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Linguistic modeling with weighted double-consequent fuzzy rules based on cooperative coevolutionary learning

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This paper presents an evolutionary learning process for linguistic modeling with weighted double-consequent fuzzy rules. These kinds of fuzzy rules are used to improve the linguistic modeling, with the aim of introducing a trade-off between interpretability and precision. The use of weighted double-consequent fuzzy rules makes more complex the modeling and learning process, increasing the solution search space. Therefore, the cooperative coevolution, an advanced evolutionary technique proposed to solve decomposable complex problems, is considered to learn these kinds of rules. The proposal has been tested with different problems achieving good results.
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Linguistic Modeling with Weighted
Double-Consequent Fuzzy Rules Based on
Cooperative Coevolutionary Learning
Rafael Alcal´a1, Jorge Casillas2, Oscar Cord´on2, Francisco Herrera2
1Dept. of Computer Science
University of Ja´en, E-23071 Ja´en, Spain
e-mail: alcala@ujaen.es
2Dept. of Computer Science and Artificial Intelligence
University of Granada, E-18071 Granada, Spain
e-mail: {casillas,ocordon,herrera}@decsai.ugr.es
Abstract
This paper presents an evolutionary learning process for linguistic modeling with weighted
double-consequent fuzzy rules. These kinds of fuzzy rules are used to improve the linguistic
modeling, with the aim of introducing a trade-off between interpretability and precision.
The use of weighted double-consequent fuzzy rules makes more complex the modeling and
learning process, increasing the solution search space. Therefore, the cooperative coevolution,
an advanced evolutionary technique proposed to solve decomposable complex problems, is
considered to learn these kinds of rules. The proposal has been tested with different problems
achieving good results.
Keywords: Fuzzy linguistic modeling, double-consequent fuzzy rules, weighted fuzzy rules,
genetic algorithms, cooperative coevolution.
Supported by the Spanish CICYT, project PB98-1319.
R. Alcal´a, J. Casillas, O. Cord´on, F. Herrera. Linguistic modeling with weighted double-consequent fuzzy rules
based on cooperative coevolutionary learning. Integrated Computer Aided Engineering 10 (4), 2003, 343-355.
1
1 Introduction
One of the problems associated with Linguistic Modeling (LM) is its lack of accuracy when modeling
some complex systems. It is due to the inflexibility of the concept of linguistic variable, which
imposes hard restrictions to the fuzzy rule structure [1]. A way to improve the LM accuracy losing
interpretability but not to a high degree is to extend the usual linguistic model structure to be
more flexible.
Two specific possibilities for modifying the rule structure have been considered in the literature:
double-consequent fuzzy rules [5, 15], where each combination of antecedents may have two
consequents,
and weighted fuzzy rules [4, 16, 25], where an importance degree is considered for each rule
in the fuzzy reasoning process.
In the same way, even more flexible fuzzy rules may be obtained combining both approaches to
design linguistic models based on weighted double-consequent fuzzy rules, thus involving a potential
improvement of the accuracy and maintaining an acceptable description level. This task could be
performed by Genetic Fuzzy Systems (GFSs) [6], usually based on Genetic Algorithms (GAs) [14].
However, the use of weights and double-consequents makes more complex the modeling process
as it increases the solution search space since new parameters are considered in addition to the
traditional approach.
Recently, an advanced evolutionary technique, cooperative coevolution [17, 20], has arised to
solve problems with a large search space by independently evolving two or more species which
together comprise solution structures. We will use this novel technique to generate linguistic
models with the said extended structure by using a preliminary simple linguistic model with a
large number of simple and double-consequent rules and coevolving two species, the subset of rules
best cooperating and the weights associated to them.
Notice that, this contribution proposes the use of weighted double-consequent fuzzy rules to
improve simple linguistic fuzzy models by means of a cooperative coevolutionary algorithm. This
can be intended as a meta-method over any other linguistic rule generation method, developed to
improve simple linguistic fuzzy models by only reinforcing the modeling of those problem subspaces
with more difficulties while the use of rule weights improves the way in which they interact. De-
pending on the combination of this technique with different inductive methods, different learning
approaches arise. In this work, we will consider the Wang and Mendel’s method [23] (WM) for
this purpose.
The paper is organized as follows. In Section 2, the said specific ways to relax the linguistic
model structure are presented in depth. Section 3 briefly introduces the concepts of GAs, GFSs
and cooperative coevolution. In Section 4, the weighted double-consequent fuzzy rule structure
together with the coevolutionary genetic learning method to derive these kinds of rules are pro-
posed. Experimental results are shown in Section 5, whilst some concluding remarks are pointed
out in Section 6. Appendix A presents a table with the used acronyms.
2 Flexibilizing the Fuzzy Linguistic Model
Nowadays, system modeling is one of the most important applications in the framework of the
Fuzzy Rule-Based Systems (FRBSs) [12, 13, 18, 26]. It may be considered as an approach used to
model a system making use of a descriptive language based on fuzzy logic with fuzzy predicates
[22]. In this kind of modeling the accuracy and the interpretability of the obtained model are
contradictory properties directly depending on the learning process and/or the model structure.
Traditionally, according to the rule structure it is possible to distinguish between two modeling
approaches clearly opposed: LM and Fuzzy Modeling, with the interpretability and the accuracy
of the model being their main requirement, respectively.
