Evolving Fuzzy Classification Systems from Numerical Data by ijcsiseditor


									                                                    (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                        Vol. 9, No. 6, June 2011

      Evolving Fuzzy Classification Systems from
                   Numerical Data

     Pardeep Sandhu                     Shakti Kumar                       Himanshu Sharma                    Parvinder Bhalla
 Department of Electronics        Computational Intelligence           Department of Electronics         Computational Intelligence
     & Communication,                     Laboratory                      & Communication,                       Laboratory
 Maharishi Markandeshwar           Institute of Science and            Maharishi Markandeshwar            Institute of Science and
    University, Mullana,            Technology, Klawad,                  University, Mullana,              Technology, Klawad,
      Haryana, INDIA                   Haryana, INDIA                      Haryana, INDIA                     Haryana, INDIA
er.pardeepsandhu@gmail.co            shaktik@gmail.com                 himanshu.zte@gmail.com           parvinderbhalla@gmail.com

Abstract — Fuzzy Classifiers are an important class of fuzzy           systems are also called as Fuzzy Rule Based Systems
systems. Evolving fuzzy classifiers from numerical data has            (FRBSs) [4]. These systems have been successfully
assumed lot of significance in the recent past. This paper             applied to a wide range of problems from different areas
proposes a method of evolving fuzzy classifiers using a three          presenting uncertainty and vagueness in different ways
step approach. In the first step, we applied a modified Fuzzy          [5], [6], [7]. These FRBS‘s can be categorized as
C–Means Clustering technique to generate membership
                                                                       knowledge based systems and data driven systems. There
functions. In the second step, we generated rule base using
Wang and Mendel algorithm. The third step was used to                  are two ways of providing knowledge to the systems. In
reduce the size of the generated rule base. This way rule              first type of systems called knowledge driven modeling,
explosion issue was successfully tackled. The proposed                 the rule base is provided by an expert who has the
method was implemented using MATLAB. The approach                      complete knowledge of the domain while in second type
was tested on four very well known multi dimensional                   of models called data driven models, this rule base is
classification data sets. The bench mark classification data           generated from available numerical data [8].
sets contain: Iris Data, Wine Data, Glass Data and Pima
Indian Diabetes Data sets. The performance of the proposed                In data driven systems to automatically generate the
method was very encouraging. We further implemented our                rule base, a number of classical approaches like Hong and
algorithm on a Mamdani type control model for a quick                  Lee‘s Algorithm [9], Wang and Mendel Algorithm [4],
fuzzy battery charger data set. This integrated approach was           [6], [10], [11], [12], Online Learning Algorithm [13],
able to evolve model quickly.                                          Multiphase Clustering Approach [14] and soft computing
   Keywords — Linguistic rules, Fuzzy classifier, Fuzzy logic,         techniques like Artificial Neural Networks [15], [16], [17],
Rule base.
                                                                       Genetic Algorithm [18], [19], Swarm Intelligence based
                     I.   INTRODUCTION                                 techniques [20], Ant Colony Optimization [21], Particle
                                                                       Swarm Optimization [22], Biogeography based
   The theory of fuzzy sets and fuzzy logic was introduced             Optimization [23], Big Bang – Big Crunch Optimization
by Lotfi A. Zadeh through his seminal paper in 1965 [1].               technique [24] are available in the literature [25].
Both these, fuzzy set theory and fuzzy logic act as a
powerful methodology for dealing with imprecision and                     This paper is based on an integrated approach that
nonlinearity in an efficient way [2], [3]. As far as the need          makes use of a modified Fuzzy C–Means Clustering
of fuzzy set theory is concerned, there are numerous                   approach (FCM) [26] and Wang and Mendel method [6].
situations in which classical set theory of 0‘s and 1‘s is not         The approach was implemented in MATLAB for fuzzy
sufficient to describe human reasoning. Thus, for such                 classification problems [27] of Iris data of Fisher [28],
situations we need a more appropriate theory that can also             Wine data, Glass data, Pima Indian Diabetes (PID) data
define membership grades in between ‗0‘ and ‗1‘ thereby                and Battery Charger data (control problem) [29]. A system
providing better results in terms of human reasoning.                  was evolved using set of training examples and system‘s
Fuzzy set theory attempts to do this.                                  performance was then evaluated using test data set for the
                                                                       given system. The system performances were evaluated in
   Further this theory of fuzzy logic leads to the                     terms of Average Classification Rate (for classification
development of fuzzy logic based systems, the systems                  problems) and Mean Square Error (for control problem).
which are capable of making a decision on the basis of
knowledge or intelligence provided to the system through                 The paper is organized as follows: Section II introduces
linguistic rule bases. As a particular combination of input            Fuzzy Logic Based Systems. Section III discusses the
is given to the system, system on the basis of knowledge               proposed integrated approach and WM method for rule
embedded into it in the form of linguistic rules makes a               base generation. In section IV the result analysis along
decision and processes those inputs. As the intelligence of            with the comparative study for above mentioned standard
these systems depends upon linguistic rule base, these                 data sets are shown and section V includes conclusions.

