert,
131
DDV
.
Algorithm 2
Input: The set of elements that we want
to learn to classify, .
Output: A set of three similar elements,
.
1) Form a group for each element belonging
to
and introduce it in the set .
2) Calculate the separation between each
pair of groups in .
3) Merge the groups with the least distance verifying that:
• The distance is lower than a
(for example,
pre-set limit,
).
• There are at least two elements
and
belonging to different groups
such that the entry ( , ) in matrix
is equal to “Nil,”
4) If a merge has been carried out, include this merge in . Go to step 2.
5) If the cardinality of the maximal
equivalence group is equal to or
greater than three, select three elements from this subgroup that verify
that all pairs of elements belonging to
this set of three elements have entry
, . Return
equal to “Nil” in matrix
them.
6) Otherwise, return “Stop,”
7) End of algorithm.
The distance between two groups of elements
be calculated in (9).
and
will
(9)
The parameter
takes its value from the [0,1] interval and
will be defined by us. The greater the value that the parameter
takes, the greater is the possibility of combine two groups to
form one and consequently greater is the risk of merge groups
which elements will be clearly distinguished.
After algorithms 1 and 2 have been executed, we
will have the definition domains
for these variables
. The machine learning techniques can use this
information.
The excessive freedom granted to the expert when introducing continuous values results in situations where there are
that verify that
values ,
(Fig. 9). As this can cause multiple interpretations when the
machine learning is applied [9], it will be avoided.
It is necessary to develop an algorithm (Algorithm 3) that obtains for each variable , the set of values without intersections
in their cores
and that will be derived from the values
. The proposed algorithm converts
provided by the expert
the areas where there are intersections into fuzzy areas. The corbelonging to any
responding fuzzy sets
form a fuzzy partition of the domain in the strict sense, that is,
. with
and
being the right and left boundaries on the domain X.
Algorithm 3
Input: A set of trapezoidal values,
.
Output: A set of trapezoidal values
without intersections in their cores,
.
1) Order the values
according to
the parameter . Let us call this or.
dered list
2) For each value
2.1. Check if the value
has nonnull
intersection in its core with any
.
other value
2.1.1
.
If this is so then:
and
• Introduce the parameters
of
into the list of over.
lapped values,
is the farthest right
• If
value of
then also introduce
the parameter .
is the farthest left value
• If
then also introduce the
of
parameter .
2.1.2If this is not so then:
into
.
• Introduce
, then take values in
• If
fours, beginning with the first
and conelement of the list
of
tinuing from the parameter
the previous value. Each four
132
Fig. 10.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 1, FEBRUARY 2004
DDV
of the values shown in Fig. 9.
values constitute a new value
of
.
then take values in fours,
3) If
beginning with the first element of the
and continuing from the paramlist
of the previous value. Each four
eter
of
values constitute a new value
.
4) End of algorithm.
The intersection areas in the cores are transformed into fuzzy
areas since they can belong to the two values. In Fig. 10 we show
which values are obtained after executing Algorithm 3 over the
values shown in Fig. 9.
Once we have obtained the values with null intersections in
their cores, which constitute definition domains of each of the
, we apply an alinput variables, used by the expert
gorithm to obtain rules that relate inputs variables with output
elements.
The proposed algorithm interprets the fuzzy repertory table
and distinctions matrix and obtains the rules that model the expert’s classification activity.
The algorithm that we propose establishes an IF–THEN rule for
each element that we want to learn how to classify. The form of
our rules will be the same as that proposed in [9]:
with being an input variable and the set of values that can
in that rule. The values of
are
be taken by the variable
taken from the definition domain of the input variable . The
definition domains of each input variable (or attribute) , i.e.,
, will be formed by the values that the expert has used to
value each element according to this variable.