2
LM is developed by means of linguistic FRBSs, typically called Mamdani-type FRBSs [12, 13],
which are composed of input and output linguistic variables [27] taking values from a linguistic
term set with a real-world meaning. Therefore, each rule may be clearly interpreted by human
beings.
Improvements in the LM can be accomplished to make more flexible the learning and/or the
model structure [3]. Two specific possibilities to relax the model structure are the following:
Use of double-consequent fuzzy rules, which involves allowing the model to present rules where
each combination of antecedents may have two consequents associated when it improves the
model accuracy [5, 15].
Consideration of weighted fuzzy rules in which modifying the linguistic model structure an
importance factor (weight) is considered for each rule [4, 16, 25]. By means of this technique,
the way in which these rules interact with their neighbor ones could be indicated.
It is clear that both possibilities will improve the capability of the model to perform the inter-
polative reasoning and, thus, its performance. This is one of the most interesting features of FRBSs
and plays a key role in their high performance, being a consequence of the cooperative action of
the linguistic rules existing in the fuzzy rule base.
On the other hand, simply new implicit granularity levels are added with these kinds or rules.
Therefore, weighted double-consequent fuzzy models are less interpretable than the classical lin-
guistic ones but, in any case, these kinds of FRBSs can be interpreted to a high degree, and also
make use of human knowledge and a deductive process. Moreover, notice that this paper will
propose a linguistic fuzzy modeling technique that only consider the use of double-consequents
when it is really needed to improve the accuracy of the system.
In the following subsections, these two approaches to relax the model structure will be analyzed.
2.1 Double-Consequent Fuzzy Linguistic Rules
More flexible linguistic models may be obtained by allowing them to present fuzzy rules where each
combination of antecedents may have two consequents (linguistic terms of the output variable)
associated. The consideration of these kinds of rules may be intended as a local reinforcement of
the problem space zones presenting high complexity. Therefore, as shown in [5, 15], considering
some rules with multiple consequents could improve the global system behavior. The rule structure
so obtained is:
IF X1is A1and . . . and Xnis AnTHEN Yis {B1, B2},
with Xi(Y) being the linguistic input (output) variables, Aibeing the linguistic label used in the
i-th input variable, and B1and B2the two linguistic terms associated to the output variable.
The use of two consequents has no influence on the linguistic model inference system. Since each
double-consequent fuzzy rule can be decomposed into two simple rules with the same antecedent
and different consequent, the usual plain fuzzy inference system can be applied [5]. The only
restriction imposed is to use the FITA (First Infer, Then Aggregate) scheme [7] and considering
the matching degree of the rules fired. In other case, by using the FATI (First Aggregate, Then
Infer) scheme [7] or a deffuzzification strategy not considering the matching, the influence of one
of the two rule consequents will be canceled. For example, the center of gravity weighted by the
matching degree defuzzification strategy [7] may be used:
Pi=RVY·µB0
i(Y)·dY
RVµB0
i(Y)·dY ,
y0=Pimi·Pi
Pimi
,
3
with y0being the crisp value obtained from the defuzzification process, mibeing the matching
degree of the i-th rule, Vbeing the universe of discourse of the output variable Y, and Pibeing
the characteristic value center of gravity of the output fuzzy set inferred from the i-th rule,
B0
i.
We should note these kinds of rules do not constitute an inconsistency from the LM point of
view but only a shift of the main labels making the final output of the rule lie in an intermediate
zone between both consequents. In this case, simply new implicit granularity levels are added with
these kinds or rules (similar to the use of linguistic modifiers where the interpretability is slightly
lost but being interpretable to an acceptable degree). Indeed, the said double-consequent fuzzy
rule structure may be interpreted as follows [5]:
IF X1is A1and . . . and Xnis AnTHEN Yis between B1and B2.
whose output is exactly the middle point between these two consequents,
y0=mi·PB1+mi·PB2
mi+mi
=PB1+PB2
2.
Of course, notice that when these double-consequent rules interact with their neighbor ones, the
output is not the middle point between both consequents but it is shifted according to the firing
strengths of those neighbor rules.
An example of learning process
The consideration of this structure to generate advanced linguistic models was initially proposed
in [15]. Another approach, according to the Accurate Linguistic Modeling (ALM) methodology, is
introduced in [5]. In this work, the use of double-consequent is considered when it is really needed
to improve the accuracy of the system. This methodology consists of two steps:
1. Firstly, two rules, the primary and secondary in importance, are obtained in each fuzzy in-
put subspace considering a specific generation process. In this contribution, the generation
process proposed by Wang and Mendel [23] is considered. Thus, the process involves divid-
ing the input and output spaces into fuzzy regions, generating the rule best covering each
example, and finally selecting the two rules with the highest covering degree for each fuzzy
input subspace (if there is more than a single rule on it).
2. Then, after decomposing each double-consequent rule into two independent simple ones 1, the
selection process proposed in [10] is employed to select the subset of rules best cooperating.
It is based on a binary-coded GA where each gene indicates if the corresponding rule is
considered or not to belong to the final fuzzy rule base.
2.2 Weighted Fuzzy Linguistic Rules
Using rule weights [4, 16, 25] has been usually considered to improve the way in which the rules
interacts, improving the accuracy of the learned model. In this way, rule weights suppose an
effective extension of the conventional fuzzy reasoning system that allow the tuning of the system
to be developed at the rule level [4, 16].
