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To gain an intuitive understanding of how a classification value is computed, let us sketch a simple example. Referring to the system of Fig. 4, assume that the following values are presented as inputs (these represent case # 145 of the WBCD database):

Figure 4 To gain an intuitive understanding of how a classification value is computed, let us sketch a simple example. Referring to the system of Fig. 4, assume that the following values are presented as inputs (these represent case # 145 of the WBCD database):

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Fig. 1. Example of a fuzzy variable with two possible fuzzy values labeled Low and High, and orthogonal membership functions, plotted above as degree of membership versus input values. P and d define the start point and the length of membership function edges, respectively. The orthogonality condition means that the sum of all membership functions at any point is one. In the figure, an example value wu is assigned the membership values ,y(u) = 0.8 and Myign(u) = 0.2 (as can be seen py 54(u) + Lipign(U) = 1).
Fig. 1. Example of a fuzzy variable with two possible fuzzy values labeled Low and High, and orthogonal membership functions, plotted above as degree of membership versus input values. P and d define the start point and the length of membership function edges, respectively. The orthogonality condition means that the sum of all membership functions at any point is one. In the figure, an example value wu is assigned the membership values ,y(u) = 0.8 and Myign(u) = 0.2 (as can be seen py 54(u) + Lipign(U) = 1).
Fig. 2. Basic structure of a fuzzy inference system. A fuzzy inference system is a rule-based system that uses fuzzy logic, rather than Boolean logic, to reason about data [33]. Its basic structure includes four main components, as depicted in Fig. 2: (1) a fuzzifier, which translates crisp (real-valued) inputs into fuzzy values; (2) an inference engine that applies a fuzzy reasoning mechanism to obtain a fuzzy output; (3) a defuzzifier, which translates this latter output into a crisp value; and (4) a knowledge base, which contains both an
Fig. 2. Basic structure of a fuzzy inference system. A fuzzy inference system is a rule-based system that uses fuzzy logic, rather than Boolean logic, to reason about data [33]. Its basic structure includes four main components, as depicted in Fig. 2: (1) a fuzzifier, which translates crisp (real-valued) inputs into fuzzy values; (2) an inference engine that applies a fuzzy reasoning mechanism to obtain a fuzzy output; (3) a defuzzifier, which translates this latter output into a crisp value; and (4) a knowledge base, which contains both an
Parameter classification of fuzzy inference systems Table 1 ensemble of fuzzy rules, known as the rule base, and an ensemble of membership functions, known as the database.
Parameter classification of fuzzy inference systems Table 1 ensemble of fuzzy rules, known as the rule base, and an ensemble of membership functions, known as the database.
Fig. 4. Proposed diagnosis system. Note that the fuzzy subsystem displayed to the left is in fact the entire fuzzy inference system of Fig. 2.
Fig. 4. Proposed diagnosis system. Note that the fuzzy subsystem displayed to the left is in fact the entire fuzzy inference system of Fig. 2.
