Random Fuzzy Sets: Theory & Applications

Random Fuzzy Sets: Theory & Applications

Hung T. Nguyen (New Mexico State University, USA), Vladik Kreinovich (University of Texas at El Paso, USA) and Gang Xiang (University of Texas at El Paso, USA)
DOI: 10.4018/978-1-59904-982-3.ch002
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It is well known that in decision making under uncertainty, while we are guided by a general (and abstract) theory of probability and of statistical inference, each specific type of observed data requires its own analysis. Thus, while textbook techniques treat precisely observed data in multivariate analysis, there are many open research problems when data are censored (e.g., in medical or bio-statistics), missing, or partially observed (e.g., in bioinformatics). Data can be imprecise due to various reasons, for example, due to fuzziness of linguistic data. Imprecise observed data are usually called coarse data. In this chapter, we consider coarse data which are both random and fuzzy. Fuzziness is a form of imprecision often encountered in perception-based information. In order to develop statistical reference procedures based on such data, we need to model random fuzzy data as bona fide random elements, that is, we need to place random fuzzy data completely within the rigorous theory of probability. This chapter presents the most general framework for random fuzzy data, namely the framework of random fuzzy sets. We also describe several applications of this framework.
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From Multivariate Statistical Analysis To Random Sets

What is a random set? An intuitive meaning. What is a random set? Crudely speaking, a random number means that we have different numbers with different probabilities; a random vector means that we have different vectors with different probabilities; and similarly, a random set means that we have different sets with different probabilities.

How can we describe this intuitive idea in precise terms? To provide such a formalization, let us recall how probabilities and random vectors are usually defined.

How probabilities are usually defined. To describe probabilities, in general, we must have a set of possible situations , and we must be able to describe the probability P of different properties of such situations. In mathematical terms, a property can be characterized by the set of all the situations which satisfy this property. Thus, we must assign to sets , the probability value .

According to the intuitive meaning of probability (e.g., as frequency), if we have two disjoint sets and , then we must have . Similarly, if we have countably many mutually disjoint sets , we must have .

A mapping which satisfies this property is called -additive.

It is known that even in the simplest situations, for example, when we randomly select a number from the interval , it is not possible to have a -additive function which would be defined on all subsets of . Thus, we must restrict ourselves to a class of subsets of . Since subsets represent properties, a restriction on subsets means restriction on properties. If we allow two properties and , then we should also be able to consider their logical combinations , , and – which in set terms correspond to union, intersection, and complement. Similarly, if we have a sequence of properties , then we should also allow properties and which correspond to countable union and intersection. Thus, the desired family should be closed under (countable) union, (countable) intersection, and complement. Such a family is called a -algebra.

