A Classifier with the Function-based Decision Tree
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1 A Classifier with the Function-based Decision Tree Been-Chian Chien and Jung-Yi Lin Institute of Information Engineering I-Shou University, Kaohsiung 84008, Taiwan, R.O.C Abstract Classification is one of the important problems in the research area of knowledge discovery and machine learning. In this paper, an accurate classifier with multi-category based on the genetic programming is proposed. The classifier consists of the discriminant functions that are generated by genetic programming. We propose the function-based decision tree (FDT) to resolve the problem of ambiguity between discriminant functions, and the experiments show that the proposed methods are accurate. (Keywords: Knowledge discovery, Machine learning, Genetic programming, Classification, Decision Tree) 1. Introduction Classification plays an important role in knowledge discovery and machine learning. Many applications can be viewed as extensions of classification problem. For example, pattern recognition, disease diagnosis, and business decision-making. To reduce the training time and increase the classification rate, many different methods have been proposed for building an efficient and effective classifier in the past decades. For example, the Bayesian decision theory [5], neural network [17], k-nearest neighbor [2], genetic algorithm (GA) [15][16] and genetic programming (GP) [4][6]. In genetic programming, the task of classification is accomplished by a set of discriminant functions[6] or classification rules. Classifying by discriminant functions has more advantages because they are concise, efficient, and the number of functions is fewer. However, the ambiguity may occur when an obect is recognized by two or more discriminant functions at the same time or an obect is not recognized by all functions, thus a complete classification approach should provide a refined mechanism used to resolve the problems. In this paper, we proposed a method to resolve the ambiguity: the function-based decision tree (FDT). In this paper, two well-known datasets are selected to show the performance of the proposed classifier: the Fisher s Iris dataset and Wisconsin Breast Cancer dataset. The remainder of this paper is organized as follows. Section 2 briefly reviews the genetic programming. In Section 3, we propose the distance-based fitness function of GP. In Section 4, we propose the FDT. The experimental results are described in Section 5. Finally, conclusions are made in Section GENETIC PROGRAMMING The technique of genetic programming (GP) was proposed by Koza [7][8]. Genetic programming can discover underlying relationships between data and presents such
2 relationships by expressions. The expression is constructed by possible terminals and functions. Several types of functions can be applied within genetic programming, for example, arithmetic operations and trigonometric functions. Genetic programming begins with a set called population. The population consists of randomly created individuals. Each individual is an expression that stands for a potential solution. A fitness function is given for evaluating the fitness of each individual. For the period of a generation, individuals are modified to get better fitness by genetic operators. In the end of a generation, a new population contains individuals with better fitness will replace the original one. The modifications of individuals are made by genetic operators: reproduction, crossover, and mutation [1]. After the evolution of a number of generations, an individual with best fitness value can be taken as the solution. 3. The Learning of Discriminant Functions based on Genetic Programming 3.1 A Formal Description of Classification Problem Given a data set S, for each data x S has n attributes. Let x be represented as x = (v 1, v 2,, v t,, v n ), 1 t n, where v t R stands for the t-th attribute of x. Let C = {C 1, C 2,..., C K } be the set of K predefined classes, we say that <x, c > is a sample if the obect x has been assigned to a specified class c, c C. We then define a training set (T) to be a set of samples, T = {<x, c > x = (v 1, v 2,..., v n ), c C, 1 m} where m is the number of samples in T, and T = m. We denote the number of samples in T belonging to the class C i as m i, 1 i K. A discriminant function f i is a function mapping from R n to R. For a sample <x, c >, the functions should satisfy the following conditions, fi ) a,, where 1 i K, 1 m. fi ) < a, A set of discriminant functions F is defined by follows, F = { f i f i : R n R,1 i K}. 