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              II.       FUZZY RULE BASED SYSTEMS                       Rule: IF antecedent……THEN consequent…….                                 (2)
   Fuzzy logic is a mathematical approach to emulate the                  The antecedent part provides the input variable
human way of thinking and learning [30]. This logic is an              conditions using IF statements and consequent provides
extension of classical set theory which says a fuzzy set is a          the output using THEN statements. For example, if X and
class of objects with a continuum of grades of                         Y are the input and output universes of discourse of a
membership. Such a set is characterized by a membership                fuzzy system with a rule base of size ‗N‘, then the rule
mapping the elements of a domain, space or universe of                 will be of the form as shown by equation (3):
discourse ‗U‘ to the interval {0, 1}. If ‗U‘ is a collection           Rule ith: IF x is Ai THEN y is Bi                                      (3)
of objects denoted by x, then a fuzzy set ‗A‘ in the
universe of discourse ‗U‘ can be defined as a set of                      Where, x and y represent input and output fuzzy
ordered pairs as shown in equation (1) [5], [8]:                       linguistic variables respectively, and Ai Є X and Bi Є Y
                                                                       (1≤ i ≤N) are fuzzy sets representing linguistic values of x
A  xi ,  A ( xi)        x  A                          (1)         and y [5].
  Here x refers to ith element of the set and µA (xi) is the              In Mamdani type systems the consequent is represented
membership grade of xi in set ‗A‘.                                     using fuzzy sets while in Sugeno type systems, it is a
  Fuzzy Logic Based Systems or Fuzzy Rule Based                        fuzzy singleton. Also in TSK type systems, it is a function
Systems (FRBS) are intelligent systems those are based on              of inputs [23].
mapping of input spaces to output spaces where the way of
                                                                                          III.     PROPOSED APPROACH
representing this mapping is known as fuzzy linguistic rules.
These intelligent systems provide a framework for representing            We first broke the system identification problem into
and processing information in a way that resembles human               three sub–problems and solved these one by one as
communication and reasoning process.                                   follows:
                                                                         1. Classify all the relevant input and output domains
                                                                            into various membership functions using modified
                                                                            FCM method [26].
                                                                         2. Apply Wang and Mendel algorithm [6] for creating
                                                                            a fuzzy rule base, evolved as a combination of rules
                                                                            generated from numerical examples and linguistic
                                                                            rules supplied by human experts.
                                                                         3. Keep the number of rules to bare minimum. We
                                                                            used a rule reduction technique as proposed in [32],
                     Figure 1. Fuzzy Logic System                           [33] to keep the rule base as compact as possible.
   Each fuzzy rule based system, typically possesses a                    The backbone of this approach is the Wang and Mendel
fuzzy inference system (shown in Figure 1) composed of                 algorithm [6] which has proved to be very effective.
four major modules: Fuzzification module, Inference
Engine, Knowledge Base and Defuzzification module                            Suppose the given set of desired input–output data pairs
[31]. The fuzzification module performs the                            is:
transformation of crisp inputs into fuzzy domain values. It
is mainly done to find the belongingness of data sets to                     x(1) (1) (1)
                                                                               1 , x2 ; y    x
                                                                                             ,   ( 2 ) ( 2) ( 2 )
                                                                                                 1 , x2 ; y         ,.......                 (4)
different membership functions. The fuzzification can be
                                                                          Here x1, x2 are inputs and y is the output. The problem
performed by either with the help of domain experts or
                                                                       formulation consists of generating fuzzy rules and to use
directly from the available numerical data. These fuzzy
                                                                       these rules to determine a mapping from inputs (x1, x2) to
domain values are then processed by inference engine
                                                                       output (y).
which is composed of composition, implication and
aggregation processes. The method of processing the                          The following steps present our integrated approach:
inputs is supplied by the knowledge base and rule base                   Step 1: Divide the input output spaces into fuzzy
module as it contains the knowledge of the application                 regions:
domain and the procedural knowledge. Finally, the
processed output of inference engine is transformed from                 We divide input spaces into desired number of
fuzzy domain to crisp domain by defuzzification module.                membership functions using modified FCM [26].
   One of the biggest challenges in the field of modeling                 Assuming that the domain intervals of inputs x1, x2 and
fuzzy rule based systems is the designing of rule base as it           output y (equation (4)) lies in [x1-, x1+], [x2-, x2+] and
is characterized by a set of IF–THEN linguistic rules. This            [y-, y+]. Here, the domain interval means the values for a
rule base can be defined either by an expert or can be                 particular variable will lie in this interval. Each of these
extracted from numerical data using any computerized                   input and output, spaces are partitioned into (2N+1)
techniques as mentioned in section I. A rule in fuzzy                  regions. The number N can be different for each of the
domain can be represented by equation (2):                             variables. E.g. if the value of N = 2, then there will be five

