Supuestos de procedencia de la acción de extinción de dominio

A genetic operator is a process used in GAs to maintain genetic diversity. Genetic diversity is a necessity for the process of evolution. Several genetic operators are proposed in the study of GAs. The most common genetic operators among them are Selection, Crossover, and Mutation.

Selection is used in REGAL, a GA based classification system that produces a set of first order predicate logic rules from a given data set [Giordana and Saitta 1993]. The simplest selection scheme is the roulette-wheel selection, also called stochastic sampling with replacement [Baker 1987; Davis 1987]. This scheme is analogous to spinning a roulette wheel with each member of the population having a slice of the wheel that is proportional to its fitness. In practice, the individual chromosomes are mapped to contiguous segments of a line, such that each chromosome’s segment is equal in size to its fitness. A random number is generated and the chromosome whose segment spans the random number is selected. The process is repeated until the desired number of individuals are obtained for either cross over or mutation. The roulette-wheel selection example is shown in Figure 3.4. To select three chromosomes, the appropriate number of uniformly distributed random numbers (between 0.0 and 1.0) is independently generated as 0.45, 0.11, and 0.56. After selection, chromosome 3,1, and 4 are selected.

Some other selection schemes are; Rank Selection and Steady-State Selection. In rank selection, the chromosomes are ordered according to their fitness values and the desired number of chromosomes is selected from the top. In Steady-State Selection, the fittest individuals are selected first and new individuals are created from these selected ones. The least fit individuals are replaced with the new ones, the other ones become members of the next generation without changes. When creating a new generation, there is always a risk of losing the most fit individuals. Elitism is another example of Selection procedure in GAs where the most fit individuals are copied to the next generation and the rest undergo crossover and mutation. Elitism selection improves the efficiency of GAs and prevents losing the best results.

Crossover is a genetic operation by which two different chromosomes can combine to form two new chromosomes known as offspring. The common way of implementing crossover is to choose the same random position of both chromosomes and exchange their bit information. These two newly formed chromosomes inherit some characteristics from both parents. There are many methods to implement crossover for GAs. The common methods are 1-point crossover, 2-point crossover, arithmetic crossover, and uniform crossover. In 1-point crossover, a binary string from the beginning of the chromosome to the crossover point is copied from the first parent chromosome, the rest is copied from the other parent chromosome. The following example illustrates 1-crossover in two chromosomes.

A more unbiased method for constructing offspring from two chromosomes is known as 2-point crossover. Cutting both the chromosomes at not just one place, but at two places performs this. The offspring is formed by taking first crossover point is copied from the first parent, the part from the first to the second crossover point is copied from the other parent and the rest is copied from the first parent again. An example of a 2-point crossover is shown below.

Chromosome

0.01 0.19 0.30 0.51 0.591 0.85

2 3

1 4 5

Trial 2 1 3

Figure 3.4 Roulette wheel selection

+ =

Chromosome A Chromosome B Offspring

Arithmetic crossover is another way of performing genetic change in offspring. In this crossover scheme some arithmetic operations (e.g. and, or logical operation) on the parent chromosomes are used to form the offspring, rather than an exchange. An example of Arithmetic crossover with the OR logical operation is shown below.

Uniform crossover is another crossover method. This is achieved by creating a random bit mask of the same length as the chromosome. The new chromosome is formed by taking the values in the nth position of the first chromosome if the nth position bit value is 1, otherwise taking the value in the nth position of the second chromosome and again, vice versa. The uniform cross over is illustrated below.

In early theoretical and empirical studies of different cross over methods performance, it was found that the number of crossover points should be fixed at a very low constant value of 1 or 2 for better results [Holland 1975; Jong 1975]. Later, a

+ =

Chromosome A Chromosome B Offspring

10001011 + 11011111 = 10011011

+ =

Chromosome A Chromosome B Offspring

10101011 + 11011111 = 11111111

+ =

Chromosome A Chromosome B Offspring

similar type of study found that a higher number of crossover points is beneficial to achieve better performance in GAs [Syswerda 1989]. Recently, a most surprising result was published claiming that uniform crossover is even more effective than any n-point crossover in handling optimization problem with GAs [Syswerda 1989].

Selection and crossover genetic operators may be enough to create new generations of chromosomes in GAs. However, it may fail to produce enough varieties of chromosomes to ensure that the GA is directing the search to the right solution space. Without diversity and with a bad initial population, a GA may find itself converging to chromosomes that will not produce optimum solutions. These problems can be overcome by introducing a mutation operator into the GA. During mutation, each element value (bit) of the chromosome changes with a given probability to a new value. The chromosome and the element selection are also performed randomly in mutation. The mutation probability should be kept very low (usually about 0.001%) since a high mutation rate will destroy a chromosome’s fitness and degenerate the GA into a random walk with all its associated problems [Ochoa et al 1999]. The following example demonstrates a mutation process in a GA where 0 is changed to 1 in a chromosome and it produces an off spring.