3.2 The Distance-Based Fitness Function In the learning procedure, we first prepare the training set T, which includes positive instances and negative instances [11]. The learning procedure can be started after the training set T is prepared[1][11]. We denote an individual in GP as h i. Consider a discriminant function f i for a class C i, we give a specified constant a and urge that f i (x ) a for the positive instance and f i (x ) a for negative instance. Two parameters p and q are defined as p > a and q < a. For the individual h i, the fitness measure for a positive instance is defined to be 0 ) a Dp, c ) = 2, (1) [ p hi )] ) < a and the fitness measure for a negative instance is defined to be 2 [ hi ) q] ) a Dn, c ) =. (2) 0 ) < a The fitness value of an individual h i for the training set T is defined as m Fitness(h i (x ), T)= ( Dp, c) + Dn, c)). (3) = 1 where <x, c > T, 1 m. Since we use the negative of measure to be the fitness value
3 of an individual, the best fitness value is zero. 4. The Function-based Classifiers In this paper, we proposed the FDT method to resolve the two ambiguous problems. In FDT, each node stands for a discriminant function. An obect through the FDT will be taken out from the dataset if it has recognized by a node, therefore, the ambiguity is avoided. In the FDT, we generate only K-1 discriminant functions for a K-class problem. Since there are obects not recognized by f K-1, they will be classified into the class C K. To construct the FDT, a good permutation of the nodes is important to perform accurate classification. The performance of a node is measured by precision and recall[1]. Higher precision means lower misclassification rate gher recall means higher recognition rate. We now define the permutation of the FDT. At first, from the classification results of the training obects with the generated K-1 discriminant functions, the precision and recall of the discriminant function f i can be evaluated and denoted as <p i, r i >. We define the permutation of discriminant functions as f (1), f (2),, f (i), f (i+1),, f (K-1), where f (i) stands for a function f i with <p i, r i > and p i > p i+1, or p i = p i+1 and r i r i+1, otherwise. 5. EXPERIMENTAL RESULTS In experiments, we modify the GP Quick 2.1[14] to fit the requirements of proposed approaches and perform the experiments. The test datasets are the Fisher s Iris dataset (IRIS)[3] and Wisconsin Breast Cancer dataset (WBCD)[12]. The accuracy of the classification results is evaluated by average accuracy and overall accuracy [1]. The average accuracy stands for the average recognition rate of discriminant functions. The overall accuracy presents the performance of recognition for a classifier. In Table 1, we show the parameters of genetic programming used in the two experiments Fisher s Iris In Iris [3], there are 150 data separated into three classes: Setosa, Versicolor and Virginica. Each class contains 50 data that has four numerical attributes. We denote the attributes as SL, SW, PL and PW, respectively. We randomly select 25 data from each class to construct the training set. After the learning procedure, three discriminant functions are generated and are shown as following: f Setosa = SW-PL, f Versicolor = ((((((((((((((PL + PL) - (-33 / PL)) / -22) - 11) / (SW - PL)) * 99) - 121) - PW) * PL) / -31) + 45) / -21) - PW) / 43), f Virginica = ((((PW * PL) + (-11 / PW)) - ((((-26-92) / (PW * 59)) / -7) / PL))/SL). The classification results of training set and test set by using above three functions are shown in Table 2. The permutation of discriminant functions in the FDT can be achieved: f (1) = f Setosa, f (2) = f Versicolor. The classification result of FDT is shown in Table 2. We found that only one obect causes misclassification. The average accuracy and overall accuracy are high. Finally, we compare with the results of previous researches in Table Wisconsin Breast Cancer dataset WBCD [12] contains 699 data that separated into two classes, Malignant and Benign.