When weights are applied to complete rules, the corresponding weight is used to modulate the
firing strength of a rule in the process of computing the defuzzified value. From human beings, it is
very near to consider this weight as an importance degree associated to the rule, determining how
this rule interacts with its neighbor ones. We will follow this approach, since the interpretability
of the system is appropriately maintained. In addition, we will only consider weight values in [0,1]
1The preliminary rule base is derived to simple rules only to be considered in the selection process. If one of
the two simple rules obtained from decomposing a double-consequent rule is removed by the selection process, this
fuzzy input subspace will have just a single consequent associated.
4
since it preserves the model readability. In this way, the use of rule weights represents an ideal
framework for extended LM when we search for a trade-off between accuracy and interpretability.
In order to do so, we will follow the weighted rule structure and the inference system proposed
in [16]:
IF X1is A1and . . . and Xnis AnTHEN Yis Bwith [w],
where wis the real-valued rule weight, and with is the operator modeling the weighting of a rule.
With this structure, the fuzzy reasoning must be extended. The classical approach is to infer
with the FITA (First Infer, Then Aggregate) scheme [7] and to compute the defuzzified output as
the following weighted sum:
y0=Pimi·wi·Pi
Pimi·wi
,
with mibeing the matching degree of the i-th rule, wibeing the weight associated to it, and Pibeing
the characteristic value of the output fuzzy set corresponding to that rule. In this contribution,
the center of gravity will be considered as characteristic value [7] (see the previous subsection).
An example of learning process
A simple approximation for weighted rule learning would consist of the following two steps —we
will use this process in our experiments for comparison purposes, calling it WRL—:
1. Firstly, a preliminary fuzzy rule set is derived considering a specific generation process. In
this work, the generation process proposed by Wang and Mendel [23] is considered.
2. Then, a learning algorithm is used to derive the associated weights of the previously obtained
rules. A real-coded GA where each gene indicates the corresponding rule weight may be
considered as learning algorithm.
3 GAs, GFSs and Coevolution
Considering both approaches —weighted and double-consequent rules— together, makes more
complex the modeling process thus increasing the solution search space.
GFSs have been successfully applied to learn fuzzy systems in the last years. They have been
usually based on GAs although other evolutionary algorithms have been also considered. On
the other hand, more sophisticated evolutionary approaches, as cooperative coevolution [17, 20],
could be considered to solve this complex modeling process. These concepts are introduced in this
section.
3.1 Genetic Algorithms
GAs are general-purpose global search algorithms that use principles inspired by natural population
genetics to evolve solutions to problems. The basic principles of the GAs were first laid down
rigorously by Holland [11] and are well described in many texts as [14].
The basic idea is to maintain a population of knowledge structures that evolves over time
through a process of competition and controlled variation. Each structure in the population repre-
sents a candidate solution to the specific problem and has an associated fitness to determine which
structures are used to form new ones in the process of competition.
In this way, a subset of relatively good solutions are selected for reproduction to give offspring
that replace the relatively bad solutions which die. Usually, offspring replace their parents for the
next generation (generational approach). These new individuals are created by using genetic oper-
ators such as crossover and mutation. The crossover operator combines the information contained
into the parents increasing the average quality of the population (exploitation), while the muta-
tion operator randomly changes the new individuals helping the algorithm to avoid local optima
(exploration).
5
3.2 Genetic Fuzzy Systems
During the 90s, a large amount of work has been devoted to add learning capabilities to FRBSs.
The automatic design of FRBSs can be considered in many cases as an optimization or search
process on the space of potential solutions. Since GAs are well known and widely used global
search techniques, a large number of publications explored the use of GAs for designing FRBSs,
thus obtaining the so-called GFSs [6]. Figure 1 depict this idea.
Output Interface
Fuzzy Rule-
Based System
Input Interface
Evolutionary algorithm
based learning process
Fuzzy rule base
Computation with fuzzy systemsEnvironment Environment
DESIGN PROCESS
Figure 1: Genetic fuzzy systems.
Nowadays, as the field of GFSs matures and grows in visibility, there is an increasing concern
about the integration of these two topics from a novel more sophisticated perspective. Indeed, as
David Goldberg stated in [9], the integration of single methods into hybrid intelligent systems goes
beyond simple combinations. For him, the future of Computational Intelligence lies in the careful
integration of the best constituent technologies and subtle integration of the abstraction power of
fuzzy systems and the innovating power of genetic systems requires a design sophistication that
goes further than putting everything together. This is the case of our contribution, where we use
a cooperative coevolutionary model for learning the weighted double-consequent fuzzy rules.
3.3 Cooperative Coevolutionary Algorithms
Coevolutionary algorithms [17] are advanced evolutionary techniques proposed to solve decompos-
able complex problems. They involve two or more species (populations) that permanently interact
among them by a coupled fitness. Thereby, in spite of each species has its own coding scheme and
reproduction operators, when an individual must be evaluated, its goodness is calculated consid-
ering some individuals of the other species. This coevolution makes easier to find good solutions
to complex problems.