This algorithm establishes one rule per element. The aim of
the algorithm is to include in the antecedent part of the rule the
smallest number of attributes that are necessary to classify the
element of the consequent part of the rule. The algorithm checks
if the element is sufficiently separated from the rest of the elements in the input group, that is to say, there are attributes that
distinguish it from more than half of the elements of . In order
to do so, it analyzes the distinctions matrix . If this is so, the
algorithm will take from each input ( , ) of the distinctions matrix the attribute which most separates elements and . When
there are several attributes that separate in the same way and
which are the best, then it takes the most significant. By significant, we are referring to the variable which is most frequently
used to distinguish the consequent element of any other element.
If there are several equally significant attributes, we introduce
the one which is not yet part of the antecedent in the rule. This
is useful information which we need to store for the future. If
there are several attributes that are not in the rule antecedent,
then all are included in it.
We use a function in the algorithm which makes correspond
, which are introduced by the expert
to the values
belonging
in the construction process of the FRT, the values
to
that verifies
. It is noted
as DDV. When an element is not sufficiently separated from the
other elements, the method establishes a rule whose antecedent
part will be formed by all the attributes for which this element
has been valued.
Algorithm 4
Input: The set of elements that we want
to learn to classify, ; FRT
and
Distinctions matrix ( );
Output: A set of rules, .
.
1) Let be
rules
, one per element.
2) To build
3) For
to
do:
to
do:
3.1. For
3.1.1
“Nil” then discriminant++
.
If
3.2. If (
) then
3.2.1
to
do
.
For
3.2.1.1
“Nil” and
.
If
then introduce the attribute
and
its value DDV(A[z,i]) in the rule
.
3.2.1.2
“Nil” and
then
.
If
, that
introduce the attribute
most separates between the elements
i and j and their value DDV(A[z,i])
.
in rule
3.2.1.2.1
.
If there are two or more attributes
that separate with the same value,
introduce the most significant atand its value DDV(A[z,i])
tribute
in the rule.
CASTRO-SCHEZ et al.: FUZZY REPERTORY TABLE
133
Fig. 11.
Fuzzy repertory table after introducing the first variable.
Fig. 12.
Distinctions matrix after introducing the variable
3.2.1.2.2
.
If there are two or more attributes
that separate with the same value
and all are equally significant,
and the
introduce the attribute
value DDV(A[z,i]) that is not still
.
included in the rule
3.2.1.2.3
.
If no attribute is included then introduce all of them.
3.3. Otherwise establish a rule which
has the values DDV(A[z,i]) for
..number of rows of
as its
antecedent.
4) End of the algorithm.
In Section V, we explain a practical example for showing how
the suggested algorithms perform.
V. EXAMPLE OF THE SUGGESTED METHOD
This section examines two examples and cite briefly other
concrete and real applications where the FRT has been used. In
the first example we have kept the discussion down to a minimum in order to give a description of how the FRT technique
works rather to provide a solution to the complex problem to
which is applied. Our focus in the second example is to show
how a machine learning algorithm [9] uses the results obtained
by the FRT. Next, we summarize a set of applications where the
FRT is being successfully applied.
A. First Example, how the FRT Works
Let us apply the FRT technique to the first application. We
want to find a set of variables that allows us to distinguish between good, ordinary and bad teachers, that is to say, to carry out
a teacher quality assessment. In order to do so, we have a domain expert who is a member of the teaching quality unit of the
University of Castilla-La Mancha, Spain. This unit is concerned
with their teachers’ competence. For simplicity reasons, we will
be looking at Computer Science Teachers and more specifically
the teachers of the first course (it is only an example to show
the performance of the FRT technique). If we want to discover
Fig. 13.
C
(Teacher preparation).
Definition of the apply classroom practices variable.
which rules this unit uses to assess its teachers, our first job is
to find out the variables that it uses.
