Biomedical Signal Analysis: Contemporary Methods and Applications

7. Fuzzy Clustering and Genetic Algorithms

7 Fuzzy Clustering and Genetic Algorithms Besides artificial neural networks, fuzzy clustering and genetic algorithms represent an important class of processing algorithms for biosignals . Biosignals are characterized by uncertainties resulting from incomplete or imprecise input information, ambiguity, ill–defined or overlapping boundaries among the disease classes or regions, and indefiniteness in extracting features and relations among them. Any decision taken at a particular point will heavily influence the following stages. Therefore, an automatic diagnosis system must have sufficient possibilities to capture the uncertainties involved at every stage, such that the system’s output results should reflect minimal uncertainty. In other words, a pattern can belong to more than one class. Translated to clinical diagnosis, this means that a patient can exhibit multiple symptoms belonging to several disease categories. The symptoms do not have to be strictly numerical. Thus, fuzzy variables can be both linguistic and/or set variables. An example of a fuzzy variable is the heart-beat of a person ranging from 40 to 150 beats per minute, which can be described as slow, normal, or fast. The main difference between fuzzy and neural paradigms is that neural networks have the ability to learn from data, while fuzzy systems (1) quantify linguistic inputs and (2) provide an approximation of unknown and complex input-output rules. Genetic algorithms are usually employed as optimization procedures in biosignal processing, such as determining the optimal weights for neural networks when applied, for example, to the segmentation of ultrasound images or to the classification of voxels. This chapter reviews the basics of fuzzy clustering and of genetic algorithms. Several well-known fuzzy clustering algorithms and fuzzy learning vector quantization are presented. 7.1 Fuzzy Sets Fuzzy sets are an important tool for the description of imprecision and uncertainty. A classical set is usually represented as a set with a crisp boundary. For example, 218 Chapter 7 X = {x|x > 8} (7.1) where 8 represents an unambiguous boundary. On the other hand, a fuzzy set does not have a crisp boundary. To represent this fact, a new concept is introduced, that of a membership function describing the smooth transition from the fact “belongs to a set” to “does not belong to a set”. Fuzzyness stems not from the randomness of the members of the set but from the uncertain nature of concepts. This chapter will review some of the basic notions and results in fuzzy set theory. Fuzzy systems are described by fuzzy sets and operations on fuzzy sets. Fuzzy logic approximates human reasoning by using linguistic variables and introduces rules based on combinations of fuzzy sets by these operations. The notion of fuzzy set way introduced by Zadeh [295]. Crisp sets Definition 7.1: Crisp set Let X be a non empty set considered to be the universe of discourse. A crisp set A is defined by enumerating all elements x ∈ X, A = {x1, x2, · · · , xn} (7.2) that belong to A. The universe of discourse consists of ordered or nonordered discrete objects or of the continuous space. Definition 7.2: Membership function The membership function can be expressed by a function uA, that maps X on a binary value described by the set I = {0, 1}: uA : X → I, uA(x) =  1 if x ∈ A 0 if x ∈ A. (7.3) Here, uA(x) represents the membership degree of x to A. Thus, an arbitrary x either belongs to A or it does not; partial member- Fuzzy Clustering and Genetic Algorithms 219 1 A(x) 25 60 years middle aged young old Figure 7.1 A membership function of temperature. ship is not allowed. For two sets A and B, combinations can be defined by the following operations: A ∪ B = {x|x ∈ A or x ∈ B} (7.4) A ∩ B = {x|x ∈ A and x ∈ B} (7.5) Ā = {x|x ∈ A, x ∈ X}. (7.6) Additionally, the following rules have to be satisfied: A ∪ Ā = ∅, and A ∩ Ā = X (7.7) Fuzzy sets Definition 7.3: Fuzzy set Let X be a non–empty set considered to be the universe of discourse. A fuzzy set is a pair (X, A), where uA : X → I and I = [0, 1]. Figure 7.1 is an example of a possible membership function. The family of all fuzzy sets on the universe x will be denoted by L(X). Thus L(X...