Uncertainty theory. An introduction to its axiomatic foundations.
type: Item
instance of: scholarly article
title: Uncertainty theory. An introduction to its axiomatic foundations.
zbMATH Open document ID: 1072.28012
author: Baoding Liu
published in: Studies in Fuzziness and Soft Computing
publication date: 2004-12-07T00:00:00Z
review text: Uncertainty theory is used by the author as generic term of probability theory, fuzzy set theory and rough set theory. The author presents axiomatic foundations of these three approaches (Chapters 2--4) and of all nine possible combinations of two such approaches (Chapters 5--13). For instance random variables with fuzzy values are subject of fuzzy random theory in Chapter 5, random fuzzy theory (Chapter 6) deals with fuzzy sets on a universe of probability distributions, fuzzy rough theory stands for rough sets on a universe of fuzzy sets. To complete this list, we find in a more or less similar presentation: bifuzzy theory (Chapter 7), birandom theory (8), rough random theory (9), rough fuzzy theory (10), random rough theory (11) and birough theory (13). The typical procedure of the author is the following: Definition of an appropriate variable (e.g., fuzzy variable, fuzzy random variable a.s.o.) and an associated (not necessarily additive) measure (e.g., probability, credibility, trust, chance measure) with its distribution function. So called optimistic and pessimistic values of these measures (some kind of quantiles) and an always real-valued expectation are introduced. Finally, convergence concepts are investigated. The presentation is clear but somewhat tedious because of the structural similarity of all Chapters. The value of these formal approaches should be discussed. Classical fuzzy set theory and rough set theory live from the excluded middle law which leads to essentials like possibility and necessity, upper and lower bounds. Using the arithmetic mean of these essentials, obviously we obtain an at least self dual measure and all the ``probability-like'' investigations presented in this book are possible. It seems to me, however, e.g. fuzzy theory with credibility measures is a ``defuzzified'' theory.
reviewed by: Wolfgang Näther
Mathematics Subject Classification ID: 28E10
Mathematics Subject Classification ID: 68T37
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60A05
Mathematics Subject Classification ID: 03E72
zbMATH DE Number: 2121482
zbMATH Keywords: fuzzy sets
zbMATH Keywords: credibility measures
zbMATH Keywords: rough sets
zbMATH Keywords: probability
zbMATH Keywords: chance measures
zbMATH Keywords: fuzzy measures
zbMATH Keywords: optimistic and pessimistic values
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2121482