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                                                    (IJCSIS) International Journal of Computer Science and Information Security,
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membership functions [6]. A number of other methods                                       IV.   RESULT ANALYSIS
are also available to divide the input output spaces into                 This section presents the performances obtained by our
fuzzy regions.                                                         integrated approach that uses modified Fuzzy C–Means
  Step 2: Generate fuzzy rules from given input–output                 Clustering [26] and Wang and Mendel algorithm [6] to
data pairs:                                                            evolve fuzzy rule based systems. We applied our approach
                                                                       on four very well known classification data sets from
  In this step, first the degree of a given data set (x1(i),           machine learning repository and one control data set. In
x2 ; y(i)) into different fuzzy membership functions are               each experiment, the input and output domain intervals are
determined.                                                            fuzzified using modified FCM approach. The training data
  Second, assign a given data set (x1(i), x2(i); y(i)) to the          samples are selected from available data sets in
region with maximum degree and obtain one rule from                    correspondence with the peaks of the input membership
one data set.                                                          functions. This sequence is used to train the systems
                                                                       which are then tested using testing data sets.
  Step 3: Assign a degree to each rule:
                                                                       A. Example 1: Iris Data Classification Problem
   A degree to each generated rule can be assigned using
following formula of equation (5):                                         The proposed approach has been applied on Iris Data
                                                                       classification problem. The Iris data set is a widely used
    Drule   A ( x1 )   B ( x2 )   C ( y)         (5)           benchmark for classification and pattern recognition
                                                                       studies [27], [28]. The dataset contains 150 samples of
   That is the product of membership grade of input x1 in              data (50 samples for each species) with four attributes as
fuzzy set ‗A‘, membership grade of input x2 in fuzzy set               inputs, Sepal Length, Sepal Width, Petal Length and Petal
‗B‘ and membership grade of output y in fuzzy set ‗C‘.                 Width and three classes of iris plants namely: Iris Setosa,
Also at this point if an expert is available and he assigns            Iris Versicolor and Iris Virginica as output. All the input
his degree of belief in the correctness of a particular data           variables have measurement units in centimeter while the
set then that degree ‗m‘ must be multiplied with the above             output is the type of iris plant. The learning sequence
expression.                                                            includes 24 data samples while the system is tested on all
  Step 4: Create a combined fuzzy rule base:                           150 data samples. By applying the proposed method on
                                                                       the learning sequence, a set of 24 classification rules (one
   The combined fuzzy rule base is assigned rules from                 rule per training data sample) is obtained. From this
either those generated from numerical data or linguistic               combined rule base, the redundant rules are then removed
rules (we assume that a linguistic rule also has a degree              using rule reduction algorithm [32], [33] and the final rule
that is assigned by the human experts and reflects the                 base composing 4 rules are shown in Table I.
expert‘s belief of the importance of the rule). Also, if there
is more than one rule having same antecedents but                      TABLE I.      CLASSIFICATION RULE BASE FOR IRIS DATA CLASSIFIER
different or same consequents then rule with maximum
                                                                          Sepal        Sepal        Petal        Petal
degree is to be selected. In this way, both numerical and                Length        Width       Length        Width
linguistic information are represented by a common
framework– the combined fuzzy rule base.                                  SL–L         SW–M         PL–L          PW–L          Setosa