Different kinds of interactions may be considered among the species according to the depen-
dencies existing among the solution subcomponents. Generally, we can distinguish two different
kinds of interaction:
Competitive coevolutionary algorithms [21]: Those where each species competes with the
remainder. In this case, increasing the fitness of an individual in a species implies decreasing
the fitness of the ones in the other species, i.e., the success of somebody else entails the
personal failure.
6
Cooperative or symbiotic coevolutionary algorithms [20]: Those where all the species cooperate
to build the problem solution. In this case, the fitness of an individual depends on its ability
to cooperate with individuals from other species.
The use of cooperative coevolutionary algorithms is recommendable when the following issues
arise [19]:
1. the search space is huge,
2. the problem may be decomposable in subcomponents,
3. different coding schemes are used, and
4. there are strong interdependencies among the subcomponents.
In Figure 2, the evolutionary process for Sspecies of a cooperative coevolutionary system is
illustrated. Each individual being evaluated could be combined with one or more cooperators of
the other species.
Cooperators
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Evolutionary
Algorithm
Population
Species 1
1
1
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Individual
to be
evaluated
Fitness .
.
.
1
Cooperators
. . .
Evolutionary
Algorithm
Population
Species 2
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. . .
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S
Individual
to be
evaluated
Fitness
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2
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12
Problem solution
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2. . . S
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1
Cooperators
. . .
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S
Evolutionary
Algorithm
Population
Species S
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Individual
to be
evaluated
Fitness .
.
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SSS
S
S
S
S
S
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1
Problem solution
. . .
. . .
Figure 2: Cooperative coevolutionary system for Sspecies.
This sophisticated technique can be used within the field of GFSs. Indeed, in [19] this technique
has been already applied to learn FRBSs coevolving two species, the membership functions and
the fuzzy rules.
4 Genetic Fuzzy Systems to Generate Weighted Double-
Consequent Fuzzy Rules
In this section the structure of the proposed weighted double-consequent fuzzy rule as well as a
cooperative coevolutionary-based GFS to learn these kinds of linguistic models are explained in
detail. Finally, in order to check the behavior of the coevolutionary approach a genetic model
based on a standard GA is presented as a first approximation to learn these kinds of models.
4.1 Rule Structure and Learning Process
To improve the linguistic model accuracy, the use of a more flexible linguistic model structure that
combine the two said approaches is proposed. Thus, the weighted double-consequent fuzzy rules
present the following structure:
7
IF X1is A1and . . . and Xnis AnTHEN Yis {B1with [w1], B2with [w2]},
with w1and w2being the weights associated to the consequents B1and B2, respectively. Therefore,
a weighted double-consequent fuzzy rule can be seen as two weighted single-consequent fuzzy rules
with the same antecedent and different consequents (and so, still considering two consequents in the
corresponding subspace). Thus, the fuzzy reasoning must be extended as in the case of weighted
fuzzy rules, considering the matching degree of the rules fired (see Sections 2.1 and 2.2).
These kinds of rules could be interpreted by adding the characteristics of double-consequent
and weighted fuzzy rules. Therefore, when double consequent fuzzy rules are obtained, the output
can be interpreted as a shift of the main labels making the final output of the rule lie in an
intermediate zone between both consequents. However, in this case the way in which these rules
interact is known since the correspondent weights can be interpreted as their importance degree
(see Section 2.2).
L1 L2 L3 L4 L5 L6 L7
mM
0.5
Figure 3: Graphical representation of a possible fuzzy partition.
To generate linguistic models with this new structure, we may follow an operation mode similar
to the ALM methodology [5] introduced in Section 2.1, but including the weight learning. To do
that, we will consider symmetrical fuzzy partitions of triangular-shaped membership functions (see
Figure 3). Therefore, after performing the first step of the ALM methodology, where an initial
set of numerous double-consequent rules is generated, and decomposing them to simple ones (see
Section 2.1), the two following tasks must be performed:
Genetic selection of a subset of rules presenting good cooperation.
Genetic derivation of the weights associated to these rules.
These interdependent tasks significantly increase the search space with respect to the original
methodology making the choice of the considered search technique crucial.
4.2 Evolutionary Learning of Weighted Double-Consequent Fuzzy Rules
with a Cooperative Coevolutionary Model
As we have seen, the problem that concern us can be easily decomposed into two subtasks, the
rule selection and the weight derivation. Therefore, it can be solved by coevolving two species
cooperating to form the complete solution by learning a set of weighted fuzzy rules. In the following
subsections, the main characteristics of the proposed cooperative coevolutionary algorithm are
presented.
4.2.1 Interaction Scheme Between Species
The objective will be to minimize the well-known Mean Square Error (MSE):
MSEij =1
2·N
N
X
l=1
(Fij (xl)yl)2,
8
with Nbeing the number of training data, Fij (xl) being the output inferred from the model
obtained by combining the individuals iand jof the species 1 (rule selection) and 2 (weight
derivation) when the input xl= (xl
1, . . . , xl
n) is presented, and ylbeing the known desired output.
Thus, individuals in the species 1 and 2 are respectively evaluated with the fitness functions f1
and f2, defined as follows:
f1(i) = min
jR2P2
MSEij
f2(j) = min
iR1P1
MSEij ,
with iand jbeing individuals of species 1 and 2 respectively, R1and R2being the sets of the r
fittest individuals in the previous generation of the species 1 and 2 respectively, and P1and P2
being the sets of the pindividuals selected at random from the previous generation of the species
1 and 2 respectively. This evaluation process is graphically shown in Figure 4.