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Table of Contents
Hsiao-Fan Wang
Hsiao-Fan Wang
Chapter 1
Martin Spott, Detlef Nauck
This chapter introduces a new way of using soft constraints for selecting data analysis methods that match certain user requirements. It presents a... Sample PDF
Automatic Intelligent Data Analysis
Chapter 2
Hung T. Nguyen, Vladik Kreinovich, Gang Xiang
It is well known that in decision making under uncertainty, while we are guided by a general (and abstract) theory of probability and of statistical... Sample PDF
Random Fuzzy Sets: Theory & Applications
Chapter 3
Gráinne Kerr, Heather Ruskin, Martin Crane
Microarray technology1 provides an opportunity to monitor mRNA levels of expression of thousands of genes simultaneously in a single experiment. The... Sample PDF
Pattern Discovery in Gene Expression Data
Chapter 4
Erica Craig, Falk Huettmann
The use of machine-learning algorithms capable of rapidly completing intensive computations may be an answer to processing the sheer volumes of... Sample PDF
Using "Blackbox" Algorithms Such AS TreeNET and Random Forests for Data-Ming and for Finding Meaningful Patterns, Relationships and Outliers in Complex Ecological Data: An Overview, an Example Using G
Chapter 5
Eulalia Szmidt, Marta Kukier
We present a new method of classification of imbalanced classes. The crucial point of the method lies in applying Atanassov’s intuitionistic fuzzy... Sample PDF
A New Approach to Classification of Imbalanced Classes via Atanassov's Intuitionistic Fuzzy Sets
Chapter 6
Arun Kulkarni, Sara McCaslin
This chapter introduces fuzzy neural network models as means for knowledge discovery from databases. It describes architectures and learning... Sample PDF
Fuzzy Neural Network Models for Knowledge Discovery
Chapter 7
Ivan Bruha
This chapter discusses the incorporation of genetic algorithms into machine learning. It does not present the principles of genetic algorithms... Sample PDF
Genetic Learning: Initialization and Representation Issues
Chapter 8
Evolutionary Computing  (pages 131-142)
Thomas E. Potok, Xiaohui Cui, Yu Jiao
The rate at which information overwhelms humans is significantly more than the rate at which humans have learned to process, analyze, and leverage... Sample PDF
Evolutionary Computing
Chapter 9
M. C. Bartholomew-Biggs, Z. Ulanowski, S. Zakovic
We discuss some experience of solving an inverse light scattering problem for single, spherical, homogeneous particles using least squares global... Sample PDF
Particle Identification Using Light Scattering: A Global Optimization Problem
Chapter 10
Dominic Savio Lee
This chapter describes algorithms that use Markov chains for generating exact sample values from complex distributions, and discusses their use in... Sample PDF
Exact Markov Chain Monte Carlo Algorithms and Their Applications in Probabilistic Data Analysis and Inference
Chapter 11
J. P. Ganjigatti, Dilip Kumar Pratihar
In this chapter, an attempt has been made to design suitable knowledge bases (KBs) for carrying out forward and reverse mappings of a Tungsten inert... Sample PDF
Design and Development of Knowledge Bases for Forward and Reverse Mappings of TIG Welding Process
Chapter 12
Malcolm J. Beynon
This chapter considers the role of fuzzy decision trees as a tool for intelligent data analysis in domestic travel research. It demonstrates the... Sample PDF
A Fuzzy Decision Tree Analysis of Traffic Fatalities in the US
Chapter 13
Dymitr Ruta, Christoph Adl, Detlef Nauck
In the telecom industry, high installation and marketing costs make it six to 10 times more expensive to acquire a new customer than it is to retain... Sample PDF
New Churn Prediction Strategies in the Telecom Industry
Chapter 14
Malcolm J. Beynon
This chapter demonstrates intelligent data analysis, within the environment of uncertain reasoning, using the recently introduced CaRBS technique... Sample PDF
Intelligent Classification and Ranking Analyses Using CARBS: Bank Rating Applications
Chapter 15
Fei-Chen Hsu, Hsiao-Fan Wang
In this chapter, we used Cumulative Prospect Theory to propose an individual risk management process (IRM) including a risk analysis stage and a... Sample PDF
Analysis of Individual Risk Attitude for Risk Management Based on Cumulative Prospect Theory
Chapter 16
Francesco Giordano, Michele La Rocca, Cira Perna
This chapter introduces the use of the bootstrap in a nonlinear, nonparametric regression framework with dependent errors. The aim is to construct... Sample PDF
Neural Networks and Bootstrap Methods for Regression Models with Dependent Errors
Chapter 17
Lean Yu, Shouyang Wang, Kin Keung Lai
Financial crisis is a kind of typical rare event, but it is harmful to economic sustainable development if occurs. In this chapter, a... Sample PDF
Financial Crisis Modeling and Prediction with a Hilbert-EMD-Based SVM Approachs
Chapter 18
Chun-Jung Huang, Hsiao-Fan Wang, Shouyang Wang
One of the key problems in supervised learning is due to the insufficient size of the training data set. The natural way for an intelligent learning... Sample PDF
Virtual Sampling with Data Construction Analysis
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