X U/Q

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R X

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where R is also an equivalence relation on U. This function can then be used to calculate dependencies and thus determine -reducts. The dependency function becomes R, (Q) = |POSR, (Q)| |U|

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ROUGH SET BASED APPROACHES TO FEATURE SELECTION

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It can be seen that the QuickReduct algorithm outlined previously can be adapted to incorporate the reduction method built upon VPRS theory. By supplying a suitable value to the algorithm, the -lower approximation, -positive region, and -dependency can replace the traditional calculations. This more approximate nal reduct may be a better generalization when encountering unseen data. Additionally, setting to 0 forces such a method to behave exactly like standard rough set theory. Extended classi cation of reducts in the VPRS approach may be found in [27,28,189]. However, the variable precision approach requires the additional parameter that has to be speci ed from the start. By repeated experimentation, this parameter can be suitably approximated. Nevertheless, problems arise when searching for true reducts as VPRS incorporates an element of imprecision in determining the number of classi able objects.

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DYNAMIC REDUCTS

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Reducts generated from an information system are sensitive to changes in the system. This sensitivity can be seen when removing a randomly chosen set of objects from the original object set. The reducts frequently occurring in random subtables can be considered to be stable; it is these reducts that are encompassed by dynamic reducts [23]. Let A = (U, C d) be a decision table; then any system B = (U , C d) (U U) is called a subtable of A. If F is a family of subtables of A, then DR(A, F) = Red(A, d) {

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Red(B, d)}

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de nes the set of F-dynamic reducts of A. From this de nition it follows that a relative reduct of A is dynamic if it is also a reduct of all subtables in F. In most cases this is too restrictive, so a more general notion of dynamic reducts is required. By introducing a threshold, 0 1, the concept of (F, )-dynamic reducts can here be de ned: DR (A, F) = {C Red(A, d) : sF (C) } where sF (C) = |{B F : C Red(B, d)}| |F|

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is the F-stability coef cient of C. This lessens the previous restriction that a dynamic reduct must appear in every generated subtable. Now a reduct is considered to be dynamic if it appears in a certain proportion of subtables, determined by the value . For example, by setting to 0.5 a reduct is considered to be

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DYNAMIC REDUCTS

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DYNAMICRED(A, ,its) Input: A, the original decision table; , the dynamic reduct threshold; its, the number of iterations Output: dynamic reducts (1) R {} (2) A calculateAllReducts(A) (3) foreach j = 1...its Aj deleteRandomRows(A) (4) (5) R R calculateAllReducts(Aj ) (6) foreach C A (7) if sF (C, R) then output C

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Figure 5.9 Dynamic reduct algorithm

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dynamic if it appears in at least half of the subtables. Note that if F = {A}, then DR(A, F) = Red(A, d). Dynamic reducts may then be calculated according to the algorithm given in Figure 5.9. First, all reducts are calculated for the given information system, A. Then, the new subsystems Aj are generated by randomly deleting one or more rows from A. All reducts are found for each subsystem, and the dynamic reducts are computed using sF (C, R), which denotes the signi cance factor of reduct C within all reducts found, R. Returning to the example decision table (call this A), the rst step is to calculate all its reducts. This produces the set of all reducts A = {{b, d}, {c, d}, {a, b, d}, {a, c, d}, and {b, c, d}}. The reduct {a, b, c, d} is not included as this will always be a reduct of any generated subtable (it is the full set of conditional attributes). The next step randomly deletes a number of rows from the original table A. From this, all reducts are again calculated. For one subtable the result might be R = {{b, d}, {b, c, d}, {a, b, d}}. In this case the subset {c, d} is not a reduct (though it was for the original dataset). If the number of iterations is set to just one, and if is set to a value less than 0.5 (implying that a reduct should appear in half of the total number of discovered reducts), then the reduct {c, d} is deemed not to be a dynamic reduct. Intuitively, the hope in nding stable reducts is they will be more representative of the real world, meaning it is more likely that they will be reducts for unseen data. A comparison of dynamic and nondynamic approaches can be found in [22], where various methods were tested on extracting laws from decision tables. In the experiments the dynamic method and the conventional RS method both performed well. In fact it appears that the RS method has on average a lower error rate of classi cation than the dynamic RS method. A disadvantage of this dynamic approach is that several subjective choices have to be made before the dynamic reducts can be found (e.g., the choice of the value of ); these values are not contained in the data. Also the huge complexity of nding all reducts within subtables forces the use of heuristic techniques such

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