   Step 5: Determine a mapping based on the combined                      SL–M         SW–L         PL–M         PW–M         Versicolor
fuzzy rule base:
   Defuzzification strategy is used to determine the output               SL–M         SW–L         PL–H         PW–M         Virginica
control for given inputs. This step performs nothing but
                                                                          SL–M         SW–L         PL–H         PW–H         Virginica
the same operation as defuzzification module performs in
a fuzzy inference system.
                                                                          Here, L – Low, M – Medium, H – High
  Step 6: Rule reduction:
   This step is used to reduce the number of redundant                   TABLE II.     CLASSIFICATION RATES FOR IRIS DATA CLASSIFIER
                                                                                            (PROPOSED APPROACH)
rules from the rule base. Thus the main objective of this
step has been to deal with rule explosion issue which if                Number                                                Average
left untackled may lead to a rule base with unmanageable,                            Setosa     Versicolor    Virginica
                                                                        of Rules                                               Rate
large number of rules in the rule base.
                                                                           4         98.00%      100.00%        94.00%         97.33%
   This procedure can easily be extended to general multi–
input multi–output cases. So, the approach can be viewed
                                                                           3         98.00%      100.00%        90.00%         96.00%
as a very general ‗model–free trainable fuzzy system‘ for
a wide range of applications, where model free means no
mathematical model is required for the problem and                        Table II shows the class wise classification rates along
trainable means the system learns from examples and                    with the effect of variations in the size of the rule base.
expert rules, and can adaptively change the mapping when               Table III presents a comparative analysis of different
new examples and expert rules are available.                           algorithms with the proposed integrated approach for Iris

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data set. The different parameters taken for comparison                 high performance Iris data classifier can be designed using
include number of input fuzzy sets, number of rules, and                a smaller length learning sequence and a compact set of
classification rates. The table clearly demonstrates that a             rules as shown in Table I.

