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Fig. 3 An example of the evolving SOM tree along with two chromosomes encoding two hypothetical sub-trees, the leaf nodes of which are connected by the bold solid and dashed lines. Structure, features, and parameters of the membership functions are to be encoded. The structure is determined by a sub-tree and is encoded as a connected graph. Fig. 3 presents an example of the evolving SOM tree along with two chromosomes encoding two hypothetical sub-trees. The hypothetical leaf nodes of the two sub-trees are shown connected by the bold solid and the dashed line, respectively, in Fig. 3. There are as many sections in the chromosome, as there are leaf nodes in the corresponding sub-tree (the number of fuzzy rules in the classifier).

Figure 3 An example of the evolving SOM tree along with two chromosomes encoding two hypothetical sub-trees, the leaf nodes of which are connected by the bold solid and dashed lines. Structure, features, and parameters of the membership functions are to be encoded. The structure is determined by a sub-tree and is encoded as a connected graph. Fig. 3 presents an example of the evolving SOM tree along with two chromosomes encoding two hypothetical sub-trees. The hypothetical leaf nodes of the two sub-trees are shown connected by the bold solid and the dashed line, respectively, in Fig. 3. There are as many sections in the chromosome, as there are leaf nodes in the corresponding sub-tree (the number of fuzzy rules in the classifier).

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Fig. 1 Four decision classes in the two-dimensional space. the next section, the fuzzy model is described. The approach proposed is outlined in Section 3. Section 4 discusses the results of the experimental investigations. Section 5 presents conclusions of the work.
Fig. 1 Four decision classes in the two-dimensional space. the next section, the fuzzy model is described. The approach proposed is outlined in Section 3. Section 4 discusses the results of the experimental investigations. Section 5 presents conclusions of the work.
Fig. 2 An example of the evolving SOM tree generated to represent the two-dimensional data.
Fig. 2 An example of the evolving SOM tree generated to represent the two-dimensional data.
Fig. 5 A part of the evolving SOM tree illustrating the mutation operation in structure evolution. Mutation in structure evolution amounts to taking one step up or down (the direction is selected randomly) along a randomly selected branch of the tree. To select the direction, an integer ¥ is selected randomly from the set {—1,1} for each node. The node undergoing the mutation is replaced with its parent node, if 3 < 0 or is split into children, if ¥ > 0. Fig. 5 illustrates the mutation operation in structure evolution.
Fig. 5 A part of the evolving SOM tree illustrating the mutation operation in structure evolution. Mutation in structure evolution amounts to taking one step up or down (the direction is selected randomly) along a randomly selected branch of the tree. To select the direction, an integer ¥ is selected randomly from the set {—1,1} for each node. The node undergoing the mutation is replaced with its parent node, if 3 < 0 or is split into children, if ¥ > 0. Fig. 5 illustrates the mutation operation in structure evolution.
Fig. 4 Three sub-sections (f;, ¢;, and s;) of the jth node encoding features, centers and width of the membership functions, respectively.
Fig. 4 Three sub-sections (f;, ¢;, and s;) of the jth node encoding features, centers and width of the membership functions, respectively.
Fig. 6 A part of the evolving SOM tree illustrating the verification operation in structure evolution. After the genetic operations, the sub-trees are verified for consistency. The redundant active nodes are deactivated during the verification. Fig. 6 illustrates the verification operation. Nodes being active before the verification operation, are shown in grey and connected by the dashed line, in Fig. 6. Nodes being active after the verification operation are shown connected by the bold line. As can be seen from Fig. 6, nodes 4, 6, 8, 9, 12, and 13 are left unaffected, while nodes 16 and 17 are deactivated, since node 11, predecessor of nodes 16 and 17, was activated during the mutation operation The deactivated nodes are labelled by x, in Fig. 6.
Fig. 6 A part of the evolving SOM tree illustrating the verification operation in structure evolution. After the genetic operations, the sub-trees are verified for consistency. The redundant active nodes are deactivated during the verification. Fig. 6 illustrates the verification operation. Nodes being active before the verification operation, are shown in grey and connected by the dashed line, in Fig. 6. Nodes being active after the verification operation are shown connected by the bold line. As can be seen from Fig. 6, nodes 4, 6, 8, 9, 12, and 13 are left unaffected, while nodes 16 and 17 are deactivated, since node 11, predecessor of nodes 16 and 17, was activated during the mutation operation The deactivated nodes are labelled by x, in Fig. 6.
Fig. 7 A part of the evolving SOM tree illustrating the crossover operation in structure evolution.
Fig. 7 A part of the evolving SOM tree illustrating the crossover operation in structure evolution.
Fig. 8 Parameters of two parent chromosomes are exchanged at the crossover point. The mutation operation is accomplished by reversing the value of a bit in the “feature mask” (f sub-section) and by adding a random value from a given, symmetric around zero, interval to parameters stored in the slot (gene) selected for mutation in the c and s sub-sections. Selection of genes for mutation is performed independently in the three sub-sections.
