Computational Intelligence

The origins of artificial intelligence can be traced back to early centuries, even to the times of ancient philosophers, especially if we consider the philosophical aspects of this field of science. Less distant in time is the first half of the 19th century when a professor of the University of Cambridge, Charles Babbage, came up with an idea of the so-called “analytical machine” which could not only perform arithmetic operations of a certain type but was also capable of performing operations according to pre-defined instructions. What played an essential role in that project was a punched card which one hundred years later turned out to be a very important element of communication between man and computer. In 1950 Alan Turing came up with a test, the purpose of which was to check whether a given program is intelligent. Soon afterwards a number of works appeared and research projects were carried out in order to understand the natural language and solving complex problems.

This book focuses on various techniques of computational intelligence, both single ones and those which form hybrid methods. Those techniques are today commonly applied to classical issues of artificial intelligence, e.g. to process speech and natural language, build expert systems and robots, search for information as well as for the needs of learning by machines. Below are specified the main threads of this book.

When considering the issues of artificial intelligence, we need to have a point of reference. This point of reference may be the definition of human intelligence. The literature contains many different definitions, but most of them come down to the conclusion that intelligence is the ability to adapt to new tasks and living conditions or a way in which humans process information and solve problems. Intelligence is also the ability to associate and to understand. It is influenced by both hereditary factors and by nurture. The most important processes and functions making up human intelligence are learning and using knowledge, ability to generalize, perception and cognitive abilities, e.g. ability to recognize a given object in any context. Moreover, we can list such elements as memorizing, setting and achieving objectives, ability to cooperate, formulation of conclusions, ability to analyze, creativity as well as conceptual and abstractive thinking. Intelligence is also related to such factors as self-consciousness, emotional and irrational states of human being.

The so-called man-made intelligent machines may be programmed to imitate only in a very limited scope, a few of above listed elements making up human intelligence. Thus, we have still a long way to go before we understand the functioning of the brain and are able to build its artificial counterpart. In this chapter, we shall briefly present the selected issues concerning artificial intelligence, beginning with the historical Turing test and the issue of the “Chinese room”.

In the physical world around us, it is impossible to find any two object (things) that are identical. By comparing any two objects, even if very similar, we will always be able to find differences between them, in particular if we consider a sufficiently large number of their features (attributes) with a sufficiently great accuracy. Of course, such a detailed description of the world is not always needed. If we decrease the precision of description, it may happen that some or even several objects that were distinguishable before become indiscernible. For example, all cities in Poland may be discernible with respect to the exact number of inhabitants. If we are interested in cities with the number of inhabitants within a given interval, e.g. from 100 to 300 thousand people, then some cities will be indiscernible with respect to the feature (attribute) “number of inhabitants”. Moreover, in the description of any given object, we only consider a limited number of features, adequate to a given purpose. Quite often, we want to reduce that number to the necessary minimum. These are the problems dealt with by the theory of rough sets.

In this chapter the rough sets theory will be presented in the form of a series of definitions illustrated by examples. Table 3.1 will allow the reader an easier handling of them.

In everyday life, we come across phenomena and notions, the nature of which is ambiguous and imprecise. Using the classical theory of sets and bivalent logic, we are unable to formally describe such phenomena and notions. We are supported by the fuzzy sets theory, which in the last dozen of years has found many interesting applications.

In this chapter, we shall present, in a Reader friendly manner, the basic terms and definitions of fuzzy sets theory (points 4.2 – 4.7). We shall then discuss the issues of approximate reasoning, i.e. the reasoning on the basis of fuzzy antecedents (point 4.8). The next point relates to the problem of construction of fuzzy inference systems (point 4.9). The chapter is finalized by some examples of application of fuzzy sets in the issues of forecasting, planning and decision making.

