Ralph Vince - Portfolio Mathematics

The Leverage Space Model 319

has a scaled fitness score of .05, the second has one of .1, and the

third .08, then they are set up in a selection scheme as follows:

First candidate 0 to .05

Second candidate .05 to .15

Third candidate .15 to .23

This continues until the last candidate has its upper limit at 1.0.

Now, two random numbers are generated between 0 and 1, with

the random numbers determining from the preceding selection

scheme who the two parents will be. Two parents must now be

selected for each candidate solution of the next generation.

c.Crossover. Go through each bit of thechild, the new population

candidate. Start by copying the first bit of the first parent to the

first bit of the child. At each bit carryover, you must also generate

a random number. If the random number is less than or equal to

(probability of crossover/gene length), then switch to copying the

bits over from the other parent. Thus, if we have three scenario

spectrums and 12 bits per each variable, then the gene length is 36.

If we use a probability of crossover of .6, then the random number

generated at any bit must be less than .6/36, or less than .01667, in

order to switch to copying the other parent’s code for subsequent

bits. Continue until all the bits are copied to the child. This must be

performed for all new population candidates.

Typically, probabilities of crossover are in the range .6 to .9. Thus,

a .9 probability of crossover means there is a 90% chance, on average,

that there will be crossover to the child, that is, a 10% chance the

child will be an exact replicant of one of the parents.

d.Mutation. While copying over each bit from parent to child, gener-

ate a second random number. If this random number is less than

or equal to the probability of mutation, then toggle that bit. Thus, a

bit which is 0 in the parent becomes 1 in the child and vice versa.

Mutation helps maintain diversity in the population. The probability

of mutation should generally be some small value (i.e.,<=.001); oth-

erwise the algorithm tends to deteriorate into a random search. As

the algorithm approaches an optimum, however, mutation becomes

more and more important, since crossover cannot maintain genetic

diversity in such a localized space in then+1 terrain.

Now you can go back to step three and perform the process for the next
generation. Along the way, you must keep track of the highest objective
function returned and its corresponding gene. Keep repeating the process