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CHECK-DISCERNIBILITY(DVD, v, c) Input: DVD, the current Discernibility Vector Dictionary; v, a data vector to test; c, the equivalence class of v Output: DVD, the updated dictionary (1) if DVD[v] c, f, i = DVD[v] (2) (3) if c = c (4) DVD[v] c, f + 1, i (5) return DVD (6) else (7) DVD[v] c, f + 1, false (8) return DVD (9) else (10) DVD[v] c, 1, true (11) return DVD
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5.2.2.3 Further Optimizations Additional speed may be gained by making further observations about the operation of RSAR. Increasingly sophisticated optimizations were implemented as different versions of the QuickReduct algorithm, numbered as follows:
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QuickReduct. The original algorithm, augmented by using DVD-based calculation. QuickReduct II, also known as ReverseReduct. A backward-eliminating version of QuickReduct. QuickReduct III. Unlike the original QuickReduct, this version of the algorithm introduces a minor optimization: the reduction stops immediately as soon as the rst reduct is found. QuickReduct IV. Augmented version of QuickReduct III. In adding further features to the attribute subset, dataset samples are not examined if they are known to be discernible given the current attribute subset. Only heretofore indiscernible samples are used in the calculation of . This reduces the effective size of the dataset as QuickReduct IV approaches a solution. For example, assume a dataset of 150 samples is being processed, and the rst evaluation of dependency yields 120 discernible samples. In adding further attributes to the attribute subset, will only be evaluated for the remaining 30 samples. This depends on the monotonicity of : if samples d and d are discernible given the attribute set A, then they will also be discernible given the attribute set A , where A A QuickReduct V. In selecting an attribute to be added to the candidate reduct, the attribute whose addition leads to the highest is used. This version of
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the algorithm immediately stops calculating if the result is guaranteed to be below the highest result so far. This is tested incrementally by summing the frequencies of indiscernible pairs in the DVD. This optimization saves considerable time, especially where datasets have a signi cant degree of redundancy and/or noise.
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5.2.2.4 Experimental Results The aim of the following experiments is to gauge the implementation s performance when operating on worst-case and average-case scenarios. The worst-case scenario is one in which all remaining attributes have to be examined because they provide almost identical gains. Given a dataset with n conditional attributes, is evaluated n + (n 1) + + 2 + 1 = n (1 + n) times, a O(n2 ) time complexity. Evaluating 2 involves processing each dataset item using check-discernibility clearly, O(l) with respect to the number of dataset items. The average case is described as a dataset in which high-signal and noise/redundant attributes are homogeneously mixed in a dataset. The best, nontrivial case is one where high signal attributes are the rst to be chosen by the algorithm. For brevity and because of its simplicity, the last case is not addressed here. Experimental results were obtained by running different versions of the improved RSAR on arti cially constructed datasets with varying dimensionality and number of samples to gauge ef ciency and complexity. To smooth out timing errors, numerous runs were performed and averaged. Approximate curves are shown interpolating the experimental data. The worst-case runtimes are shown in Figure 5.4, where RSAR performed FS on pure binary noise (each attribute is one bit of noise). The rst graph shows the results of varying the number of attributes (with 20 dataset items); the second graph plots runtime over the number of dataset items (with 20 attributes). O(n2 ) and O(l) complexities are obvious. As expected, QuickReduct V is the best choice for this type of problem. Two different average-case experiments were conducted. In both cases signal-rich attributes (drawn from the Iris dataset [38]) and attributes consisting of pure binary noise are combined so that the signal-rich attributes are separated by noise attributes. The two experiments differ in the layout of the datasets:
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