In order to obtain this set of variables, we will use the suggested fuzzy repertory table. The set of elements that we want
to learn to classify will be good teacher, ordinary teacher and
bad teacher. Referring to abstract teachers, however, complicates the comparisons and it is for this reason that we ask the expert to assess the six teachers of the first course according to the
three categories of good, ordinary and bad. The first course only
has six teachers. From now on, we will refer to these teachers
as: , , , , , and . The expert classifies the and
as good teacher, and as ordinary teachers and , as bad
teacher.
The technique randomly selects three elements of
, and when the expert was asked for one variable
that distinguishes between these elements, she/he replied that
“teacher preparation” can be useful for this purpose. Next,
the technique gives support based on examples to the expert
for selecting the type of this variable, the expert states that the
type of this variable is ranking. Moreover, s/he states that this
variable takes its values from a scale of 1 to 5. The minimum
value of the scale means that the teacher is well prepared, that
is s/he has an in-depth knowledge of content, and the maximum
value means that the teacher is not well prepared. In Fig. 11, we
show the values that the expert introduces to value each teacher
belonging to . The distinctions matrix is shown in Fig. 12.
The analysis of the fuzzy repertory table shown in Fig. 11
discovers the following maximal equivalence subgroup
. The algorithm asked the expert to distinguish
. The expert answered that
between the teachers
good teachers explain by applying effective classroom practices
while the ordinary and bad teachers do not apply classroom
practices. The variable apply classroom practices is a continuous fuzzy variable, which takes the values shown in Fig. 13.
134
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 1, FEBRUARY 2004
Fig. 14. Fuzzy repertory table after introducing the second variable.
Fig. 15.
Distinctions matrix after introducing the variable C (apply classroom practices).
Fig. 16. Fuzzy repertory table after introducing the third variable.
Fig. 17.
Distinctions matrix after introducing the variable C (skill at teaching).
The apply classroom practices variable indicates the number
of minutes in a class hour devoted to this purpose. When the
expert introduces a fuzzy value (by means of a linguistic term),
she/he set out its core, that is the values which clearly belong
to the linguistic term, the interval [ , ]. When all the values
have been introduced, the values and of each fuzzy value
are determined on the basis of the values [ , ] of its right and
left neighbor values. For example, the expert affirms that the
teacher applies a lot of classroom practices, and she/he set
out its core to [17,20]. After, the expert estimate will the value
of the teacher , the parameters and of this value will be
determined from the a few and too much values to 13 and 25
respectively. The value too much is suggested to be introduced
by the FRT assistant for completing the domain.
In Fig. 14, we show the fuzzy repertory table after introducing
the apply classroom practices variable. In Fig. 15, we show the
distinctions matrix.
The analysis of the fuzzy repertory table shown in Fig. 14
discovers the following maximal equivalence subgroup
. The algorithm asks the expert to distinguish
between the teachers
, s/he answers that the skill at
teaching is the variable that allows them to be distinguished
between. The type of this variable is ranking and takes its
values from a scale of 1–5. The minimum scale value means
that the teacher is not skilful, and the maximum value means
that the teacher is skilful. In Fig. 16, we show the values that
are introduced by the expert to assess each teacher belonging
to . The distinctions matrix is shown in Fig. 17.
CASTRO-SCHEZ et al.: FUZZY REPERTORY TABLE
Fig. 18.
135
Fuzzy repertory table after introducing the fourth variable.
Fig. 19. Distinctions matrix after introducing the variable C (plan productive lessons).
Fig. 20.
Distinctions matrix associated to the problem.
Analysis of the fuzzy repertory table shown in Fig. 16 dis.
covers the following maximal equivalence subgroup
The algorithm asks the expert to distinguish between these elements. S/he answers that the ordinary teacher plans productive lessons whereas the bad teachers and does not plan
them. The expert affirms that the type of this variable is Boolean.
In Fig. 18, we show the repertory table after introducing the plan
productive lessons variable. In Fig. 19, we show the distinctions
matrix.