(# = p)
Species 1 Species 2
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Cooperators Cooperators
Population t-1
Population t-1
r = 2
p = 1
R
P
1
1
R
P
2
2
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Fittest individuals
Randomly selected
individuals
Population t
.
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.
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




fitness
Population t
.
.
.
.
.
.
(# = r)
min
MSE
min
fitness
MSE
Figure 4: Fitness evaluation for species 1 and 2.
Whilst the sets R1|2allow the best individuals to influence in the process guiding the search
towards good solutions, the sets P1|2introduce diversity in the search. The combined use of both
kinds of sets makes the algorithm have a trade-off between exploitation (R1|2) and exploration
(P1|2). The cardinalities of the sets R1|2and P1|2have to be previously defined by the designer.
Agenerational [14] scheme is followed in both species. Baker’s stochastic universal sampling
procedure [2] together with an elitist mechanism (that ensures to maintain the best individual of
the previous generation) are used.
The specific operators considered in every species are described in the following.
4.2.2 Species 1: Fuzzy rule selection
For the species 1, we will use the genetic rule selection method proposed in [5]. The coding scheme
generates binary-coded strings of length m(with mbeing the number of single-consequent fuzzy
rules in the previously derived rule set, obtained in the first step of ALM). Depending on whether
9
a rule is selected or not, the alleles ‘1’ or ‘0’ will be respectively assigned to the corresponding
gene. Thus, the p-th chromosome for the species 1, Cp
1, will be a binary vector representing the
subset of rules finally obtained.
The initial pool is generated at random except for the first individual, which represents the
complete previously obtained fuzzy rule set:
k {1, . . . , m}, C1
1[k] = 1.
For this species, the standard two-point crossover operator is used. The two-point crossover
involves interchanging the fragments of the parents contained between two points selected at ran-
dom. As regards the mutation operator, it flips the value of the gene.
4.2.3 Species 2: Weight derivation
The coding scheme generates real-coded strings of length m. The value of each gene indicates
the weight used in the corresponding rule. They may take any value in the interval [0,1]. Now,
the p-th chromosome for the species 2, Cp
2, will be a real-valued vector representing the weights
associated to the fuzzy rules considered.
The initial pool for this species is generated with the first chromosome having all the genes
with the value ‘1’, and the remaining individuals taking values randomly generated within the
variation interval [0,1]:
k {1, . . . , m}, C1
2[k] = 1.0.
The max-min-arithmetical crossover operator [10] is considered. Using the max-min-arithmetical
crossover, if Cv
2= (c1, . . . , ck, . . . , cn) and Cw
2= (c0
1, . . . , c0
k, . . . , c0
n) are going to be crossed, the
resulting descendents are the two best of the next four offspring:
C1
2=aCw
2+ (1 a)Cv
2,C2
2=aCv
2+ (1 a)Cw
2,
C3
2with c3k= min{ck, c0
k},C4
2with c4k= max{ck, c0
k},
with a[0,1] being a constant parameter chosen by the GA designer.
As regards the mutation operator, it simply involves changing the value of the selected gene
by other value obtained at random within the interval [0,1].
4.3 A Standard Genetic Algorithm
A standard GA performing the rule selection together with the derivation of weights was developed
as a first approximation to the problem. The classical generational [14] scheme together with the
Baker’s stochastic universal sampling procedure [2] and an elitist mechanism were considered in
this algorithm.
Coding scheme and initial gene pool
A double coding scheme (C=C1+C2) for both rule selection and weight derivation is considered:
The coding scheme for the C1part was introduced in Section 4.2 as the coding scheme for
species 1. Thus, the corresponding part Cp
1for the p-th chromosome will be a binary vector
representing the subset of rules finally obtained.
The coding scheme for the C2part was also introduced in Section 4.2 as the coding scheme
for species 2. Now, the corresponding part Cp
2for the p-th chromosome will be a real-valued
vector representing the weights associated to the fuzzy rules considered.
The initial pool is obtained with an individual having all genes with value ‘1’ in both parts,
and the remaining individuals generated at random.
10
Evaluating the chromosome
The fitness function considered is the MSE. With Fi(xl) being the model inferred output for the
i-th chromosome, this measure is represented by the following expression:
f(i) = MSEi=1
2·N
N
X
l=1
(Fi(xl)yl)2,
Genetic operators
The crossover operator will depend on the chromosome part where it is applied: in the C1
part, the standard two-point crossover is used, whilst in the C2part, the max-min-arithmetical
crossover [10] is considered. Both operators were explained in the previous section. In this case,
eight offspring are generated by combining the two ones from the C1part (two-point crossover)
with the four ones from the C2part (max-min-arithmetical crossover). The two best offspring so
obtained replace the two correspondent parents in the population.
As regards the mutation operator, it flips the gene value in the C1part while takes a value at
random within the interval [0,1] for the corresponding gene in the C2part. Both operators were
also explained in the previous section.
