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(8.8)
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TABLE 8.4 Experimental comparison of the two formulations for the calculation of the positive region Dataset Number of Equation (8.8) Equation (8.9) Optimized Features (seconds) (seconds) (seconds) Glass 10 29.5 26.7 7.18 14 5.41 3.05 2.20 Wine Olitos 26 47.6 21.9 13.0 27 19.2 5.75 2.72 JobSat Ionosphere 35 204.5 107.8 76.9 Selwood 54 57.5 15.9 5.64 618 368.4 131.9 47.2 Isolet Phenetyl 629 740.7 145.0 70.3 714 2709.3 213.5 114.1 Caco
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where P is a subset of the conditional attributes, Q the decision attribute(s). In order to speed up computation time, equation (8.8) can be rewritten as POS P (Q) (x) = sup min( F (x), sup { inf max(1 F (y), X (y))})
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(8.9) This reformulation helps to speed up the calculation of the fuzzy positive region by considering each fuzzy equivalence class F in U/P rst. If the object x is found not to belong to F , the remainder of the calculations for this class need not be evaluated, due to the use of the min operator. This can save substantial time, as demonstrated in Table 8.4, where the two de nitions of the positive region are used to determine reducts from several small to large datasets. The times here are the times taken for each version of FRFS to nd a reduct. Each version of FRFS will follow exactly the same route and will locate identical reducts, hence the results are comparable. All the datasets are from the Machine Learning Repository [38] and contain real-valued conditional features with nominal classi cations. Additionally in Table 8.4 average runtimes are given for the optimized implementation of the fuzzy-rough feature selector. This includes the use of the algorithm presented in Figure 8.4, which is designed to result in the faster computation of the fuzzy-rough metric for small feature subsets. Excess computation is avoided at lines (4) and (6), which exploit the nature of t-norms and s-norms.
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In order to evaluate the utility of the new fuzzy-rough measure of feature signi cance, a series of arti cial datasets were generated and used for comparison with 5 other leading feature ranking measures. The datasets were created by
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CALCULATEGAMMA (C,D,P) Input: C, the set of all conditional features; D, the set of decision features; P, feature subset for evaluation Output: P (D) (1) m[], 0 (2) foreach F U/P (3) deg = supX U/D {infy U max{1 F (y), X (y)}} (4) if deg = 0 (5) foreach o U (6) if m[o] = 1 and deg > m[o] (7) m[o] = max(min( F (o), deg), m[o]) (8) foreach o U (9) += m[o] (10) return
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generating around 30 random feature values for 400 objects. Two or three features (referred to as x, y, or z) are chosen to contribute to the nal Boolean classi cation by means of an inequality. For example, in Table 8.6, if the inequality (x + y)2 > 0.25 holds for an object, then it is classi ed as 1, with a classi cation of 0 otherwise. The task for the feature rankers was to discover those features that are involved in the inequalities, ideally rating the other irrelevant features poorly in contrast. The tables presented in the metric comparison section show the ranking given to the features that are involved in the inequality that determines the classi cation. The nal row indicates whether all the other features are given a ranking of zero. The full results can be seen in Appendix A. For the data presented in Table 8.5, the rst feature, x, is used to determine the classi cation. The values of features y and z are derived from x: y = x, z = x 2 .
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