## Handbook of Fuzzy Sets Comparison

Theory, Algorithms and Applications

### GCSR Volume 6

### ISBN 978-618-82628-1-2 (print) – ISBN 978-618-82628-2-9 (e-book)

### DOI 10.15579/gcsr.vol6

**Edited by**

**George A. Papakostas, Anestis G. Hatzimichailidis and Vassilis G. Kaburlasos**

#### Front Matter

#### Chapter 1 – On Constructing Distance and Similarity Measures based on Fuzzy Implications

Anestis G. Hatzimichailidis, George A. Papakostas and Vassilis G. Kaburlasos

(pages 1-21) DOI: 10.15579/gcsr.vol6.ch1

**Abstract**

This chapter deals with the construction of distance and similarity measures by utilizing the theoretical advantages of the fuzzy implications. To this end the basic definitions of fuzzy implications are initially discussed and the conditions of typical distance and similarity measures that need to be satisfied are defined next. On the basis of this theory a straightforward methodology for building fuzzy implications based measures is analysed. The main advantage of the proposed methodology is its generality that makes it easy to be adopted in several types of fuzzy sets.

#### Chapter 2 – Toward a Synergy of a Lattice Implication Algebra with Fuzzy Lattice Reasoning – A Lattice Computing Approach

Yi Liu, Vassilis G. Kaburlasos, Anestis G. Hatzimichailidis and Yang Xu

(pages 23-42) DOI: 10.15579/gcsr.vol6.ch2

**Abstract**

Automated reasoning can be instrumental in real-world applications involving “intelligent” machines such as (semi-)autonomous vehicles as well as robots. From an analytical point of view, reasoning consists of a series of inferences or, equivalently, implications. In turn, an implication is a function which obtains values in a welldefined set. For instance, in classical Boolean logic an implication obtains values in the set {0, 1}, i.e. it is either true (1) or false (0); whereas, in narrow fuzzy logic an implication obtains values in the specific complete mathematical lattice unit-interval, symbolically [0, 1], i.e. it is partially true/false. A lattice implication algebra (LIA) assumes implication values in a general complete mathematical lattice toward enhancing the representation of ambiguity in reasoning. This work introduces a LIA with implication values in a complete lattice of intervals on the real number axis. Since real numbers stem from real-world measurements, this work sets a ground for real-world applications of a LIA. We show that the aforementioned lattice of intervals includes all the enabling mathematical tools for fuzzy lattice reasoning (FLR). It follows a capacity to optimize, in principle, LIA-reasoning based on FLR as described in this work.

#### Chapter 3 – Relationships Among Several Fuzzy Measures

Yingfang Li, Xingxing He and Keyun Qin

(pages 43-78) DOI: 10.15579/gcsr.vol6.ch3

**Abstract**

In fuzzy set theory, similarity measure, divergence measure, subsethood measure and fuzzy entropy are four basic concepts. They surface in many fields, such as image processing, fuzzy neural networks, fuzzy reasoning, fuzzy control, and so on. The similarity measure describes the degree of similarity of fuzzy sets A and B. The divergence measure describes the degree of difference of fuzzy sets A and B. The subsethood measure (also called inclusion measure) is a relation between fuzzy sets A and B, which indicates the degree to which A is contained in B. The entropy of a fuzzy set is the fuzziness of that set.

This chapter focuses on discussing relationships among these four fuzzy measures. All of the fuzzy measures are discussed on discrete universes here; the cases for continuous universes can be researched similarly.

#### Chapter 4 – Pattern Classification using Generalized Recurrent Exponential Fuzzy Associative Memories

Marcos Eduardo Valle and Aline Cristina de Souza

(pages 79-102) DOI: 10.15579/gcsr.vol6.ch4

**Abstract**

Generalized recurrent exponential fuzzy associative memories (GRE-FAMs) are biologically inspired models designed for the storage and recall of fuzzy sets. They can be viewed as a recurrent multilayer neural network that employs a fuzzy similarity measure in its first hidden layer. In this chapter, we provide theoretical results concerning the storage capacity and noise tolerance of a single-step GRE-FAM. Furthermore, we describe how a GRE-FAM model can be applied for pattern classification. Computational experiments show that the accuracy of certain GRE-FAM classifiers is competitive with some well-known classifiers from the literature.

#### Chapter 5 – Fuzzy Set Similarity using a Distance-Based Kernel on Fuzzy Sets

Jorge Guevara, Roberto Hirata Jr. and Stephane Canu

(pages 103-120) DOI: 10.15579/gcsr.vol6.ch5

**Abstract**

Similarity measures computed by kernels are well studied and a vast literature is available. In this work, we use distance-based kernels to define a new similarity measure for fuzzy sets. In this sense, a distance-based kernel on fuzzy sets implements a similarity measure for fuzzy sets with a geometric interpretation in functional spaces.

When the kernel is positive definite, the similarity measure between fuzzy sets is an inner product of two functions on a Reproducing kernel Hilbert space. This new view of similarity measures for fuzzy sets given by kernels leverages several applications in areas as machine learning, image processing, and fuzzy data analysis. Moreover, it extends the application of kernel methods to the case of fuzzy data. We show an application of our method in a kernel hypothesis testing on fuzzy data.

#### Chapter 6 – FSSAM: A Fuzzy Rule-Based System for Financial Decision Making in Real-Time

**Abstract**

This chapter looks into some problems financial managers face when they have to make decisions in real time while confronted with restrictions, such as coping with imprecise information or processing enormous amount of financial data. However, it is not concerned with existing software systems for supporting investment decisions, neither the ones based on fundamental analysis nor those based on technical analysis of stock markets.

What the chapter describes in detail is a real-time software application – Fuzzy Software System for Asset Management (FSSAM). FSSAM collects and processes the data autonomously, and produces outputs that support the process of financial management.

Thus, the fuzzy rule-based systems (FRBS) are presented as a type of technology which provides tools for overcoming the above-mentioned difficulties, with its unique features, such as the capacity for implementing human knowledge; error tolerance and the ability to, relatively easily, create models of complex dynamic and non-deterministic systems with volatile and/or uncertain parameters.

#### Chapter 7 – Application of Fuzzy Rule Base Design Method

**Abstract**

In many classification tasks the final goal is usually to determine classes of objects. The final goal of fuzzy clustering is also the distribution of elements with highest membership functions into classes. The key issue is the possibility of extracting fuzzy rules that describe clustering results. The paper develops a method of fuzzy rule base designing for the numerical data, which enables extracting fuzzy rules in the form IFTHEN. To obtain the membership functions, the fuzzy c-means clustering algorithm is employed. The described methodology of fuzzy rule base designing allows one to classify the data. The practical part contains implementation examples.