5 Experiments and Results
In this section, we will analyze the performance of the two GFSs presented in Sections 4.2 and 4.3,
the proposed cooperative coevolutionary algorithm (calling it as WALM-CC) and the standard
steady-state GA-based method (calling it as WALM), when solving two real-world electrical engi-
neering distribution problems [8]. They will be compared to the models designed by the following
methods: the well-known ad hoc data-driven method proposed by Wang and Mendel (calling it as
WM) [23], the original ALM method [5], and the WRL method presented in Section 2.2. Except
WM, all of them follow a generational [14] scheme.
With respect to the fuzzy reasoning method used, we have selected the minimum t-norm playing
the role of the implication and conjunctive operators, and the center of gravity weighted by the
matching strategy acting as the defuzzification operator [7].
Finally, the following values have been considered for the parameters of each method 2:
Rule selection step of ALM: 61 individuals, 1000 generations, 0.6 as crossover probability,
and 0.2 as mutation probability per chromosome.
WRL: 61 individuals, 1,000 generations, 0.6 as crossover probability, 0.2 as mutation proba-
bility per chromosome, and 0.35 for the factor ain the max-min-arithmetical crossover.
WALM: 61 individuals, 1,000 generations, 0.6 as crossover probability, 0.2 as mutation prob-
ability per chromosome, and 0.35 for the factor ain the crossover operator.
WALM-CC: 62 individuals (31 for each species), 1,000 generations, 0.6 and 0.2 for the
crossover and mutation probabilities in both species respectively, 0.35 for the factor ain
the crossover operator in the species 2, the three fittest individuals (|R1|2|= 3) and two
random individuals (|P1|2|=2) of each species are considered for the coupled fitness.
2With these values we have tried easy the comparisons selecting standard common parameters that work well
in most cases instead of searching very specific values for each method. Moreover, we have set a large number of
generations (1000) in order to allow the compared algorithms an appropriate convergence. No significant changes
were achieved by increasing that number of generations.
11
5.1 Estimating the Length of Low Voltage Lines
Sometimes, there is a need to measure the amount of electricity lines that an electric company owns.
This measurement may be useful for several aspects such as the estimation of the maintenance costs
of the network, which was the main goal in this application [8]. Since a direct measure is very
difficult to obtain in some cases 3, the consideration of models becomes useful. In this way, the
problem involves finding a model that relates the total length of low voltage line installed in a rural
town with the number of inhabitants in the town and the mean of the distances from the center of
the town to the three furthest clients in it. This model will be used to estimate the total length of
line being maintained.
To do so, a sample of 495 rural nuclei has been randomly divided into two subsets, the training
set with 396 elements and the test set with 99 elements, the 80% and the 20% respectively. Both
data sets considered are available at http://decsai.ugr.es/casillas/fmlib/. The linguistic partitions
considered are comprised by seven linguistic terms with triangular-shaped fuzzy sets giving meaning
to them (see Figure 3). The corresponding labels, {L1, L2, L3, L4, L5, L6, L7}, stand for extremely
small, very small, small, medium, large, very large, and extremely large respectively.
The results obtained by the five analyzed methods are presented in Table 1, where #Rstands
for the number of rules, and MSEtra and MSEtst for the error obtained over the training and test
data respectively. The best results are shown in boldface.
Table 1: Results obtained in the length of low voltage lines estimation problem
Method #R (SC+DC) MSEtra MSEtst
WM 24 222,654 239,962
ALM 17 (14+3) 155,898 178,534
WRL 24 191,577 221,583
WALM 17 (14+3) 145,124 186,704
WALM-CC 18 (14+4) 144,290 176,057
SC = Single Consequent, DC = Double Consequent.
In this case, WM —performing classical LM— presents the worst results obtaining more rules
than the remaining ones. ALM and WRL —performing improved LM— present significantly more
accurate models than WM in both, approximation and generalization, with improvements of about
30% and 25% respectively.
WALM does not achieve the desired results, only improving the result obtained from ALM
in the approximation capability. This fact evidences some lacks in the optimization technique
considered to learn weighted double-consequent fuzzy rules, since theoretically, better results than
the ones obtained by ALM should be obtained.
In this case, the model obtained by WALM-CC presents the best results with improvement
rates near of the 10% in generalization (MSEtst) with respect to the results obtained by WALM.
Therefore, the necessity of considering a sophisticated technique to solve this complex learning
problem seems to be interesting.
Figure 5 represents the decision table of the model obtained from WALM-CC. In the left
side of this figure, each cell of the table represents a fuzzy subspace and contains its associated
output consequent(s) —the primary (C1) and/or the secondary (C2) in importance—, i.e., the
correspondent label(s) together with its(their) respective rounded rule weight(s). The absolute
importance weight for each fuzzy rule has been graphically showed by means of the grey colour
scale, from black (1.0) to white (0.0). In this way, we can easily see the importance of a rule with
respect to their neighbor ones which could help the system experts to identify important rules. On
the other hand, in the right side of this figure, an expert interpretation of the relative importance
of the rules is presented as regards their influence in the modeling of the respective problem space
zone. Three kinds of rules are represented in the figure:
3Low voltage lines installation is often very intricate since they are contained in little villages and rural nuclei.
