FUZZY-ROUGH DECISION TREES in VS .NET

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FUZZY-ROUGH DECISION TREES
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RSAR-SAT requires more preprocessing time. Included in this table are the number of clauses appearing in the resultant discernibility function for the RSAR-SAT method. The average times of the execution of these algorithms are also presented in Table 15.1. The time taken for RSAR-SAT is split into two columns. The rst indicates the average length of time taken to nd the minimal subset, the second how long it takes to verify that this is indeed minimal. For RSAR an asterisk next to the time indicates that it found a nonminimal reduct. The results show that RSAR-SAT is comparable to RSAR in the time taken to nd reducts. However, RSAR regularly fails to nd the smallest optimal subset, because of its being misled in the search process. For larger datasets the time taken for RSAR-SAT veri cation exceeds that of RSAR. Note that the veri cation stage involves simple chronological backtracking. There are ways in which this can be made more effective and less time-consuming. DPLL resorts to chronological backtracking if the current assignment of variables results in the unsatis ability of F . Much research has been carried out in developing solution techniques for SAT that draws on related work in solvers for constraint satisfaction problems (CSPs) [21,370]. Indeed the SAT problem can be translated to a CSP by retaining the set of Boolean variables and their {0, 1} domains, and to translate the clauses into constraints. Each clause becomes a constraint over the variables in the constraint. Unit propagation can be seen to be a form of forward checking. In CSPs more intelligent ways of backtracking have been proposed such as backjumping, con ict-directed backjumping, and dynamic backtracking. Many aspects of these have been adapted to the SAT problem solvers. In these solvers, whenever a con ict (deadend) is reached, a new clause is recorded to prevent the occurrence of the same con ict again during the subsequent search. Nonchronological backtracking backs up the search tree to one of the identi ed causes of failure, skipping over irrelevant variable assignments. With the addition of intelligent backtracking, RSAR-SAT should be able to handle datasets containing large numbers of features. As seen in the preliminary results, the bottleneck in the process is the veri cation stage the time taken to con rm that the subset is indeed minimal. This requires an exhaustive search of all subtrees containing fewer variables than the current best solution. Much of this search could be avoided through the use of more intelligent backtracking. This would result in a selection method that can cope with many thousands of features, while guaranteeing resultant subset minimality something that is particularly sought after in feature selection.
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15.2 15.2.1
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FUZZY-ROUGH DECISION TREES Explanation
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A decision tree can be viewed as a partitioning of the instance space. Each partition, represented by a leaf, contains the objects that are similar in relevant
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respects and thus are expected to belong to the same class. The partitioning is carried out in a data-driven manner, with the nal output representing the partitions as a tree. An important property of decision tree induction algorithms is that they attempt to minimize the size of the tree at the same time as they optimize a certain quality measure. The general decision tree induction algorithm is as follows: The signi cance of features is computed using a suitable measure (in C4.5 this is the information gain metric [281]). Next the most discriminating feature according to this measure is selected and the dataset partitioned into sub-tables according to the values this feature may take. The chosen feature is represented as a node in the currently constructed tree. For each sub-table the procedure above is repeated, namely to determine the most discriminating feature and split the data into further sub-tables according to its values. This is a similar process in the fuzzy case. However, a measure capable of handling fuzzy terms (instead of crisp values) must be used. Data are partitioned according to the selected feature s set of fuzzy terms. There must also be a way of calculating the number of examples that belong to a node. In the crisp case, this is clear; objects either contain a speci c attribute value or they do not. In the fuzzy case, this distinction can no longer be made, as objects may belong to several fuzzy terms. A suitable stopping condition must also be chosen that will limit the number of nodes expanded. Clearly, one important aspect of this procedure is the choice of feature signi cance measure. This measure in uences the organization of the tree directly and will have profound effects on the resulting tree s accuracy. Given the utility of the fuzzy-rough dependency measure as an estimator of feature importance, it should be useful as a feature signi cance measure in decision tree construction. This section presents some initial investigations comparing the standard fuzzy ID3 approach [157,363] (F-ID3) using fuzzy entropy with an adapted fuzzy ID3 algorithm that employs the fuzzy-rough measure (FR-ID3).
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