The fuzzy sets, discussed in the previous chapter, are called type-1 fuzzy sets. They are characterized by the membership function, while the value of this function for a given element x is called the grade of membership of this element to a fuzzy set. In case of type-1 fuzzy sets, the membership grade is a real number taking values in the interval [0,1]. This chapter will present another concept of a fuzzy description of uncertainty. According to this concept, the membership grade is not a number any more, but it has a fuzzy character. Figure 5.1 shows a graphic illustration of type-1 fuzzy sets A₁,…,A₅ and corresponding type-2 fuzzy sets Ã₁,…, Ã₅. It should be noted that in case of type-2 fuzzy sets, for any given element x, we cannot speak of an unambiguously specified value of the membership function. In other words, the membership grade is not a number, as in case of type-1 fuzzy sets.

In subsequent points of this chapter, basic definitions concerning type-2 fuzzy sets will be presented and operations on these sets will be discussed. Then type-2 fuzzy relations and methods of transformation of type-2 fuzzy sets into type-1 fuzzy sets will be introduced.

In the last part of this chapter, the theory of type-2 fuzzy sets will serve for the construction of the fuzzy inference system. Particular blocks of such system will be discussed in details, including type-2 fuzzification, type-2 rules base, type-2 inference mechanisms and the two-stage defuzzification consisting of type-reduction and defuzzification.

For many years, scientists have tried to learn the structure of the brain and discover how it works. Unfortunately, it still remains a fascinating riddle not solved completely. Based on observation of people crippled during different wars or injured in accidents, the scientists could assess the specialization of particular fragments of the brain. It was found, for example, that the left hemisphere is responsible for controlling the right hand, whereas the right hemisphere – for the left hand. The scientists still do not have any detailed information on higher mental functions.We can assume hypothetically that the left hemisphere controls speech function and scientific thinking, whereas the right hemisphere is its opposite as it manages artistic capabilities, spatial imagination etc. The nervous system is made of cells called neurons. There are about 100 billion of them in the human brain. The functioning of a single neuron consists in the flow of so-called nerve impulses. The impulse induced by a specific stimulus encountering a neuron causes its spreading along all its dendrones. As a result, a muscle contraction can occur or another neuron can be stimulated. Why, then, appropriately connected artificial neurons could not, instead of controlling muscles, manage, for example, the work of a device or solve various problems requiring intelligence? This chapter discusses artificial neural networks. We will present a mathematical model of a single neuron, various structures of artificial neural networks and their learning algorithms.

The beginning of research into evolutionary algorithms was inspired by the imitation of nature. All the living organisms live in certain environment. They have a specific genetic material containing information about them and allowing them to transfer their features to new generations. During reproduction, a new organism is created, which takes certain features after its parents. These features are coded in genes, and these are stored in chromosomes, which in turn constitute genetic material – genotype. During the transfer of features, genes become modified. Then the crossover of different paternal and maternal chromosomes occurs. Mutation often occurs additionally, which is the exchange of single genes in a chromosome. An organism is created which differs from that of its parents and contains genes of its predecessors but also has certain features specific to itself. This organism starts to live in a given environment. If it turns out that it is well fit to the environment, in other words – if the combination of genes turns out to be advantageous – it will transfer its genetic material to its offspring. The individual that is poorly fit to the environment will find it difficult to live in this environment and transfer its genes to subsequent generations.

With the use of appropriately defined fitness function, we check to what extent they are adapted to the environment. Individuals exchange genetic material with each other, crossover and mutation operators are introduced in order to generate new solutions. Among potential solutions, only the best fit ones “survive”.

This chapter will discuss the family of evolutionary algorithms, i.e. the classical genetic algorithm, evolution strategies, evolutionary programming, and genetic programming.We are also going to present advanced techniques used in evolutionary algorithms. The second part of the chapter will discuss connections between evolutionary techniques and neural networks and fuzzy systems.

In daily life as well as in different fields of science we encounter big, sometimes enormous volume of information. One look is enough for humans to distinguish the shapes of objects being of interest to us from a specific image. Intelligent machines, however, are still incapable of prompt and unerring distinguishing of objects in the image, due to the lack of universal algorithms which would work in every situation.