4 Each data in WBCD has 9 numerical attributes. However, there are 16 data have missing values, thus we use the remainder 683 data to accomplish this experiment, where the Malignant contains 239 data and the Benign contains 444 data. The training set is constructed by 119 data from Malignant and 222 data from Benign. The remainder data are used to be the test set. The 9 attributes of a sample are denoted as F1, F2,, F9, which are used to be the terminals in GP. We show the classification results of training set and test set in Table 4. From Table 4, the permutation of the two discriminant functions can be obtains as f (1) =f Benign, f (2) =f Malignant. The classification results of FDT are shown in Table 4. Finally, we compared the previous classified results with the proposed methods in Table 5. Table 1: Parameters used in the experiments. Parameter Value Parameter Value Node mutate weight 43.5% Mutation weight annealing 40% Mutate constant weight 43.5% Population size 1000 Mutate shrink weight 13% Generations in Iris 5000 Selection method Tournament Generations in WBCD Mutation weight 8% Crossover weight 28% Terminal set in Iris SL, SW, PL, PW Crossover weight annealing 20% Terminal set in WBCD F1, F2 F9 Function set +,,, Table 2: The classification results. Method Classification with the discriminant functions Classification with FDT Dataset Training set Test set Test set Assigned class Setosa Versicolor Virginica Setosa Versicolor Virginica Setosa Versicolor Virginica Setosa Versicolor Virginica Precision 100% 100% 88.9% 100% 100% 89.3% 100% 100% 96.2% Recall 100% 80.0% 96.0% 100% 96.0% 100% 100% 96.0% 100% Average 98.6% Overall 98.6% Table 3: The accuracy comparison of IRIS dataset. Models or Methods Recognition Rate GPCE[6] 96.0% FUNLVQ+GFENCE [9] 96.3% FEBFC with 4 features [10] 96.7% FEBFC with 2 selected features[10] 97.1% FDT 98.6% Table 4: The classification results of WBCD. Method Classification with the discriminant functions Classification with FDT Dataset Training set Test set Test set Assigned class Malignant Benign Malignant Benign Malignant Benign Malignant Benign Precision 95.93% 99.54% 95.16% 98.63% 95.12% 98.63% Recall 99.16% 96.85% 98.33% 97.30% 97.50% 97.30% Average 97.40% Overall 97.37%
5 Table 5: The comparison of classification results on WBCD. Models or methods Recognition rate C4.5 (cited from [10]) 93.1% NNFS with 9 features [13] 93.94% NNFS with avg features [13] 94.15% FEBFC with 9 features [10] 94.67% FEBFC with selected 6 features [10] 95.14% FDT 97.37% 6. CONCLUSIONS This paper presents an efficient distance-based fitness function of genetic programming is proposed to learn efficient discriminant functions, and two methods are proposed to resolve ambiguity, the FDT classification approach. The experimental results show that the proposed method performs high accuracy. However, the attributes of the obects in the dataset are all numerical. The categorical attribute seems to have no simple way to be applied for classification by discriminant functions. The future work on text classification based on discriminant functions is worth to be investigated. REFERENCES [1] B. C. Chien and J. Y. Lin, Learning Discriminating Functions Based on Genetic Programming for Classification, in Proc. TAAI2001, Taiwan (2001) [2] R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis, John Wiley & Sons, New York, [3] R. A. Fisher, The Use of Multiple Measurements in Taxonomic Problems, Ann. Eugenics, pt. II 7 (1936) [4] A. A. Freitas, A Genetic Programming Framework for Two Data Mining Tasks: Classification and Generalized Rule Induction, in Proc. 2 nd Annual Conference Morgan Kaufmann (1997) [5] D. Heckerman and M. P. Wellman, Bayesian networks, Communications of the ACM 38, No. 3 (1995) [6] J. K. Kishore et al., Application of Genetic Programming for Multicategory Pattern Classification, IEEE Trans. on Evolutionary Computation 4, No. 3, (2000) [7] J. R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT Press, Cambridge, [8] J. R. Koza et al. (eds.), Genetic Programming 1996, MIT Press, Cambridge, [9] H. M. Lee, A Neural Network Classifier with Disunctive Fuzzy Information, Neural Networks 11, No. 6 (1998) [10] H. M. Lee et al., An Efficient Fuzzy Classifier with Feature Selection Based on Fuzzy Entropy, IEEE Trans. on Systems, Man, and Cybernetics-part b: Cybernetics 31, No. 3, (2001) [11] J. Y. Lin and B. C. Chien, Learning Classification Rules With Fuzzy Attributes Using Genetic Programming, in Proc. of Ninth National Conference on Fuzzy Theory and Its Applications, Taiwan (2001) [12] O. L. Mangasarian and W. H. Wolberg, Cancer Diagnosis via Linear Programming, SIAM News 23, No. 5 (1990) [13] R. Setiono et al., Neural-Network Feature Selector, IEEE Trans. on Neural Networks 8 (1997) [14] A. Singleton, Genetic Programming with C++, Byte (Feb 1994) [15] C. H. Wang et al., A Fuzzy Inductive Learning Strategy for Modular Rules, Fuzzy Set and Systems 103 (1999) [16] C. H. Wang et al., Integrating Fuzzy Knowledge by Genetic Algorithms, IEEE Trans. on Evolutionary Computation 2, No. 4 (1998) [17] G. P. Zhang, Neural Networks for Classification: A Survey, IEEE Trans. on Systems, Man, and Cybernetics-Part C: Applications and Reviews, 30, No. 4 (2000)
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