The pairs ( , ) and ( , ) can be shown to the expert in a
successive step (there is not triple of elements). If the expert is
not able to distinguish between them, then these teachers are
denoted as nondistinguishable (Fig. 20).
Once we have set up the repertory table and we have checked
that is not necessary to apply the Algorithm 3 (see Table I), we
obtain some rules that model how the expert classifies (Algorithm 4) in this specific case. The rules obtained are as follows.
: If his/her preparation is 1 and his/her skill at
•
teaching is 1 and plans productive lessons then she/he is a
Good teacher.
: If his/her skill at teaching is 1 and she/he does not plan
•
productive lessons then she/he is an Ordinary teacher.
: If his/her preparation is 4 and s/he plans productive
•
lessons then s/he is an Ordinary teacher.
: If his/her skill at teaching is 5 and she/he does
•
not plan productive lessons then she/he is a Bad teacher.
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 1, FEBRUARY 2004
TABLE I
INPUT VARIABLES INFORMATION ACQUIRED MAKING USE OF THE FUZZY REPERTORY TABLE
Next, we ask to the appraiser a set of properties valued with
(Table III), and we apply the maregard to the set of variables
chine learning algorithm suggested in [9]. This algorithm yields
the following set of rules.
•
•
Fig. 21.
Definition of the output variable, price (Euro).
•
When a variable does not appear in the rule it means that
the variable can represent any value. These rules have the same
structure as those presented in [9]. The rules generated should
have a maximal structure in the sense of both having fewer components in the antecedent part of the rule and containing knowledge that is more general.
and their definition domains
The input variables
obtained by the FRT can be used to assess other sets
of elements. In this way, we will obtain a set of examples that
can be used by the machine learning algorithm proposed in [9]
to acquire a set of linguistic rules. The examples will have the
with
being
following structure:
takes and
being the
the value that the input variable
classification of the system.
•
•
•
•
•
B. Second Example, Integration With a Machine Learning
Algorithm
•
In the following example, we show how could be the integration among the FRT and a machine learning algorithm. We applied FRT to know what are the variables and their values that an
appraiser agent working in a property valuation company uses
when fixes the price of the property (Low price, Medium price
and High price; see Fig. 21). Next, we ask for a set of element
valued according with this set of variables. Then, we apply a
machine learning algorithm [9] to how he carries out his work.
The elements of this FRT are a set of properties , , , ,
and which are selected randomly from a set of already valued
properties. In the process of construction of the FRT, the appraiser describes and analyzes the differences and similarities
among these five properties (Fig. 22). The set of variables
and their definitions domains
obtained by means of the
FRT is shown in Table II.
and
were given as fuzzy variables and
The variables
they were described by a set of linguistic terms. The variable
is a ordinal variable, the appraiser attaches number to each
location according to its preferences. The membership function
associated to each value is obtained from direct interaction with
the appraiser during the FRT performance.
•
•
•
•
: If its age is Normal or Antique and is located in the
Outskirts, then the price is Low.
: If its surface is Low or Medium and the age is Normal
or Antique and is located in the city Center, then price is
Low.
: If its surface is Low and the age is Newly built and is
located in the Outskirts, then the price is Low.
: If its surface is Low and the age is Normal or Antique
and is located in the Residential area, then the price is Low.
: If its surface is Medium or High and the age is Newlybuilt and is located in city Center, then price is Low.
: If its surface is Medium and the age is Antique and is
located in the Outskirts or Residential areas, then the price
is Medium.
: If its surface is Medium and the age is Newly built or
Normal and is located in the Outskirts, then the price is
Medium.
: If its surface is Low and the age is Antique or Newly
built and is located in the city Center, then the price is
Medium.
: If its surface is Medium and the age is Normal and is
located in the Outskirts or Residential areas, then the price
is High.
: If its age is Newly-built or Normal and is located in
the Residential area, then the price is High.
: If its surface is High and the age is Antique, then the
price is High.
