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# Application of an Adaptive Differential Evolution Algorithm ... - Koszalin

Application of an Adaptive Differential Evolution Algorithm ... - Koszalin

## SLOWIK: APPLICATION OF

3162 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 8, AUGUST 2011where F ∈ [0, 2), and r1,r2,r3,i∈ [1,PopSize] fulfill theconstraintr1 ≠ r2 ≠ r3 ≠ i. (2)Fig. 1. Part of (a) ANN, corresponding to its (b) chromosome containing theweight values; weights w i,0 represent bias weights [15].the optimized function [1], [10]. Another important property ofthis algorithm is a local limitation of the selection operator toonly two individuals (parent (x i ) and child (u i )), and, owingto this property, the selection operator is more effective andfaster [10]. Also, to accelerate the convergence of the algorithm,it is assumed that the index r 1 (occurring in the algorithmpseudocode) points to the best individual in the population.B. DE Algorithm in ANN TrainingIn the literature, we can find several applications of the DEalgorithm to ANN training. For example, we can mention [30]and [31]. In these papers, the DE algorithm without adaptiveselection of control parameters was used in ANN training.Therefore, the main problem in these papers was the tuningof the algorithm parameters. This problem was overcome in[15], in which the adapted DE algorithm [11] was used in ANNtraining (DE-ANNT algorithm). Due to the use of DE-ANNT,the tuning of primary DE parameters, such as F and CR, is notneeded.IV. DE-ANNT+ METHODThe proposed DE-ANNT+ method is based on the previouslyelaborated DE-ANNT method [15] and operates accordingto the following steps:In the first step, a population of individuals is randomlycreated. The number of individuals in the population is storedin parameter PopSize. Each individual x i consists of k genes(where k represents the number of weights in the trained ANN).In Fig. 1(a), a part of an ANN with neurons from n to mis shown. Additionally, in Fig. 1(b), the coding scheme forweights in an individual x i connected to neurons from Fig. 1(a)is shown.Each jth (j ∈ [1,k]) gene of individual x i can have valuesfrom a determined range of variability (closed double-sided)from min j to max j . In the proposed method, the values ofmin j = −1 and max j =1are assumed.In the second step, the NT (number of trial vectors) mutatedindividuals (trial vectors) V i,m (m ∈ [1,NT]) are created foreach individual x i in the population, according to the formulaV i,m = x r1 + F · (x r2 − x r3 ) (1)Indexes r2 and r3 point to individuals randomly chosen fromthe population. Index r1 points to the best individual in thepopulation, which has the lowest value of the training errorfunction ERR(.). This function is described as follows:ERR = 1 2 ·T∑(Correct i − Answer i ) 2 (3)i=1where i is the actual number of training vector, T is the numberof all training vectors; Correct i is the required correct answerfor the ith training vector, and Answer i is the answer generatedby the neural network for the ith training vector applied toits input. The DE-ANNT+ method minimizes the value of theobjective function ERR(.).From the created set of mutated vectors V i,m , only one vectorV i,m (individual), having the lowest value of the objectivefunction ERR(.), is chosen for each individual x i , and it isassigned as vector v i .In the third step, all individuals x i are crossed over with theirmutated individuals v i . As a result of this crossover operation,an individual u i is created. The crossover operates as follows:for chosen individual x i =(x i,1 ,x i,2 ,x i,3 ,...,x i,j ) and individualv i =(v i,1 ,v i,2 ,v i,3 ,...,v i,j ); for each gene j ∈ [1; k]of individual x i , randomly generate a number rand j from therange [0; 1), and use the following rule:If rand j < CR then u i,j = v i,jElse u i,j = x i,jwhere CR ∈ [0; 1).In this paper, an adaptive selection of control parametervalues F and CR is introduced (similarly as in [11]) accordingto the formulasA =T heBest iT heBest i−1(4)F =2· A · random (5)CR = A · random (6)where random—the random number with a uniform distributionin the range [0; 1); T heBest i —the value of theobjective function for the best solution in ith generation;T heBest i−1 —the value of the objective function for the bestsolution in the i − 1th generation.From (5) and (6), we can see that, in the case of a stagnation(lack of changes of the best solution), the F parameter takesrandom values from the range [0; 2), and the CR parametertakes random values from the range [0; 1). In such a case, thesearching of the solution space has a more global character, andthe DE algorithm may “get out” more easily from the localextreme that is causing its stagnation. However, in the casewhere the results obtained by the DE algorithm are improvingin subsequent generations, then the F parameter accepts randomvalues from the range [0; 2 · A), and the CR parameteraccepts random values from the range [0; A). Obviously, the

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