The objective of data clustering is a partition of data set into clusters of similar data. Objects in the data set may be e.g. bank customers, figures or things in a photograph, sick and healthy persons. A human being may effectively group only one- and two-dimensional data, while three-dimensional data may cause serious difficulties. The scale of the problem is intensified by the fact that the number of samples in real tasks may amount to thousands and millions. In the light of those facts it would be very useful to have algorithms for automatic data clustering. Operation of those algorithms would result in a fixed structure of data partition, i.e. location and shape of the clusters and membership degrees of each sample to each cluster. Data clustering is a complicated issue as the structures hidden in the data set may have any shapes and sizes. Moreover, the number of clusters is usually unknown. Unfortunately, the literature so far does not provide any algorithm which would work in the case of any shapes of clusters.

Within the last dozen of years, different structures of neuro-fuzzy networks have been presented, often referred to in the world literature as neuro-fuzzy systems. They combine the advantages of neural networks and classic fuzzy systems. In particular, the neuro-fuzzy networks are characterized – in contrast with neural networks – by a interpretable representation of knowledge represented by fuzzy rules. As generally known, the knowledge in neural networks is represented by the values of synaptic weights, and therefore is completely not interpretable, for instance, for a user of a medical expert system that uses neural networks. Moreover, neuro-fuzzy networks can be trained, using the idea of error backpropagation method, which is the basis of learning of multilayer neural networks. The learning usually applies to membership function parameters of the IF and THEN part of the fuzzy rules. As shown in Chapter 7, there is also the possibility to apply the evolutionary algorithms to learn not only the parameters of the membership functions but also the fuzzy rules themselves. The above discussed advantages of the neuro-fuzzy networks are the reason for their common application in classification, approximation and prediction problems. Most of neuro-fuzzy structures described in the world literature utilizes the Mamdani type inference or the Takagi-Sugeno schema. As mentioned in Chapter 4, the Mamdani type inference consists in connecting the antecedents and the consequents of rules using a t-norm (most often the t-norm of the min type or of the product type). Then the aggregation of particular rules is made using a t-conorm. In case of the Takagi-Sugeno schema, the consequents of the rules are not fuzzy in nature, but are functions of the input variables. Less often the logical inference is applied, which consists in connecting the antecedents and the consequents of rules using a fuzzy implication that satisfies the conditions of Definition 4.47. In case of an inference of logical type the aggregation of particular rules is made using a t-conorm. It is obvious that the designers and users of neuro-fuzzy systems would like to obtain a possibly high accuracy of these systems operation in the sense of the chosen quality criterion. In approximation and prediction problems, such quality criterion is the mean squared error, and in classification problems – the number of erroneously classified samples. In both problems, the experiments are made on learning sequences and testing sequences. It should be stressed that the satisfactory results obtained on a learning sequence do not guarantee a correct system operation on a testing sequence. In other words, the neuro-fuzzy system should have good properties of the so-called generalization. In particular, neuro-fuzzy systems designed using both the membership function and the weights describing the importance of rules and importance of linguistic variables in individual rules should be characterized by an appropriate number of all parameters which are to be subject of learning. A big number of parameters ensures a small learning error, but usually leads to wrong generalization. On the other hand, a small number of parameters in the system leads to a larger learning error. In this chapter, we will present the Mamdani, logical and Takagi-Sugeno systems, their learning algorithms and we will make a comparative analysis of their effectiveness. We will solve the issue of designing neuro-fuzzy systems, which are a compromise between accuracy and the number of parameters describing this system.

In the previous chapter we considered Mamdani and logical neuro-fuzzy systems. In the present chapter we will build a neuro-fuzzy system, the inference method (Mamdani or logical) of which will be found as a result of the learning process. The structure of such a system will be changing during the learning process. Its operation will be possible thanks to specially constructed adjustable triangular norms. Adjustable triangular norms, applied to aggregate particular rules, take the form of a classic t-norm or t-conorm after the learning process is finished. Adjustable implications, which finally take the form of a “correlation function” between premises and consequents (Mamdani approach) or fuzzy S-implication (logical approach), will be constructed in analogical way. Moreover, the following concepts will be used for construction of the neuro-fuzzy systems: the concept of soft triangular norms, parameterized triangular norms as well as weights which describe the importance of particular rules and antecedents in those rules.