We start with one of the simplest forms, the so-called Hopfield networks , which originated as physical models to describe magnetism. Hopfield networks are indeed closely related to the Ising model of magnetism. The Hopfield networks that we discussed in the preceding chapter are special recurrent networks, which have a very constrained structure. In this chapter, however, we lift all restrictions and consider recurrent networks without any constraints. Such general recurrent networks are well suited to represent differential equations and to solve them approximately in a numerical fashion.
If the type of differential equation is known that describes a given system, but the values of the parameters appearing in it are unknown, one may also try to train a suitable recurrent network with the help of example patterns in order to determine the system parameters.
The following sections treat mathematical topics that were presupposed in the text e. Evolutionary algorithms comprise a class of optimization techniques that imitate principles of biological evolution. They belong to the family of metaheuristics , which also includes, for example, particle swarm and ant colony optimization, which are inspired by other biological structures and processes, as well as classical methods like simulated annealing, which is inspired by a thermodynamical process. The core principle of evolutionary algorithms is to apply evolution principles like mutation and selection to populations of candidate solutions in order to find a sufficiently good solution for a given optimization problem.
Evolutionary algorithms are not fixed procedures, but contain several elements that must be adapted to the optimization problem to be solved. In particular, the encoding of the candidate solution needs to be chosen with care. Although there is no generally valid rule or recipe, we discuss some important properties a good encoding should have. We also turn to the fitness function and review the most common selection techniques as well as how certain undesired effects can be avoided by adapting the fitness function or the selection method.
The last section of this chapter is devoted to genetic operators, which serve as tools to explore the search space, and covers sexual and asexual recombination and other variation techniques. The preceding chapter presented all relevant elements of evolutionary algorithms, namely guidelines of how to choose an encoding for the solution candidates, procedures how to select individuals based on their fitness, and genetic operators with which modified solution candidates can be obtained.
Equipped with these ingredients we proceed in this chapter to introducing basic forms of evolutionary algorithms, including classical genetic algorithms in which solution candidates are encoded as bit strings , evolution strategies which focus on numerical optimization and genetic programming which tries to derive function expressions or even simple program structures with evolutionary principles. Finally, we take a look at related population-based approaches like ant colony and particle swarm optimization.
With this chapter we close our discussion of evolutionary algorithms by giving an overview of an application of and two special techniques for this kind of metaheuristics. In the first section we consider behavioral simulation for the iterated prisoners dilemma with an evolutionary algorithm. In the next section we study evolutionary algorithms for multi-criteria optimization, especially in the presence of conflicting criteria, which instead of returning a single solution try to map out the so-called Pareto-frontier with several solution candidates.
Finally, we take a look at parallelized versions of evolutionary algorithms. Many propositions about the real world are not either true or false, rendering classical logic inadequate for reasoning with such propositions. Furthermore, most concepts used in human communication do not have crisp boundaries, rendering classical sets inadequate to represent such concept. The main aim of fuzzy logic and fuzzy sets is to overcome the disadvantages of classical logic and classical sets. We have already discussed how set theoretic operations like intersection, union and complement can be generalized to fuzzy sets.
This chapter is devoted to the issue of extending the concept of mappings or functions to fuzzy sets.
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These ideas allow us to define operations like addition, subtraction, multiplication, division or taking squares as well as set theoretic concepts like the composition of relations for fuzzy sets. Relations can be used to model dependencies, correlations or connections between variables, quantities or attributes. In this chapter we discuss a special type of fuzzy relations, called similarity relations.
They play an important role in interpreting fuzzy controllers and, more generally, can be used to characterize the inherent indistinguishability or vagueness of a fuzzy systems. The biggest success of fuzzy systems in the field of industrial and commercial applications has been achieved with fuzzy controllers. Fuzzy control is a way of defining a nonlinear table-based controller whereas its nonlinear transition function can be defined without specifying every single entry of the table individually. Fuzzy control does not result from classical control engineering approaches.
In fact, its roots can be found in the area of rule-based systems. Fuzzy controllers simply comprise a set of vague rules that can be used for knowledge-based interpolation of a vaguely defined function. After a brief overview of fuzzy methods in data analysis, this chapter focuses on fuzzy cluster analysis as the oldest fuzzy approach to data analysis.
Fuzzy clustering comprises a family of prototype-based clustering methods that can be formulated as the problem of minimizing an objective function. As a consequence, additional means have to be employed in the objective function in order to obtain actual degrees of membership. This chapter surveys the most common fuzzification means and examines and compares their properties. A database typically consists of several tables that contain data about business objects such as customer data, sales orders or product information. Each table row represents a description of a single object with each table column representing an attribute of that object.
Relations between these objects are also modeled via tables. Please note that we use the notions table and relation interchangeably. A major part of database theory is concerned with the task to represent data with as little redundancy as possible. This chapter introduces required theoretical concepts for the definition of Bayes and Markov networks. After important elements of probability theory—especially conditional independences—are discussed, we present relevant graph-theoretic notions with emphasis on so-called separation criteria.
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Computational Intelligence: A Methodological Introduction
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