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the deposition of arti cial pheromone on the graph edges by their predecessors over time. The impact of this knowledge on an ant s traversal decision is determined by the parameter . Good paths discovered by past ants will have a higher amount of associated pheromone. How much pheromone is deposited, and when, is dependent on the characteristics of the problem. No other local or global knowledge is available to the ants in the standard ACO model. However, the inclusion of such information by extending the ACO framework has been investigated [40] with some success.
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10.2.2 Traveling Salesman Problem
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To illustrate how ACO may be applied to arti cial systems, its application to the traveling salesman problem (TSP) [198] is presented here. The TSP is a combinatorial optimization problem where, given N cities and a distance function d between cities, a minimal tour that goes through every city exactly once is to be found. The TSP is represented as a graph, with nodes representing cities and edges representing journies between cities. The heuristic desirability of edge (i, j ) is the inverse of the distance between those cities (1/d(i, j )), where i = j . Pheromone is increased proportional to the inverse of the tour length. These two measures can be thought of as providing different information about the problem: the heuristic measure provides local information and pheromone global information. The information is combined to form the so-called probabilistic transition rule, denoting the probability of an ant in city i choosing to travel to city j at time t:
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k pij (t) =
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[ ij (t)] .[ ij ] l J k [ il (t)] .[ il ]
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(10.1)
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where k is the number of ants, Jik the set of k s possible next cities, ij the heuristic desirability of choosing city j when at city i, and ij (t) the amount of virtual pheromone on edge (i, j ). The choice of and is determined experimentally. Typically several experiments are performed by varying each parameter and choosing the values that produce the best results. With this representation a number of ants may be placed at nodes (cities) in the graph and traverse edges to form a tour. The pheromone of the edges corresponding to the best tours are reinforced, and a new set of ants make their way through the graph. This process continues until an optimum has been discovered or a certain number of generations of ants has been tried.
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10.2.3 Ant-Based Feature Selection
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By following similar principles, the feature selection task may be reformulated into an ACO-suitable problem [160]. ACO requires a problem to be represented as a graph here nodes represent features, with the edges between them denoting
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FURTHER ADVANCED FS METHODS
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ACO problem representation for FS
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the choice of the next feature. The search for the optimal feature subset is then an ant traversal through the graph where a minimum number of nodes are visited that satis es the traversal stopping criterion. Figure 10.3 illustrates this setup the ant is currently at node a and has a choice of which feature to add next to its path (dotted lines). It chooses feature b next based on the transition rule, then c and then d. Upon arrival at d, the current subset {a, b, c, d} is determined to satisfy the traversal stopping criterion (e.g., a suitably high classi cation accuracy has been achieved with this subset). The ant terminates its traversal and outputs this feature subset as a candidate for data reduction. A suitable heuristic desirability of traversing between features could be any subset evaluation function, for example, an entropy-based measure [281] or the fuzzy-rough set dependency measure [164]. 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 should be proportional to goodness and inversely proportional to size. The transition rule is the same as that given in equation (10.1), but Jik in this case is the set of ant k s unvisited features. There is also the possibility of allowing the removal of features here. If feature h has been selected already, an alternative transition rule may be applied to determine the probability of removing this attribute. However, this is an extension of the approach and is not necessary to perform feature selection.
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10.2.3.1 Selection Process The overall process of ACO feature selection can be seen in Figure 10.4. The process begins by generating a number of ants, k; the ants are then placed randomly on the graph (i.e., each ant starts with one random feature). Alternatively, the number of ants to place on the graph may be set equal to the number of features within the data; each ant starts path construction at a different feature. From these initial positions, they traverse edges probabilistically until a traversal stopping criterion is satis ed. The resulting subsets are gathered and then evaluated. If an optimal subset has been found or the algorithm has executed a certain number of times, then the process halts and outputs the best feature subset encountered. If neither condition holds, then the pheromone is updated, a new set of ants are created, and the process iterates once more.
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