12
L1 L2 L3 L4 L5 L6 L7
x2
L1
L4
L3
L7
L6
L5
L2
C1
C2
#R
22
x1
L1 L2
L2 L5
L4
L4
L3






















L1 L2 L3 L4 L5 L6 L7
x2
L1
L4
L3
L7
L6
L5
L2
C1
C2
#R
22
x1
L1 - 0.8 L2 - 0.4 L2 - 0.7 L2 - 0.1
L1 - 0.6 L2 - 0.9
L3 - 0.1
L2 - 0.5
L4 - 0.5
L5 - 0.3
L3 - 0.2
L5 - 0.2
L2 - 0.3 L4 - 0.2 L3 - 0.1
L6 - 0.2 L7 - 0.2 L4 - 1.0
L4 - 0.1
L3 - 0.1
L1 - 0.5 L2 - 0.4
Significant rules
Cooperative rules
Complementary rules

Indirectly covered region



Figure 5: Decision table of the linguistic model obtained from WALM-CC for the length of low
voltage lines estimation problem.
Significant or important rules: Those in black, corresponding to rules that have a higher
weight than their neighbors or rules that are the ones of their regions.
Cooperative rules: Those in grey, representing rules that have a more or less similar weight
than their neighbor ones.
Complementary rules: Those in white (with waves), representing rules that have a lower
weight than their neighbor ones.
Due to the kind of fuzzy partition considered (see Figure 3), there are many input subspaces
which, in spite of having no rules associated, are covered by their neighbor rules, e.g., the input
subspace labeled as L3-L4. Two different zones can be clearly distinguished in the table. The first
one —located in the top-left corner— presents three significant rules with outputs about L2and
coincides with an important concentration of training examples. The second one composes a front
in the input space and presents significant rules with outputs about L4.
In this case, only four weighted double-consequent rules have been needed (locally improving the
model where it is necessary). Regarding the significant rules, we should notice that the importance
of the rules does not directly depend on the weights of the rules, but also on the weights of their
neighbors. Thus, rules as the one located in the input subspace labeled as L2-L6become significant
rules since their weights are higher than those of their neighbors.
5.2 Estimating the Maintenance Costs of Medium Voltage Lines
Estimating the maintenance costs of the medium voltage electrical network in a town [8] is an
interesting problem. Since the medium voltage lines existing in a town have been installed in-
crementally, a direct measure is very difficult to obtain and the consideration of models becomes
useful. These estimations allow electrical companies to justify their expenses. Moreover, the model
must be able to explain how a specific value is computed for a certain town. Our objective will be
to relate the maintenance costs of medium voltage line with the following four variables: sum of
the lengths of all streets in the town,total area of the town,area that is occupied by buildings, and
energy supply to the town. We will deal with estimations of minimum maintenance costs based on
a model of the optimal electrical network for a town in a sample of 1,059 towns.
To develop the different experiments, the sample has been randomly divided in two subsets,
the training and test ones, with an 80%-20% of the original size respectively. Thus, the training
set contains 847 elements, whilst the test one is composed by 212 elements. These data sets are
available at http://decsai.ugr.es/casillas/fmlib/. Five linguistic terms with triangular-
shaped fuzzy sets giving meaning to them are considered for each variable (see Figure 3). In these
13
Table 2: Results obtained in the maintenance costs of medium voltage lines estimation problem
Method #R (SC+DC) MSEtra MSEtst
WM 66 71,294 80,934
ALM 47 (44+3) 51,714 58,806
WRL 66 33,639 33,319
WALM 47 (43+4) 27,719 32,455
WALM-CC 54 (49+5) 24,961 28,225
SC = Single Consequent, DC = Double Consequent.
case, the corresponding labels, {L1, L2, L3, L4, L5}, stand for very small, small, medium, large, and
very large, respectively.
In view of the obtained results, we can see once again as the ALM and WRL methods —which
are respectively based on double-consequent fuzzy rules and weighted fuzzy rules— achieve signifi-
cant improvements over the WM method. The model obtained from ALM is the one comprised of
less fuzzy rules. However, all the methods except WM present better results than the one obtained
from ALM. Apart from ALM, the two proposed GFSs obtain the models with less number of rules.
Analyzing the model obtained by the WALM-CC method, we can conclude that it presents
the best performance in both approximation and generalization, with improvement rates of about
25% and 15% with respect to WRL, respectively. Moreover, WALM-CC presents improvement
rates of about a 10% in approximation and generalization with respect to WALM. This fact is a
consequence of the coevolutionary approach ability to tackle with decomposable complex problems.