: If its surface is High and the age is Newly built or
Normal and is located in the Outskirts, then the price is
High.
: If its surface is Medium or High and the age is Newly
built or Normal and is located in the city Center, then the
price is High.
C. Other Applications of Fuzzy Repertory Table
The range of application of the repertory grid technique is
limited only by the imagination of those who see value in the
approach [12] and are more and more people. In principle, the
range of application of FRT could be so broad, since it is an
extension of the repertory grid technique. Although its applications are still growing, the FRT is being currently applied.
CASTRO-SCHEZ et al.: FUZZY REPERTORY TABLE
Fig. 22.
137
Fuzzy repertory table in the property valuation company.
TABLE II
INFORMATION ACQUIRED MAKING USE OF THE FRT SHOWN IN FIG. 22
TABLE III
SET OF EXAMPLES USED BY THE INDUCTIVE MACHINE LEARNING [9]
• In bilateral, multi-issue negotiation, for acquiring domain
knowledge of seller and buyer agents. When we are
building models for automated negotiation based on
autonomous agents, there are many judgments and preferences that we need to capture from sellers and buyers
that can not be expressed in terms of numerical ratings
or rankings. For acquiring it, a method that involves the
construction of three fuzzy repertory tables and their
associated distinctions matrixes is being developed.
• In decision support systems, for identifying the significant
characteristics associated with each option and helping to
the decision-maker to choose the most advantageous option according to this information and his preferences. It
has been proved by the students when they are searching
for accommodation to rent and an estate agency sends
them a list containing different accommodations and data
about each of them.
VI. CONCLUSION
In this paper, we have developed a method which will help
to determine the variables and values used by the expert when
s/he faces classification problems. In the fuzzy repertory table
construction process, an interview is carried out by the expert
and consequently, new knowledge may be gained. The obtained
knowledge could be expressed by means of rules that can be
understood by the expert and/or used for establishing the input
variables and their definition domains of an inductive algorithms
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 1, FEBRUARY 2004
for learning a set of rules. This set of rules can therefore be used
to acquire more knowledge.
The major contributions of the fuzzy repertory table are as
follows.
• Obtaining a set of attributes and values that can be used
by machine learning techniques to obtain knowledge expressed in a linguistic form.
• Working with vagueness and crisp values.
• Being general, i.e. it can be used in several domains.
• Allowing none, several or all values to be introduced to
value an element according to one attribute.
• Being relatively easy to use, since the expert can express
his/her knowledge as she/he usually uses it, with a little
help from an assistant of the FRT.
• Performing a uniform analysis of the vagueness and crisp
values. This analysis uses the separation measurement that
is defined to measure the conceptual closeness between
the elements that we want to learn to classify. It is transparent, so that the expert can understand what is going on.
• Obtaining a set of linguistic rules formed by concepts, attributes and values belonging to the expert’s conceptual
structure that relates input variables with output elements.
The major difficulties of the fuzzy repertory table are that the
results obtained are very dependent on the expert’s discrimination and formalization abilities, and it is very dependent on the
set of elements and its cardinality. The elements help to control
the process of attributes elicitation. The FRT has been proved to
be more efficient the higher is the number of elements, in practice the interview uses more than three, usually eight, nine or
even twelve. Obviously, this give many more attributes the opportunity of appearing.
These difficulties are general to all repertory grids which
there are at present. The fuzzy repertory table does, however,
solve some of the limitations of the current repertory grids. Let
us look at these.
• The classic or Kelly’s repertory grid and the extendedclassic repertory grid is extended by the fuzzy repertory
table since it allows vagueness and crisp values. The expert
can value quantitative attributes by using fuzzy values.
• Bathia’s approach does not allow linguistic information;
however, the idea of a range of ratings rather than a crisp
value to value a construct is interesting. This is why the
fuzzy repertory table assimilates this idea.