Parameters encoding of an individual’s genome* * Total genome length is 54+18N,, where N, denotes the number of rules (JN, is set a priori to a value between | and 5, and is fixed during the genetic algorithm run). Table 2
Parameters encoding of an individual’s genome* * Total genome length is 54+18N,, where N, denotes the number of rules (JN, is set a priori to a value between | and 5, and is fixed during the genetic algorithm run). Table 2
Fig. 5. Example of a genome for a single-rule system, (a) Genome encoding. The first 18 positions encode the parameters P and d for the nine variables v,-vo. The rest encode the membership function applicable for the nine antecedents of the rule; (b) Interpretation. Database and rule base of the single-rule system encoded by (a). The parameters P and d are interpreted as illustrated in Fig. 1. To evolve the fuzzy inference system, we used a genetic algorithm with a fixed population size of 200 individuals, and fitness-proportionate selection (Section 1.3). The algorithm terminates when the maximum number of generations, G,,,, 1S reached (we set Gyax = 2000 + SO0(N,, i.e. dependent on the number of rules used
Fig. 5. Example of a genome for a single-rule system, (a) Genome encoding. The first 18 positions encode the parameters P and d for the nine variables v,-vo. The rest encode the membership function applicable for the nine antecedents of the rule; (b) Interpretation. Database and rule base of the single-rule system encoded by (a). The parameters P and d are interpreted as illustrated in Fig. 1. To evolve the fuzzy inference system, we used a genetic algorithm with a fixed population size of 200 individuals, and fitness-proportionate selection (Section 1.3). The algorithm terminates when the maximum number of generations, G,,,, 1S reached (we set Gyax = 2000 + SO0(N,, i.e. dependent on the number of rules used
Summary of results of 120 evolutionary runs, divided according to the three experimental categories discussed in the text (i.e. the three classes which differ in the training-set to test-set ratio)* * The table lists the average performance over all 120 runs, where the averaging is done over the best individual of each run. The performance value denotes the percentage of cases correctly classified. Three such performance values are shown, (1) performance over the training set; (2) performance over the test set; and (3) overall performance, considering the entire database. In addition, the average number of variables per rule is also shown.
Summary of results of 120 evolutionary runs, divided according to the three experimental categories discussed in the text (i.e. the three classes which differ in the training-set to test-set ratio)* * The table lists the average performance over all 120 runs, where the averaging is done over the best individual of each run. The performance value denotes the percentage of cases correctly classified. Three such performance values are shown, (1) performance over the training set; (2) performance over the test set; and (3) overall performance, considering the entire database. In addition, the average number of variables per rule is also shown.
Fig. 6. Summary of results of 120 evolutionary runs. The histogram depicts the number of systems exhibiting a given performance level at the end of the evolutionary run. The performance considered is that of the best individual of the run, measured as the overall percentage of correctly classified cases over the entire database.
Fig. 6. Summary of results of 120 evolutionary runs. The histogram depicts the number of systems exhibiting a given performance level at the end of the evolutionary run. The performance considered is that of the best individual of the run, measured as the overall percentage of correctly classified cases over the entire database.
* The first three approaches—those of Setiono [27], Setiono and Liu [28], and Taha and Ghosh [30]—involve Boolean rule bases extracted from trained neural networks; the last approach is our own previous work [25]. Shown above are the classification performance values of the top systems obtained by these approaches, along with the average number of variables-per-rule given in parentheses. Results are divided into five classes, in accordance with the number of rules-per-system, going from one-rule systems to five-rule ones. > Note that Taha and Ghosh [30] obtained slightly better results for the five-rules case by directly using their trained neural networks, rather than the extracted rule-haced eveteme Herein anr interect liew with thece latter riule-haced eveteme Comparison of the best systems evolved by our approach with the top systems obtained by four other rule-based diagnostic approaches*
* The first three approaches—those of Setiono [27], Setiono and Liu [28], and Taha and Ghosh [30]—involve Boolean rule bases extracted from trained neural networks; the last approach is our own previous work [25]. Shown above are the classification performance values of the top systems obtained by these approaches, along with the average number of variables-per-rule given in parentheses. Results are divided into five classes, in accordance with the number of rules-per-system, going from one-rule systems to five-rule ones. > Note that Taha and Ghosh [30] obtained slightly better results for the five-rules case by directly using their trained neural networks, rather than the extracted rule-haced eveteme Herein anr interect liew with thece latter riule-haced eveteme Comparison of the best systems evolved by our approach with the top systems obtained by four other rule-based diagnostic approaches*