#R: 54 (49 + 5 DC)
ECM-tra: 24962 / ECM-tst: 28226
X1 X2 X3 X4 Y with
L1 L1 L1 L1 L1 [0.838]
L1 L1 L1 L2 L2 [0.688]
L1 L2 L1 L1 L1 [0.386]
L1 L2 L2 L1 L2 [0.466]
L1 L2 L2 L2 L2 [0.355]
L2 L1 L1 L1 L1 [0.491]
L2 L1 L2 L2 L2 [0.573]
L2 L2 L1 L1 L1 [0.421]
L2 L2 L1 L2 L2 [0.388]
L2 L2 L2 L1 L1 [0.099]
L2 L2 L2 L2 L2,L3 [0.605,0.217]
L2 L3 L2 L1 L2 [0.803]
L2 L3 L2 L2 L2 [0.185]
L2 L3 L3 L2 L3 [0.945]
L3 L2 L1 L1 L1 [0.452]
L3 L2 L1 L2 L2 [0.994]
L3 L2 L1 L3 L2 [0.283]
L3 L2 L2 L3 L3 [0.322]
X1 X2 X3 X4 Y with
L3 L3 L2 L2 L2,L3 [0.684,0.108]
L3 L3 L2 L3 L3 [0.476]
L3 L3 L3 L2 L3 [0.773]
L3 L3 L3 L3 L4 [0.497]
L3 L4 L3 L2 L3 [0.256]
L3 L4 L3 L3 L3 [0.716]
L3 L4 L4 L3 L4 [0.847]
L4 L2 L2 L1 L2 [0.372]
L4 L2 L2 L2 L2,L3 [0.254,0.178]
L4 L2 L2 L3 L3,L2 [0.152,0.171]
L4 L2 L2 L4 L3 [0.267]
L4 L3 L2 L1 L2 [0.798]
L4 L3 L2 L3 L3 [0.366]
L4 L3 L2 L4 L3 [0.696]
L4 L3 L3 L2 L3 [0.200]
L4 L3 L3 L3 L4 [0.166]
L4 L3 L3 L4 L4 [0.198]
L4 L4 L3 L2 L3 [0.526]
X1 X2 X3 X4 Y with
L4 L4 L3 L3 L3 [0.257]
L4 L4 L3 L4 L4 [0.467]
L4 L4 L4 L2 L4 [0.776]
L4 L4 L4 L3 L4 [0.321]
L4 L4 L4 L4 L5 [0.786]
L4 L5 L4 L2 L3 [0.324]
L4 L5 L4 L3 L4 [0.907]
L4 L5 L4 L4 L5 [0.014]
L4 L5 L5 L2 L5 [0.434]
L4 L5 L5 L3 L5 [0.893]
L5 L2 L2 L2 L2,L3 [0.232,0.151]
L5 L2 L2 L4 L3 [0.044]
L5 L2 L2 L5 L4 [0.824]
L5 L2 L3 L2 L3 [0.985]
L5 L2 L3 L5 L4 [0.964]
L5 L4 L3 L2 L3 [0.170]
L5 L4 L3 L4 L4 [0.057]
L5 L4 L3 L5 L5 [0.399]
Figure 6: Rule set of the linguistic model obtained from WALM-CC for the maintenance costs of
medium voltage lines estimation problem.
Figure 6 represents the decision table of the model obtained from WALM-CC. In this case, each
row represents a fuzzy subspace and contains its associated output consequent(s) —the primary
and/or the secondary in importance—, i.e., the correspondent label(s) together with its(their)
respective rounded rule weight(s). Once again, the absolute importance weight for each fuzzy rule
has been graphically showed by means of the grey colour scale, from black (1.0) to white (0.0).
From the 625 (54) possible fuzzy rules, the obtained linguistic model is composed of only 54
fuzzy rules. In this case, it only contains five double-consequent rules. Notice that, all the double-
consequent rules are very near in the four-dimensional space, representing a zone with higher
complexity. Moreover, rules with weights near to 1 represents groups of important rules and do
not usually appear alone.
14
6 Concluding Remarks
In this paper, a new linguistic model structure using weighted double-consequent fuzzy rules has
been proposed with the aim of improving the performance of the so obtained models maintaining an
acceptable description level. Its main interest lies in flexibilyzing the model structure in a different
way from the usual one (e.g., learning the composition of the fuzzy membership functions).
A simple GA could be proposed as a first approximation to learn these kinds of improved
linguistic models, but the results obtained in a preliminary experimentation were not very sat-
isfactory. However, since the search space is too large and the problem easily decomposable, we
propose in his stead the use of a more sophisticated genetic model able to solve decomposable com-
plex problems, the cooperative coevolution. The very accurate results of the proposed coevolutive
learning method, compared with other related approaches, has been contrasted when solving two
electrical distribution problems.
Moreover, the improved linguistic models so obtained have presented a good description level.
In this way, models with no more than five double-consequent rules were obtained, only considering
the use of these kinds of rules when it was really needed. Moreover, significant rules have been
identified studying the weights of the rules, helping us to interpret the model behavior.
A Acronyms
Table 3 presents the list of acronyms considered in this paper.
Table 3: Acronyms considered in the paper
Acronym -Meaning Acronym -Meaning
ALM - Accurate Linguistic Modeling MSE - Mean Square Error
FRBSs - Fuzzy Rule Based Systems WALM - Weighted Accurate Linguistic Modeling algorithm
GAs - Genetic Algorithm WALM-CC - WALM algorithm with Cooperative Coevolution
GFSs - Genetic Fuzzy Systems WM - Wang and Mendel algorithm
LM - Linguistic Modeling WRL - Weighted Rule Learning
Acknowledgements
Oscar Cord´on would like to express his gratitude towards Professor Lotfi A. Zadeh for the hospi-
tality, support and inspiration he encountered during his stay with the Berkeley Initiative in Soft
Computing (BISC) at the University of California-Berkeley in 1999.
References
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of Uncertainty, Fuzziness and Knowlegde-Based Systems 2:4 (1994) 463–484.
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... of Q patterns each consisting of an Ndimensional predictor x and one response variable y , DECO 3 RUM models the input-output relationship using a total of R fuzzy rules with weighted double consequents, comprised of consecutive fuzzy sets ( Alcalá et al., 2003 ) of the form: ...
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