Intelligent systems in accounting, finance and managementassessing predictive performance of ann-based classifiers, страница 10

Reproduction

The selected chromosomes are randomly paired and recombined to produce new solutions. There are two reproduction operators: crossover and mutation. With the first, the mates are recombined and newborn solutions inherit information from both parents. With the second operator, new parts of the search space are explored and, consequently, we expect that new information is introduced into the population. In this study we have applied four types of crossover: arithmetic, one-point, multipoint and uniform crossover. Let us denote with L the length of the chromosomes and with P₁ and P₂ two parent-chromosomes:

P₁ = g₁₁, g₁₂, . . . , g_1L

P₂ = g₂₁, g₂₂, . . . , g_2L

One-point crossover. For each pair of chromosomes we generate a random integer X, X ∈ {1, L}. The two newborn children are constructed as follows:

C1 = g11, g12, . . . , g1X, g2,X+1, . . . , g2L

C2 = g21, g22, . . . , g2X, g1,X+1, . . . , g1L

Multi-point crossover. We split the chromosomes in n parts (n ≤ 5). We generate randomly the number of splitting points n. Then, n distinct random numbers (X₁, X₂, . . . , X_n) are generated with

X_i ∈ {1, L} and X₁ < X₂ < . . . < X_n. The two children are

  C1  = g11, g12,⋅ ⋅ ⋅, g1X1, g2,X1+ 1,⋅ ⋅ ⋅, g2X2, g1,X2+ 1,⋅ ⋅ ⋅, g1X3, g2,X3+ 1, ⋅ ⋅ ⋅

  C2  = g21, g22,⋅ ⋅ ⋅, g2X1, g1,X1+ 1,⋅ ⋅ ⋅, g1X2, g2,X2+ 1,⋅ ⋅ ⋅, g2X3, g1,X3+ 1,⋅ ⋅ ⋅

Arithmetic crossover. First, we split the parent chromosomes in n parts as we did for multipoint crossover. The children’s genes are convex combinations of the parents’ genes.

α  × g_1i  (  )  + 1 − α × g_2i      , i = 1 X₁

C₁  = (1  )  − α × g_1i    + α × g_2i      i = X₁  , + 1 X₂

α  × g_1i  (  )  + 1 − α × g_2i      i = X₂  , + 1 X₃

(1  )  − α × g_1i    + α × g_2i      , i = 1 X

C₂  = α  × g_1i  (  )  + 1 − α × g_2i      i = X₁  , + 1 X₂

(1  )  − α × g_1i    + α × g_2i      i = X₂  , + 1 X₃

where α ∈ [0, 1] is a random number and is generated for each chromosome-pair.

Uniform crossover. For each pair of genes of the parent chromosomes we generate a random number α ∈ [0, 1]. If α < 0.5 then the gene of the first parent goes to the first child and the gene of the second parent goes to the second child. Otherwise, the genes are inversed.

The children chromosomes are added to the population. The size of the population becomes PS′ > PS. Next, we apply the mutation operator for all the chromosomes in PS′. We used only uniform mutation.

Uniform mutation. The probability of mutation is set to P_m = 0.01, which means that approximately 1% of the genes will mutate for each chromosome. An α ∈ [0, 1] is generated for each gene of each chromosome and, if α ≤ P_m, the new gene is randomly generated within the variable domain. Otherwise, the gene remains the same. If at least one gene is changed then the new chromosome is added to the population, obtaining PS″ > PS′ > PS. As in the case of crossover probability, the proper setting of mutation probability depends on the population model, the problem to be solved and fitness function (Tuson and Ross, 1998). Tuson and Ross (1998) found no difference between fixed and adapted mutation rates: when the initial mutation rate was ‘close to theoretically optimal value—3/chrom length’ then ‘the speed to solution was improved’. DeJong (1975) considered mutation probability to be inversely proportional to the population size. Hesser and Männer (1990) included both population size and chromosome length in the calculation of mutation probability. Hoehn (1998) introduced mutation at both parental and offspring levels and implemented four GAs based on the mutation probabilities for the two levels: standard GA (no mutation for parental level and 0.001 for offspring level), low GA (0.001 for both parental and offspring levels), high GA (0.1 for parental and 0.001 for offspring levels), and variable GA (from 0.001 to 0.1 for parental and 0.001 for the offspring levels). The four GAs were compared in terms of their performances in optimizing De Jong’s (1975) functions F1–F5 (Hoehn, 1998: 222). Hoehn (1998) found that introducing parental mutation is generally advantageous when compared with the standard GA with only offspring mutation. In our experiments we used both parental and offspring mutation by applying mutation on both parents and their offspring. This operation was possible since after we apply crossover operation we add the new chroms (offsprings) to the population and keep their parents. Consequently, the mutation is applied at both levels, i.e. parental and offspring. Hoehn (1998) gives us an idea of what constitutes a very low mutation probability (0.001) and a very high one (0.1), but his results do not help in choosing between low and high mutation probabilities. For some of DeJong’s functions (F3, F4) a GA with low mutation rate performed better than a GA with high mutation rate, whereas for others (F2) it was the opposite. Correspondingly, throughout our experiments we used a ‘moderate’ mutation probability (0.01), the choice of which was, also, based upon the ‘theoretically optimal value—3/chrom length’ (Tuson and Ross, 1998), since our chroms’ lengths vary (depending on the dataset used) around value 200 (3/200 ≅ 0.01).