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The resulting variety of the recombination operations furnished by fuzzy sets is important to the design of genetic mechanisms. From this point of view, F and S types of the recombination are explorative (with the minimum and maximum being exhibiting the lowest degree of space exploration). The M type of recombination is exploitive. This taxonomy of the operations can be utilized in pursuing a dynamic type of switching between several classes of the recombination during the process of evolutionary optimization. We start from the explorative type of recombination that becomes crucial within initial stages of GA by orienting towards space exploration and preventing the method from any undesired premature convergence effect. In the course of optimization, one gradually focuses on space exploitation and then switches to some M type of the recombination.

8.4. Fuzzy metarules in genetic computing

While evolutionary algorithms are deemed universal to a high degree, they still require some extra adjustment and tuning of their parameters to perform efficiently in a given optimization environment. The parameters such as mutation rate, crossover rate, recombination intensity do play a significant role. Where do their values come from? Practically, these are experimentally adjusted and result from a series of observations of the past simulations. The experience of this form easily generalizes into a series of “if - then” statements. Moreover, the conditions and conclusions parts include fuzzy sets. The rule-based approach introduced by Herrera and Lozano (1996) uses a number of rules to describe a selection of a suitable crossover and mutation rates. Their two rule bases include size of a population and required generation time as conditions; the conclusions are the intensities of the GA operations. Thus the statements read as

  if generation and population size then pc (or pm)

where pc, and pm denote the intensity of the process of crossover and mutation, respectively. The complete collection of the metarules is shown in Fig. 8.6.


Figure 8.6  Metarules for adjustment of GA parameters (VS - Very Small, S - Small, M - Medium, L - Large, VL - Very Large)

8.5. Relational structures and their optimization

Relations are fundamental concepts capturing dependences (relationships) between variables. Relations subsume functions. Their generality comes at the expense of eventual higher demand arising at the algorithmic side. This means that a design of the efficient relation-oriented algorithms is usually more demanding. To alleviate this shortcoming, we study some mechanisms of evolutionary computation. In what follows, we discuss two interesting and representative cases of the development of relational structures for data compression (more specifically, image compression) and data mining.

8.5.1. Image compression as a problem of relation reduction

A two-dimensional fuzzy relation can be readily interpreted as a binary image. Each entry of this relation represents a pixel of the image along with its level of brightness. Let us assume, which is fully legitimate, that the range of brightness is normalized to the unit interval. As a finite model of the image, the fuzzy relation consists of a finite number of rows and columns, namely

where card(X) = n1, card(Y) = n2. Furthermore, let rkl denote the (k, 1;)-th entry of R. We consider two families of fuzzy sets (frames of cognition) and defined in X and Y, respectively. Let card , card . Moreover, p<<n1, and r<<n2.

The linguistic compression is carried out with the aid of the linguistic labels. We construct an induced fuzzy relation G defined over with the entries

i = 1, 2 … p, j = 1, 2 … r. In other words gij results as a sup-min composition (convolution) of the original image (fuzzy relation) and the filtering fuzzy sets (Ai, Bj). Owing to the size of and , the dimension of G is far lower than the original image - hence the resulting effect of the linguistic compression, Fig. 8.7.


Figure 8.7  Linguistic compression of R realized with the aid of and

With any compression comes a decompression (inverse) scheme. One can look at the, convolution formula as a fuzzy relational equation to be solved with respect to R for G and and provided.

The well-known maximal solution to this problem comes in the form of the fuzzy relation

The implication operation is induced by the original max - t convolution used in the compression scheme. The maximal character of the solution means that R is included in the above reconstruction. The compression rate is determined based on the sizes of the fuzzy relation and the number of the fuzzy filters in and . More specifically, we compare the number of the elements, of R, that is n1 n2 and the size of the second relation G, that is pr. In addition we have to take into consideration the fuzzy sets in X and Y whose dimensionality is pn1 and rn2. This leads to the compression ρ rate equal

If we confine ourselves to n1=n2 and p=r, the above expression reads as

The values of the compression rate for some selected values of n and p are summarized below

n p compression rate
100 5    9.756
100 7    6.901
100 9    5.316
200 5  19.753


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