                                                                 Number of Input             Number of          Classification Rates
                                                                   Fuzzy Sets                 Rules               (Testing Data)
           Hong and Lee‘s Algorithm [9]                                      8                 6.21                    95.57%

           Particle Swarm Optimization [22]                                 —                      —                   96.80%

           α–Cut based Fuzzy Learning Algorithm [34]                     8.21                      3                   96.21%

           Fuzzy Classifier Ensembles based Algorithm [35]                  —                      —                   90.70%

           Genetic Algorithm [36]                                           —                  10.10                   90.67%

           LEM–2 Method [37]                                                —                      —                   92.30%

           Proposed Approach                                                10                     4                   97.33%

B. Example 2: Wine Data Classification Problem                          TABLE IV. CLASSIFICATION RULE BASE FOR WINE DATA CLASSIFIER

   The Wine data set is also one of the most well–known                                            Flavan-
                                                                        Alcohol         Ash                     Hue       OD      Proline      Class
data sets in machine learning literature [27]. The data has
been obtained from the chemical analysis of wines grown                          M       L             M         M        M            H         1
in the same region in Italy but derived from three different                     M       M             M         M        M            M         1
cultivars. The chemical analysis determines the quantities
of thirteen constituents found in each of the three types of                     M       M             M         M        M            H         1
wines. These thirteen constituents are: Alcohol, Malic                           L       L             L         M         L           M         2
Acid, Ash, Alcalinity of Ash, Magnesium, Phenols,
                                                                                 L       M             M         M        M            L         2
Flavanoids, Non–Flavanoid Phenols, Proanthocyaninsm,
Color Intensity, Hue, OD280/OD315 of Diluted Wines                               M       L             M         M        M            L         2
and Proline. This dataset contains 178 samples of data (59                       M       L             L          L        L           M         3
samples for Class ‗1‘, 71 samples for Class ‗2‘ and 48
samples for Class ‗3‘ Wine). Out of these thirteen                               Here, L – Low, M – Medium, H – High
attributes, following six attributes are used to model Wine
Data Classifier: Alcohol, Ash, Flavanoids, Hue,                         TABLE V.              CLASSIFICATION RATES FOR WINE DATA CLASSIFIER
OD280/OD315 and Proline [38]. The training data set                                               (PROPOSED APPROACH)
contains 28 data samples and testing data set contains 178
samples.                                                                    Number                                                          Average
                                                                                         Class ‘1’         Class ‘2’      Class ‘3’
                                                                            of Rules                                                         Rate
   In this case, the proposed approach successfully
generated 28 rules which were reduced to 7 rules by                              7       100.00%             100.00%       95.83%           98.87%
applying rule reduction algorithm [32], [33] as shown in
                                                                                 6       100.00%             98.59%        95.83%           98.30%
Table IV. The performance of the evolved Wine data
classifier is shown in Table V in terms of classification                        5        96.61%             97.18%        95.83%           96.62%
rates. Table V also shows the variations in the
classification rate by varying the number of rules. Table                        4        96.61%             95.77%        95.83%           96.06%
VI shows the comparison of the proposed approach with
other approaches.

                                                               Number of Input           Number of              Classification Rates
                                                               Attributes Used            Rules                   (Testing Data)
           Evolutionary Approach [38]                                   6                      5                       98.90%

           eClass Classifier [39]                                       13                     7                       95.90%

           SANFIS Learning Algorithm [40]                               13                     3                       99.43%

           Hyper – Cone Membership Function Approach [41]            —                        —                        92.95%

           IPCA Algorithm [42]                                       —                        —                        87.60%
           Proposed Approach                                            6                      7                       98.87%