Fig. 8 Parameters of two parent chromosomes are exchanged at the crossover point. The mutation operation is accomplished by reversing the value of a bit in the “feature mask” (f sub-section) and by adding a random value from a given, symmetric around zero, interval to parameters stored in the slot (gene) selected for mutation in the c and s sub-sections. Selection of genes for mutation is performed independently in the three sub-sections.
Fig. 9 Examples of pavement tile surfaces: a quality surface on the left and three defective surfaces. Pavement tiles surface inspection problem. A pavement tile surface is to be assigned into a quality or defective class. Features for the classification are extracted from a camera image. Five features characterizing the image texture and the grey level distribution [63] have been used to design a classifier. Fig. 9 presents four examples of pavement tile surfaces used in the study. In total, 200 quality and 200 defective surfaces were available.
Fig. 9 Examples of pavement tile surfaces: a quality surface on the left and three defective surfaces. Pavement tiles surface inspection problem. A pavement tile surface is to be assigned into a quality or defective class. Features for the classification are extracted from a camera image. Five features characterizing the image texture and the grey level distribution [63] have been used to design a classifier. Fig. 9 presents four examples of pavement tile surfaces used in the study. In total, 200 quality and 200 defective surfaces were available.
Table 1 The average test data set classification accuracy (%) obtained from different classifiers using all available features. In the next set of experiments, feature selection was activated and features, spe- cific for each rule were found through the genetic search. Table 2 presents the average test data set classification accuracy obtained from the approach proposed using the selected features. The classification accuracy obtained using all the available features
Table 1 The average test data set classification accuracy (%) obtained from different classifiers using all available features. In the next set of experiments, feature selection was activated and features, spe- cific for each rule were found through the genetic search. Table 2 presents the average test data set classification accuracy obtained from the approach proposed using the selected features. The classification accuracy obtained using all the available features
Table 3 The number of rules and features used to classify data from the different data sets. Below given is an example of a fuzzy rule (one out of ten) generated by the propose technique for classification of the Diabetes set data.
Table 3 The number of rules and features used to classify data from the different data sets. Below given is an example of a fuzzy rule (one out of ten) generated by the propose technique for classification of the Diabetes set data.
Table 2 The average test data set classification accuracy (%) obtained from the approacl proposed using all and selected features, along with the average accuracy obtained by othe authors in recent studies.
Table 2 The average test data set classification accuracy (%) obtained from the approacl proposed using all and selected features, along with the average accuracy obtained by othe authors in recent studies.
Table 4 The average number of rules and features used to classify data from the different data sets and the average test data set classification accuracy (%) obtained using an ordinary GA-based feature selection procedure. Next, the influence of crossover and mutation probabilities, pe and pm, on classifi- ation accuracy was studied. The same pce and pm values were used for both structure nd parameter evolution. To reduce the computation time and to decouple the influ- nce of pe and/or pm, and feature selection on the classification accuracy, the studies vere performed without employing feature selection. A similar performance was ob- erved for pm values raging from 0.005 to 0.05. A value of pm = 0.01 was selected. Vhen studying the influence of pc, pm was set to pm = 0. Table 5 presents the av- rage test set classification accuracy obtained for different pe values. The interval ot c = [0.8,1.0] was studied additionally and was found that the highest and simila1 lassification accuracy is obtained for pe = 0.9 and pe = 0.95. Thus, pe values close tc nity are recommended.
Table 4 The average number of rules and features used to classify data from the different data sets and the average test data set classification accuracy (%) obtained using an ordinary GA-based feature selection procedure. Next, the influence of crossover and mutation probabilities, pe and pm, on classifi- ation accuracy was studied. The same pce and pm values were used for both structure nd parameter evolution. To reduce the computation time and to decouple the influ- nce of pe and/or pm, and feature selection on the classification accuracy, the studies vere performed without employing feature selection. A similar performance was ob- erved for pm values raging from 0.005 to 0.05. A value of pm = 0.01 was selected. Vhen studying the influence of pc, pm was set to pm = 0. Table 5 presents the av- rage test set classification accuracy obtained for different pe values. The interval ot c = [0.8,1.0] was studied additionally and was found that the highest and simila1 lassification accuracy is obtained for pe = 0.9 and pe = 0.95. Thus, pe values close tc nity are recommended.
Table 5 The average test set classification accuracy (%) obtained for different p. values. One and the same loop can be used for both structure and parameter evolution of the classifier, presumably at the expense of computation time. An experiment was performed to compare these two implementations using the Diabetes data set. A very similar test data set classification accuracy was achieved in both implementations. However, the evolution based on the two-loop implementation was about five times faster.
Table 5 The average test set classification accuracy (%) obtained for different p. values. One and the same loop can be used for both structure and parameter evolution of the classifier, presumably at the expense of computation time. An experiment was performed to compare these two implementations using the Diabetes data set. A very similar test data set classification accuracy was achieved in both implementations. However, the evolution based on the two-loop implementation was about five times faster.
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