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15.2.2 Experimentation
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To demonstrate the applicability of the proposed approach, both the fuzzy and fuzzy-rough decision tree induction methods were applied to a variety of benchmark datasets obtained from [38]. This section presents the results of experimentation carried out on these datasets. In order for both decision tree inducers to operate, fuzzy sets must rst be de ned for real-valued attributes that appear in the data. For this, a simple fuzzi cation was carried out based on the statistical properties of the attributes themselves. It is expected that the classi cation performance would be greatly improved if the fuzzi cations were optimized. The datasets were then split into two halves of equal size, one for training and the other for testing, while maintaining the original class distributions. To show the general applicability of both approaches, the inducers were also applied to nonfuzzy data. Datasets WQ2class and WQ3class are the water treatment datasets with the original 13 classes collapsed into 2 or 3, respectively [165].
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FUZZY-ROUGH DECISION TREES
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TABLE 15.2 Classi cation accuracies for fuzzy ID3 and fuzzy-rough ID3 (real-valued data) Dataset F-ID3 FR-ID3 Train Test Train Test Iris 0.973 0.947 0.973 0.947 0.514 0.495 0.523 0.476 Glass Credit 0.890 0.849 0.742 0.641 0.824 0.787 0.824 0.782 WQ2class WQ3class 0.816 0.696 0.801 0.722 Ionosphere 0.862 0.783 0.852 0.783 0.678 0.590 0.695 0.656 Olitos
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TABLE 15.3 Number of rules produced (fuzzy data) Dataset F-ID3 FR-ID3 Iris 10 13 Glass 32 44 51 50 Credit WQ2class 108 140 WQ3class 165 328 22 28 Ionosphere Olitos 9 17
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As can be seen from Table 15.2, both approaches perform similarly for the fuzzy data. The size of the resultant rulesets that produce these accuracies can be found in Table 15.3. In general, FR-ID3 produces slightly larger rulesets than the standard approach. There is a notable difference in performance for the Credit dataset, where FR-ID3 produces a reduced classi cation accuracy. From Table 15.3 it can be seen that this is the only case where FR-ID3 produces a smaller ruleset. The paired t-tests for training and testing results for F-ID3 and FR-ID3 produce the p-values 0.368 and 0.567, respectively. The results of the application of fuzzy and fuzzy-rough ID3 to crisp data can be found in Table 15.4, with the resulting ruleset size in Table 15.5. The results show that FR-ID3 outperforms F-ID3 in general, as well as producing smaller
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TABLE 15.4 Classi cation accuracies for fuzzy ID3 and fuzzy-rough ID3 (crisp data) Dataset F-ID3 FR-ID3 Train Test Train Test Derm 0.514 0.257 0.838 0.615 Derm2 0.932 0.818 0.972 0.906 0.575 0.348 0.475 0.386 DNA Heart 0.826 0.772 0.826 0.793 0.798 0.625 0.698 0.563 WQ-disc
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TABLE 15.5 Number of rules produced (crisp data) Dataset F-ID3 FR-ID3 Derm 53 46 Derm2 20 18 25 22 DNA 25 30 Heart WQ-disc 28 28
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rulesets. Here the paired t-tests for the training and testing results produce the p-values 0.695 and 0.283, respectively.
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15.3 FUZZY-ROUGH RULE INDUCTION
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One of the most successful areas of application of rough set theory is rule induction. The properties of rough sets allow the automatic derivation of rules from datasets based on the data contents alone. This is achieved primarily through the calculation of lower approximations (generating certain rules) and upper approximations (generating possible rules). However, current methods are only useful for datasets containing nominal values. The proposed fuzzy-rough method will use fuzzy-rough lower and upper approximations with extensions of other rough set concepts in order to induce fuzzy rules from continuous data. In traditional rough set theory, rules are constructed by examining attributevalue pairs in the dataset and their relation to lower and upper approximations of concepts. Concepts are de ned here as collections of objects with the same decision value in datasets. In order to locate the relevant pairs, rule inducers make use of the de nitions of minimal complex and local covering for concepts [121]. These determine the smallest set of attribute-value pairs that may cover a concept. A similar strategy can be adopted for fuzzy-rough rule induction through extensions of these de nitions. In this case the building blocks of fuzzy rules are the pairs of attribute and fuzzy linguistic values. These values need to be determined beforehand either elicited from experts or derived automatically. The fuzzy lower approximations of concepts can be calculated based on given attribute subsets. Such approximations have been found to be useful in feature selection [162]. The approximations generated by subsets associate with an object a degree of membership to the approximation. These memberships indicate the ability of a subset (and corresponding attribute-value pairs) to approximate a concept. In crisp rough set rule induction the memberships are either 0 or 1, with those attribute-value pairs that produce a maximum membership used to form certain rules. In the fuzzy-rough case, an alternative strategy must be employed as memberships lie in the range [0, 1]. One possible way of accomplishing this is to use -cuts to return crisp subsets. A more interesting approach would be to employ the information contained in the memberships to gauge the strength of the generated rules. Fuzzy upper approximations could be used to derive possible rules in a similar way.
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