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C. Example 3: Glass Data Classification Problem                        non_ float_processed, class ‗5‘ as Containers, class ‗6‘ as
                                                                       Tableware and class ‗7‘ as Headlamps. Although the
   The glass data set [27] is a nine–dimensional data set
                                                                       original data set contains seven classes but it doesn‘t have
with 214 samples from seven classes, also taken from
                                                                       any data sample from class ‗4‘. The learning sequence
Irvine Machine Learning Repository. Here, this data set
                                                                       contains 56 data samples while the testing sequence is
has been chosen because it involves many classes. The
                                                                       composed of all 214 data samples. For this classifier the
nine input attributes are: Refractive Index (RI), Sodium
                                                                       proposed method first generated a rule base of 37 rules
(Na), Magnesium (Mg) Aluminum (Al), Silicon (Si),
                                                                       which was reduced to 20 rules (shown in Table VII) by
Potassium (K), Calcium (Ca), Barium (Ba) and Iron (Fe).
                                                                       using rule reduction algorithm [32], [33]. The class wise
Out of these nine attributes the last two attributes Barium
                                                                       classification results for the modeled Glass data classifier
(Ba) and Iron (Fe) are excluded in this paper due to very
                                                                       for the given test data set are specified in Table VIII.
small variations in their sample points. The output classes
                                                                       Table IX shows a comparison of Glass classifiers for
indicate different types of the glasses: class ‗1‘ as
                                                                       different algorithms. The results show that the
Building_windows_float_processed, class ‗2‘ as Building
                                                                       classification rate of 71.49% can be achieved with lesser
_windows_non_float_processed, class ‗3‘ as Vehicle_
                                                                       training data set and with lesser number of rules.
windows_float_processed, class ‗4‘ as Vehicle_windows_

               RI           Na            Mg             Al             Si           K           Ca             Class

               M             L                H          M              H           M            L                1

               H             H                H          L             M             L           M                1

               H             M                H          L             M             L           M                1

               L             L                H          M              H           M            L                2

               L             M                H          M              H            L           L                2

               L             H                H          M             M             L           L                2

               M             L                H          M             M            M            L                2

               M             M                H          M             M            M            L                2

               H             L                L          L              H            L           H                2

               H             L                L          H              L           M            H                2

               H             H                L          L             M             L           M                2

               L             M                H          M             M            M            L                3

               L             H                M          H              L           M            L                5

               L             H                L          L              H            L           L                6

               M             H                L          H              H            L           M                6

               M             H                M          M             M             L           M                6

               L             H                L          H              H            L           L                7

               M             H                L          M              H           M            L                7

               M             H                L          H              H            L           L                7

               H             H                M          M              L            L           L                7

              Here, L – Low, M – Medium, H – High


            Number                                                                                             Average
                        Class ‘1’    Class ‘2’     Class ‘3’      Class ‘5’       Class ‘6’    Class ‘7’
            of Rules                                                                                            Rate

              20         68.57%       80.26%        17.64%         92.30%          66.66%       86.20%          71.49%

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                                                              Number of Input           Number of        Classification Rates
                                                                Fuzzy Sets               Rules             (Testing Data)

              Weighted Vote Method [43]                                15                 18, 24                68.22%

              Genetic Algorithm [44]                                   27                  14                   84.10%

              Neural Network with Pruning Algorithm [45]               —                   —                    63.28%

              Rule Weight method [46]                                  —                   61                   64.89%

              Proposed Approach                                        21                  20                   71.49%

D. Example 4: Pima Indian Diabetes Data Classification                       Skin Fold Thickness (mm) (TSFT), 2–Hour Serum
   Problem                                                                   Insulin (mu U/ml) (HSI), Body Mass Index (weight in kg/
                                                                             (height in m)˄2) (BMI), Diabetes Pedigree Function
   This data set [27] is related to the diagnosis of diabetes
                                                                             (DPF), AGE (in years) and with two output class
(with or without the disease) in an Indian population that
                                                                             variables (0 indicating tested negative for diabetes and 1
lives near the city of Phoenix, Arizona. The data base
                                                                             indicating tested positive for diabetes). The rule base
contains 768 data samples (500 samples for class ‗0‘ and
                                                                             generated by the proposed approach contains 10 rules,
268 samples for class ‗1‘) with eight input attributes as:
                                                                             shown in Table X. Table XI presents the classification
Number of Times Pregnant (NTP), Plasma Glucose
                                                                             rates obtained through our approach and Table XII
Concentration a 2 hours in an oral glucose tolerance test
                                                                             presents the comparison of this approach with other
(PGC), Diastolic Blood Pressure (mmHg) (DBP), Triceps
                                     TABLE X.        CLASSIFICATION RULE BASE FOR PID DATA CLASSIFIER