This completes the fuzzification phase (the ‘fuzzifier’ unit of Fig. 2). Having computed these membership values, the inference engine (Fig. 2) can now go on to compute the so-called truth value of each rule. This truth value is computed by applying the fuzzy ‘and’ operator (Section 1.2) to combine the antecedent clauses (the membership values) in a fuzzy manner; this results in the output truth value, namely, a continuous value which represents the rule’s degree of activation. Thus, a rule is not merely either activated or not, but in fact is activated to a certain degree represented by a value between 0 and 1. In our example, the rule activation values are as follows (remember that we ‘chucked out’ rule 2, since it was found to never fire): The membership value of each variable is then computed in accordance with the (evolved) database of Fig. 7:
This completes the fuzzification phase (the ‘fuzzifier’ unit of Fig. 2). Having computed these membership values, the inference engine (Fig. 2) can now go on to compute the so-called truth value of each rule. This truth value is computed by applying the fuzzy ‘and’ operator (Section 1.2) to combine the antecedent clauses (the membership values) in a fuzzy manner; this results in the output truth value, namely, a continuous value which represents the rule’s degree of activation. Thus, a rule is not merely either activated or not, but in fact is activated to a certain degree represented by a value between 0 and 1. In our example, the rule activation values are as follows (remember that we ‘chucked out’ rule 2, since it was found to never fire): The membership value of each variable is then computed in accordance with the (evolved) database of Fig. 7:
* Results are divided into five classes, in accordance with the number of rules-per-system, going from three-rule systems to seven-rule ones. We performed 5-7 runs per class, totaling 31 runs in all; shown above are the resulting best systems as well as the average per class. Results include the overall classification performance and the average number of variables-per-rule in parentheses. Results of evolutionary runs in which the variables-per-rule constraint has been removed*
* Results are divided into five classes, in accordance with the number of rules-per-system, going from three-rule systems to seven-rule ones. We performed 5-7 runs per class, totaling 31 runs in all; shown above are the resulting best systems as well as the average per class. Results include the overall classification performance and the average number of variables-per-rule in parentheses. Results of evolutionary runs in which the variables-per-rule constraint has been removed*
Results of evolutionary runs in which the default diagnosis is benign (rather than malignant)* * Results are divided into five classes, in accordance with the number of rules-per-system, going fron one-rule systems to five-rule ones. We performed 4-6 runs per class, totaling 27 runs in all; shown abov are the resulting best systems as well as the average per class. Results include the overall classificatior performance and the average number of variables-per-rule in parentheses. Note that we slightly modifiec the genomic representation (Section 4.2), such that if consequents are malignant, and the else consequen is benign. Table 6
Results of evolutionary runs in which the default diagnosis is benign (rather than malignant)* * Results are divided into five classes, in accordance with the number of rules-per-system, going fron one-rule systems to five-rule ones. We performed 4-6 runs per class, totaling 27 runs in all; shown abov are the resulting best systems as well as the average per class. Results include the overall classificatior performance and the average number of variables-per-rule in parentheses. Note that we slightly modifiec the genomic representation (Section 4.2), such that if consequents are malignant, and the else consequen is benign. Table 6
Summary of top 45 evolutionary runs (of 120) described in Section 5.1, divided according to the three experimental categories discussed in the text (i.e. the three classes whicl differ in the training-set to test-set ratio)* * For each of the 45 evolved systems, the table lists its fitness value, its performance, and its average number of variables-per-rule. As explained in Section 5.1, the performance value denotes the percentage of cases correctly classified. Three such performance values are shown, (1) performance over the training set; (2) performance over the test set; and 3) overall performance, considering the entire database.
Summary of top 45 evolutionary runs (of 120) described in Section 5.1, divided according to the three experimental categories discussed in the text (i.e. the three classes whicl differ in the training-set to test-set ratio)* * For each of the 45 evolved systems, the table lists its fitness value, its performance, and its average number of variables-per-rule. As explained in Section 5.1, the performance value denotes the percentage of cases correctly classified. Three such performance values are shown, (1) performance over the training set; (2) performance over the test set; and 3) overall performance, considering the entire database.
Related topics:
EngineeringGeneticsAlgorithmsArtificial IntelligenceArtificial Intelligence in MedicineEvolutionary AlgorithmFuzzy SystemBreast Cancer Diagnosis
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