Crossover is a genetic operator that combines two chromosomes (parents) to pro- duce one or two chromosomes (offspring). The idea behind crossover is that the new chromosome may be better than both of the parents if it takes the best charac- teristics from each of the parents. First, the crossover operator randomly chooses a crossover point where two parent chromosomes “break”, and then exchanges the chromosome parts after that point with a user-definable crossover probability. As a result, two new offspring are created. If a pair of chromosomes does not cross over, then the chromosome cloning takes place, and the offspring are created as exact copies of each parent [Neg02].
The most common forms of crossover are one-point, two-point, n-point, and uniform crossover showed in Figure II.2.
Figure II.2 Crossover operators 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 Parent B 1 1 0 1 1 0 0 0 1 0 1 1 1 0 1 1 Parent A A A Offspring A Offspring B 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 Parent B 1 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 Parent Offspring A Offspring B 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 Parent B 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 Parent Offspring A Offspring B Two-Point Crossover Uniform Crossover One-Point Crossover-
Mutation operator
Mutation represents a change in the gene (Figure II.3). Its role is to provide and guarantee that the search algorithm is not trapped on a local optimum. The mu- tation operator flips a randomly selected gene in a chromosome. The mutation operator uses a mutation probability pm previously set by the user, which is quite small in nature, and it is kept low for gas, typically in the range 0.001 and 0.01.
According to this probability, the bit value is changed from 0 to 1 or vice versa. This way, an offspring is produced from a single parent [Neg02].
Figure II.3 Mutation operator 1 1 0 1 1 0 1 1 1 1 1 1 1 0 1 1 Parent Offspring
The following three operators compose the chc (Cross-generational elitist
selection, Heterogeneous recombination by “incest prevention”, and Cataclysmic mutation) algorithm which has the idea that recombination should be the domi- nant search operator.
Elitist selection and incest prevention
After recombination, the N best unique individuals are drawn from the parent population and the offspring population to create the next generation. This also implies that duplicate individuals are removed from population. This form of selection is also referred to as truncation selection [Esh91].
After the truncation selection, pairs of individuals are randomly formed with the new parent population to apply the recombination, forming N/2 pairs of in- dividuals. However, the chc algorithm also employs a heterogeneous recombina-
tion restriction as a method of “incest prevention”. This is accomplished by matting only those pairs of chromosomes which differ from each other by some number of bits, i.e., a matting threshold. The initial threshold is set at L/4, where L is the length of the string. If any of the N/2 recombination could not be applied, i.e., if a generation occurs in which no offspring are inserted into the new children
population, then the threshold is reduced by one. This means that the chromoso- mes of the population have become very similar.
Half uniform crossover (hux) operator
In this operator bits are randomly and independently exchanged, but exactly half of the bits that differ between parents are swapped, see Figure II.4. The hux op-
erator [Esh91] ensures that the offspring are equidistant between the two parents. This serves as a diversity preserving mechanism.
Figure II.4 Half uniform crossover
1 1 0 1 1 0 1 1 0 0 1 1 1 0 1 0 * * * 1 1 0 1 * x * x 1 1 0 1 * 0 1 1 1 1 0 1 1 1 0 0 1 1 0 1 0 Parent A Parent B * Different allels x Allels to interchange Offspring A Offspring B Cataclysmic mutation
No mutation is applied during the regular search phase of the chc algorithm
[Esh91]. When no offspring can be inserted into the population of a succeeding generation, and the mating threshold has reached a value of zero, chc introduces
new diversity into the population via a form of restart. Cataclysmic mutation uses the best individual in the population as a template to re-initialize the population. The new population includes a copy of the best individual; the remainder of the population is generated by applying a simple mutation relatively high, for instance
35%, of the best individual. The new threshold value will be the product of the chromosome’s length (L) and the mutation’s percentage (%) used to generate the new population. There are many other ways to refresh the population, for example, to rescue the best k individuals and generating the remainder randomly, or to rescue the best k individuals and use these as templates to generate the re- maining of the population, and so on.
II.5 clusTerIngalgorIThms
The clustering algorithms form groups of objects in order to archive the greatest possible similarity between objects of a group, and at the same time to keep the dissimilarity of the objects of other groups. More formally, the clustering problem [Jai99] is dividing a given set {x1, ..., xN} of N data points into several non-over- lapping homogenous groups. Each such group or cluster should contain similar data items and data items from different groups should not be similar. We refer to a clustering in k groups as a k-clustering.
Many different approaches to the clustering problem have been developed. Some operate on data represented by their coordinates in a feature space and others operate on a matrix of pairwise similarities between data points. To give a briefly overview of the different types of methods, we list them in three groups [Ver04]:
1. Hierarchical clustering methods. These produce a hierarchy of clusters for the data. The first level of the hierarchy contains all data and at each subsequent level of the hierarchy, one of the clusters of the previous level is split in two. The last level contains all data in individual clusters. The hierarchy is based on pairwise similarities between data points and is constructed either top-down or bottom-up.
2. Partitional clustering methods. These produce a single clustering with a fixed and (often) specified number of clusters. Most partitional clustering algorithms do not operate on the basis of pairwise similarities, but with data represented in some feature space. Typically, these methods start with an initial k-clustering and apply an iterative algorithm to improve upon the initial clustering accord- ing to some criterion. Most partitional clustering methods make, sometimes implicitly, assumptions on the distribution of data within each cluster. The partitional algorithms cure, k-means [Har79] is used in this method.
3. Spectral clustering methods. These operate on a matrix with pairwise similari- ties between the data points. The optimal clustering is defined as the clustering that minimizes the ‘normalized cut’ criterion that depends on the size of the
clusters and the total sum of the similarities between points that are assigned to different clusters. Unfortunately, finding the clustering that minimizes the normalized cut is an np-complete problem). However, a relaxation of this op-
timization problem can be efficiently solved, and the solution is given by an eigenvector of the normalized similarity matrix. The solution of the relaxed problem is then further processed to find an approximate solution for the original problem. The term spectral clustering’ refers to the normalized simi- larity matrix which can be used to assess the number of clusters in the data. Spectral methods are used both to find hierarchical clustering and k-clustering for a given k. We treat them separately since their working is quite different from the other approaches.