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that the GA was more effective in the search with fewer string evaluations than the single hill climbers The reason for this result could be that each of the hill climbers searches a different space than the other hill climbers, and there is no crossover technique to distribute the good searches among the hill climbers GAs are parallel search techniques that start with a set of random potential solutions and use special search operators (evaluation, selection, crossover, and mutation) to bias the search toward the promising solutions At any given time, unlike any optimization approach, a GA has several promising potential solutions (equal to population size) as opposed to one optimal solution Each population member in a GA is a potential solution A GA starts with a random set of the population An evaluation operator is then applied to evaluate the tness of each individual In the case of learning connection weights for ANN for classi cation, the evaluation function is the number of correctly classi ed cases A selection operator is then applied to select the population members with higher tness (so that they can be assigned higher probability for survival) Under the selection operator, individual population members may be born and be allowed to live or to die Several selection operators are reported in the literature; the operators are proportionate reproduction, ranking selection, tournament selection, and steadystate selection Among the popular selection operators are ranking and tournament selection Both ranking and tournament selection maintain strong population tness growth potential under normal conditions The tournament selection operator, however, requires lower computational overhead In tournament selection, a random pair of individuals is selected and the member with the better tness of the two is admitted to the pool of individuals for further genetic processing The process is repeated in a way that the population size remains constant and the best individual in the population always survives For our case, we used the tournament selection operator After the selection operator is applied, the new population special operators, called crossover and mutation, are applied with a certain probability For applying the crossover operator, the status of each population member is determined Each population member is assigned a status as a survivor or a nonsurvivor The number of population members equal to survivor status is approximately equal to population size (1 probability of crossover) The number of nonsurviving members is approximately equal to population size probability of crossover The nonsurviving members in a population are then replaced by applying crossover operators to randomly selected surviving members Several crossover operators exist We describe and use three different crossover operators in our case The crossover operators used in our research are the one-point crossover in which two surviving parents and a crossover point are randomly selected For each parent, the genes on the right-hand side of the crossover point are exchanged to produce two children The second operator is the uniform crossover In the uniform crossover, two surviving parents are randomly selected and exchanging the genes in the two parents produces two children; probability of exchanging any given gene in a parent is 05
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Thus for every gene in a parent, a pseudorandom number is generated If the value of the pseudorandom number is greater than 05, the genes are exchanged, otherwise they are not exchanged If we have two random surviving parents P1 and P2 (as shown in one-point crossover section) then a child C1 can be produced The third operator is the arithmetic crossover, which consists of producing children in a way that every gene in a child is a convex combination of genes from its two parents Given the two parents P1 and P2 (as illustrated before), a child C1 can be produced Arithmetic crossover ensures that every gene in the child is bounded by the respective genes from both the parents Unlike uniform and one-point crossover, arithmetic crossover provides some local/hill-climbing search (if the parents are on the opposite side of the hill) capability for a GA Arithmetic crossover is a popular crossover operator when GA is used for optimization A mutation operator randomly picks a gene in a surviving population member (with the probability equal to probability of mutation) and replaces it with a real random number Since GAs are population based search procedures, at convergence of population tness, there are several promising solutions Unlike the traditional ANN where only one set of weights exists, GA has several sets of weights (equal to the population size) Since one of the objectives of our research is to minimize over tting and to increase predictive accuracy on holdout data sets, we do not use the best tness population member from the training data set to predict the group membership for the holdout sample The holdout data set contains data that are not used for training ANN, but it has similar properties (kurtosis, group means, etc) as that of the training data We use the availability of several potential solutions to minimize the impact of over tting on the training data set In order to select the population member (set of weights) to predict the group membership for the holdout sample, we identify all the population members that have a similar set of weights as that of the best tness population member on the training data set For the holdout sample, we select a population member that is the average of all the vectors This aggregation reduces the chances of over tting, where the best tness population member from the training data is used for the holdout sample In our experiments, we use all three crossover operators (one at a time) and investigate the performance of a GA when different crossover operators are used Thus, based on the crossover operator, we have three different types of GAs: genetic algorithm with arithmetic crossover called GA(A), genetic algorithm with uniform crossover operator called GA(U), and genetic algorithm with one-point crossover operator called GA(O) Our ANN architecture consists of four input nodes (three inputs one threshold), six hidden nodes ( ve hidden one threshold), and one output node We benchmark the performance of our GA based training of ANN with the results of a back-propagation algorithm based ANN For our architecture, we have a population member de ning length of 26((3 inputs 1 threshold) 5 hidden (5 hidden 1 threshold) 1 output) Data distribution characteristics determine the learning and predictive performance of different techniques for classi cation Speci cally, researchers found that
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