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FDR(C, ) Input: C, the set of all conditional features; , the allowed reduction in fractal dimension Output: R, the feature subset (1) R C; bestDim = 0; dc = calculateFractalDim(C) (2) while dc bestDim (3) T R, (4) foreach x R (5) S R {x} (6) ds = calculateFractalDim(S) (7) if ds > bestDim (8) bestDim = ds ; w x (9) R R {w} (10) return R
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Figure 4.12 FDR algorithm
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An automatic feature grouping technique is proposed in [398] that uses k-means clustering [125] beforehand in order to generate groups. From these groups one or two features are pre-selected for a typical forward search FS method. No feature grouping takes place during the search, however. As yet, no results are available for this approach. In group-wise feature selection (GFS) [255] the feature groups are again calculated beforehand. These groups are then used throughout the subset search instead of individual features. An overview of the algorithm can be seen in Figure 4.13. Here the effect of adding groups of features to the currently considered subset is evaluated at each stage. The group that produces the largest increase in performance is selected, and all features present within the group are added to the current subset. The process continues until the performance is of an appropriate quality. In the evaluation a measurement cost is also considered, although this can be omitted if no measurement information is available or required. In addition to the GFS algorithm presented in Figure 4.13, an extension to it, GNFS, that performs a nested forward search within groups has also been proposed. Instead of selecting the best group, GNFS searches for the best subset within a group and adds this to the currently selected feature subset. Both algorithms perform comparably with their standard individual feature selection method, but with the bene t of reduced computation time. All feature grouping approaches so far have relied on groups being de ned beforehand. Group membership is not changed in the selection process leading to a dependence on a suitably accurate grouping mechanism for good results. In addition the extent of group membership is not considered at all; features either belong or do not belong to single groups. Fuzzy grouping can be used to handle this problem so that features can belong to more than one group with varying
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GFS(G, ) Input: G, the set of feature groups; of subset performance Output: R, the feature subset (1) R {}; A {}; best = 0 (2) while evaluate(R) < (3) foreach group Gi (4) T all features from Gi (5) t = evaluate(R T) (6) if > best (7) A T (8) best = t (9) R R A (10) return R
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Figure 4.13 Group-wise FS algorithm
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degrees of membership. This additional membership information can then be used in the selection process. These ideas motivated the development of the new rough and fuzzy-rough set-based grouping FS technique, detailed in Section 10.1.
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4.2.1.8 Other Approaches Besides the approaches outlined above, a lter method based on ideas from probabilistic reasoning and information theory is proposed in [180]. The central motivation behind this development is the observation that the goal of an induction algorithm is to estimate the probability distributions over the class values. In the same way, feature subset selection should attempt to remain as close as possible to these original distributions. The algorithm performs a backward elimination search, at each stage removing the feature that causes the least change between the distributions. The search stops when the desired number of features remain (speci ed by the user). An important problem with this method is that it requires the features in a dataset to be binary-valued. This constraint is added to avoid the bias toward many-valued features present in entropy-based measures. Also worth mentioning is the Chi2 algorithm [211]. This is in effect a heuristic feature selector that discretizes continuous features and in the process removes irrelevant ones based on the 2 statistic [264]. Feature selection actually takes place only as a side effect of the discretization process; if a feature ends up with all its values mapped to a single discrete value, then it can be removed from the dataset without introduction of inconsistency. 4.2.2 Wrapper Methods
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The Las Vegas wrapper (LVW) algorithm [213] is a wrapper method based on the earlier LVF algorithm [212] (described in Section 4.2.1.3); it can be outlined
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