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(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 m 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. 139 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 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 140 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 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 (i) 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 Drule 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 Class 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 141 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 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. TABLE III. COMPARISON OF THE PROPOSED APPROACH WITH OTHER APPROACHES (IRIS DATA) Number of Input Number of Classification Rates Algorithm 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 oid 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. TABLE VI. COMPARISON OF THE PROPOSED APPROACH WITH OTHER APPROACHES (WINE DATA) Number of Input Number of Classification Rates Algorithm 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% 142 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 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_ TABLE VII. CLASSIFICATION RULE BASE FOR GLASS DATA CLASSIFIER 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 TABLE VIII. CLASSIFICATION RATES FOR GLASS DATA CLASSIFIER (PROPOSED APPROACH) 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% 143 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 TABLE IX. COMPARISON OF THE PROPOSED APPROACH WITH OTHER APPROACHES (GLASS DATA) Number of Input Number of Classification Rates Algorithm 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 algorithms. 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 Rules 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 144 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 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. TABLE XII. COMPARISON OF THE PROPOSED APPROACH WITH OTHER APPROACHES (PID DATA) Number of Input Number of Input Number of Classification Rates Algorithm 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 TABLE XIII. INPUT VARIABLES FOR RAPID NI–CD BATTERY TABLE XVI. COMPARISON OF THE PROPOSED APPROACH WITH CHARGER ALONG WITH THEIR UNIVERSES OF DISCOURSE OTHER ALGORITHMS (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 CHARGER ALONG WITH ITS UNIVERSE OF DISCOURSE Proposed Approach 0.060 Output Variable Minimum Value Maximum Value Charging Current (A) 0 4 TABLE XV. RULE BASE FOR BATTERY CHARGER Temperature Temperature Charging Current Gradient 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 145 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 Figure 3 is the three–dimensional graphical representation [13] R. Rovatti and R. 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