                NTP           PGC        DBP      TSFT           HSI          BMI        DPF        AGE             Class

                 L               L        M         L            L              L         L          L                 0

                 L               L        M        M             L              L         L          L                 0

                 L               L        M        M             M             M          L          L                 0

                 L               L        M        M             M             M          M          L                 0

                 M               L        L         L            L              L         L          L                 0

                 M               L        M         L            L              L         L          M                 0

                 M               L        M         L            L              L         M          M                 0

                 M               L        M        M             L             M          L          M                 0

                 L               M        M        M             L             M          M          M                 1

                 M               L        M        M             L             M          M          M                 1

                 Here, L – Low, M – Medium

 TABLE XI.        CLASSIFICATION RATES FOR PID DATA CLASSIFIER               corresponding voltage) and second is Temperature
                     (PROPOSED APPROACH)
                                                                             Gradient as obtained by taking time derivative of the
                                                                             conditioned signal as obtained from temperature sensor,
  Number of
                     Class ‘0’       Class ‘1’     Average Rate              varied from 0 to 1 mV/10s and an output Charging
                                                                             Current whose value depends on the present temperature
     10              79.40%            82.46%           80.46%               of the battery and at how much rate it is increasing. The
                                                                             input and output variables identified for rapid Ni–Cd
      9              76.80%            86.94%           80.33%               battery charger along with their universes of discourse are
                                                                             listed in Table XIII and Table XIV [24]. Here, the goal is
                                                                             to design a charger in such a manner that the required
E. Example 5: Battery Charger Design Problem
                                                                             charging current is supplied to the battery without
   Nickel Cadmium (Ni–Cd) Battery Charger is a typical                       damaging it, due to increase in temperature or excessive
example of fuzzy control problem [24], [29]. To design an                    current supply. Here, the combined rule base generated by
intelligent battery charger, two inputs have been taken,                     applying the proposed algorithm is composed of 14 rules
one is Temperature whose range is 0° to 50°C (in terms of                    which are reduced to 6 rules by removing the

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 contradictory rules and are shown in Figure 2 [24] and in                the Mean Square Errors calculated from the proposed
 Table XV as well. Table XVI presents the comparison of                   approach and other algorithms.

                                                Number of Input      Number of Input         Number of         Classification Rates
                                                  Attributes           Fuzzy Sets             Rules              (Testing Data)

          HNFB-1 Model [47]                                8                 —                     98                  78.39%

          Conventional Encoding [48]                   4.4                   22                    8.9                 74.40%

          Evolutionary Approach [48]                   4.2                  23.3                   9.7                 72.90%

          C4.5 Decision Tree [49]                       —                    —                     —                   74.70%

          Proposed Approach                                8                 16                    10                  80.46%

                                                 Figure 2. Fuzzy Model for Battery Charger


   Input Variable         Minimum Value        Maximum Value                               Algorithm                        Mean Square Error

  Temperature (◦C)                  0                 50                   Genetic Algorithm [8]                                    0.130
Temperature Gradient
                                    0                  1
      (◦C/sec)                                                             Particle Swarm Optimization [8]                          0.040

 TABLE XIV.       OUTPUT VARIABLE FOR RAPID NI–CD BATTERY                  Hybrid Learning [26]                                     0.008
                                                                           Proposed Approach                                        0.060
   Output Variable         Minimum Value        Maximum Value

Charging Current (A)                 0                 4


  Temperature                                 Charging Current
      Low                   Normal                Ultrafast

      Low                     High                 Ultrafast

    Medium                  Normal                  High

    Medium                    High                 Medium

      High                  Normal                 Trickle

      High                    High                 Trickle
                                                                                   Figure 3. Surface View of the Modeled Battery Charger

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                                                                                                         ISSN 1947-5500
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  Figure 3 is the three–dimensional graphical representation                      [13] R. Rovatti and R. Guerrieri, ―Fuzzy Sets of Rules for System
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