• Gaines’ approach takes into account the fuzzy logic when
analyzing the repertory grid but it does not explicitly hold
information linguistic, whereas the fuzzy repertory table
does.
• Hwang’s fuzzy table is suitable for treating fuzzy logic instead of using fuzzy logic with grids. This approach only
considers fuzzy values and this does not seem to be appropriate for our purpose of allowing the expert to express
him/herself in total freedom. The fuzzy repertory table allows the expert to express himself as s/he usually does. It
is suitable for uniformly manipulating the most frequent
features or input variables types (vagueness and concrete)
that are used by the expert.
The grid analysis methods are also based on distance measurements between vectors (either rows or columns in the grid)
to obtaining similarities and differences. The expert compares
analysis results in order to obtain knowledge of the problem.
These methods do not find a one-sided implication relationship
and they cannot deduce that one construct links an element. In
response to this limitation, some researchers have developed
procedures to logically analyze the connection between constructs. But the problem with these procedures is they only work
with ranking values.
In Hwang’s fuzzy table, a rule is established for each element.
The consequent part of the rule will be formed for the element
that classifies this rule. All constructs or attributes that have been
introduced in the fuzzy table construction process will form the
antecedent part of that rule. We have developed a method that
carries out a hierarchical cluster analysis but obtains a set of
one-sided implication relationships. The proposed method also
establishes one rule per element but only introduces significant
constructs for the element of the consequent part into the antecedent part of the rule.
The fuzzy repertory table suggested generates linguistic rules
which is why they can be used to acquire further knowledge.
Acquisition of new data should alternate with an analysis of
previously obtained data. The process of knowledge acquisition should be pursued incrementally. We completely agree with
Tecuci [28] and Castro [8] who show the knowledge acquisition
process to be a process of incremental extension, updating, and
improvement of an incomplete and possibly partially incorrect
knowledge base.
We suggest that the knowledge acquired from the fuzzy repertory table could be the starting point in the knowledge acquisition process. The acquired knowledge may be used to design an
interview with an expert who will extend, update and improve
that knowledge by answering interview questions.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers for
their insightful comments that have significantly improved the
quality of this paper.
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Jose J. Castro-Schez received the M.S. and Ph.D. degrees in computer science
from the University of Granada, Granada, Spain, in 1995 and 2001, respectively.
He is an Associate Professor of Computer Science at the University of
Castilla-La Mancha, Ciudad Real, Spain. His research interests include:
knowledge acquisition, machine learning, decision support, and issues of
representation in AI. He is the author of numerous papers on AI-related
subjects.
Juan Luis Castro received the M.S. and Ph.D. degrees, both in mathematics,
from the University of Granada, Spain, in 1988 and 1991, respectively. His dissertation was in logical models for artificial intelligence.
He is currently a Research Professor in the Department of Computer Science
at the University of Granada and is a Member of the Group of Intelligent Systems within this Department. He has published approximately 40 journal and 50
congress papers, and is the author of three books on computer science. His research interests include neural networks, fuzzy systems, machine learning, and
related applications.
Dr. Castro is Co-Editor-in-Chief of the journal Mathware and Soft-Computing, and serves as a reviewer for some journals and international conferences.
He is a Member of the Editorial Board of the European Society on Fuzzy Logic
and Technology (EUSFLAT), and he coordinates the Neuro-Fuzzy Working
Group of this Society.
José Manuel Zurita received the M.Sc. and Ph.D. degrees in computer science,
both from the University of Granada, Granada, Spain, in 1991 and 1994, respectively.
He is currently an Associate Professor in the Department of Computer Science and Artificial Intelligence at the University of Granada. His current main
research interests are in the fields of fuzzy and linguistic modeling, fuzzy rulesbased systems, knowledge based systems, knowledge acquisition, and fuzzy expert systems.
Dr. Zurita is a Member of the Group of Intelligent Systems with this Department and a Member of the European Society of Fuzzy Logic and Technology
(EUSFLAT).