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Figure 10.4 Overview of a general ACO-based FS system
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10.2.3.2 Complexity Analysis The time complexity of the ant-based approach to feature selection is O(IAk), where I is the number of iterations, A the number of original features, and k the number of ants. This can be seen from Figure 10.4. In the worst case each ant selects all the features. As the heuristic is evaluated after each feature is added to the reduct candidate, this will result in A evaluations per ant. After one iteration in this scenario, Ak evaluations will have been performed. After I iterations the heuristic will be evaluated IAk times. This method is attractive for feature selection as there seems to be no heuristic that can guide search to the optimal minimal subset every time. Additionally it should be the case that ants will discover best feature combinations as they traverse the graph. This information will be re ected in the pheromone matrix used by other ants to guide their own local search. 10.2.3.3 Pheromone Update Depending on how optimality is de ned for the particular application, the pheromone may be updated accordingly. For instance, subset minimality and goodness are two key factors so the pheromone update must be proportional to goodness and inversely proportional to size. To tailor this mechanism to nd fuzzy-rough set reducts, it is necessary to use the dependency measure given in equation (8.2) as the stopping criterion. This means that an ant will stop building its feature subset when the dependency of the subset reaches the maximum for the dataset (the value 1 for consistent datasets). The dependency function may also be chosen as the heuristic desirability measure,
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FURTHER ADVANCED FS METHODS
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but this is not necessary. It may in fact be of more useful to employ a nonrough set related heuristic for this purpose to avoid the pitfalls of a QuickReduct style search. An alternative measure such as an entropy-based heuristic [281] may allow the method to avoid feature combinations that may mislead the fuzzy-rough set-based heuristic. Again, the time complexity of this fuzzy-rough ant-based method will be the same as that mentioned earlier, O(IAk). The pheromone on each edge is updated according to the following formula: ij (t + 1) = (1 ) ij (t) + where
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(10.3)
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This is the case if the edge (i, j ) has been traversed; ij (t) is 0 otherwise. The value is a decay constant used to simulate the evaporation of the pheromone, and S k is the feature subset found by ant k. The pheromone is updated according to both the fuzzy-rough measure of the goodness of the ant s feature subset ( ) and the size of the subset itself. By this de nition, all ants update the pheromone. Alternative strategies may be used for this, such as allowing only the ants with the best feature subsets to proportionally increase the pheromone.
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10.2.3.4 Approach Comparison Upon inspection, a number of similarities and differences can be observed between the approaches described previously. In the genetic and ant-based FS a population of potential solutions are considered on each algorithmic cycle. In the genetic case, this population is evaluated and then used to produce the next population. However, in the ant-based case, each population is erased after generation; the only lasting effect the population has is in the updated pheromone levels that are used in the next generation cycle. Simulated annealing-based FS (discussed in Section 4.2.4) considers an individual solution candidate only. New candidates are produced by mutating the current solution in a similar manner to genetic FS. The extent of mutation decreases with time, allowing convergence to a single, hopefully optimal, solution. All of these approaches employ a certain degree of randomness in their search for optimal subsets.
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