Statistical Methods for Fuzzy Data (Wiley Series in Probability and Statistics) - Hardcover

Viertl, Reinhard

Hardcover

ISBN 10: 0470699450 ISBN 13: 9780470699454

Verlag: Wiley, 2011

Inhaltsangabe

Statistical data are not always precise numbers, or vectors, or categories. Real data are frequently what is called fuzzy. Examples where this fuzziness is obvious are quality of life data, environmental, biological, medical, sociological and economics data. Also the results of measurements can be best described by using fuzzy numbers and fuzzy vectors respectively.

Statistical analysis methods have to be adapted for the analysis of fuzzy data. In this book, the foundations of the description of fuzzy data are explained, including methods on how to obtain the characterizing function of fuzzy measurement results. Furthermore, statistical methods are then generalized to the analysis of fuzzy data and fuzzy a-priori information.

Key Features:

• Provides basic methods for the mathematical description of fuzzy data, as well as statistical methods that can be used to analyze fuzzy data.
• Describes methods of increasing importance with applications in areas such as environmental statistics and social science.
• Complements the theory with exercises and solutions and is illustrated throughout with diagrams and examples.
• Explores areas such quantitative description of data uncertainty and mathematical description of fuzzy data.

This work is aimed at statisticians working with fuzzy logic, engineering statisticians, finance researchers, and environmental statisticians. It is written for readers who are familiar with elementary stochastic models and basic statistical methods.

Über diesen Titel

Über die Autorin bzw. den Autor

Reinhard Viertl is Professor of Applied Statistics at Vienna University of Technology.
Professor Viertl has been working on statistical analysis of fuzzy data for about 20 years. He is the author of various publications including 5 books and more than 100 papers.

Statistical analysis methods have to be adapted for the analysis of fuzzy data. In this book, the foundations of the description of fuzzy data are explained, including methods on how to obtain the characterizing function of fuzzy measurement results. Furthermore, statistical methods are then generalized to the analysis of fuzzy data and fuzzy a-priori information.

Key Features:

• Provides basic methods for the mathematical description of fuzzy data, as well as statistical methods that can be used to analyze fuzzy data.
• Describes methods of increasing importance with applications in areas such as environmental statistics and social science.
• Complements the theory with exercises and solutions and is illustrated throughout with diagrams and examples.
• Explores areas such quantitative description of data uncertainty and mathematical description of fuzzy data.

This work is aimed at statisticians working with fuzzy logic, engineering statisticians, finance researchers, and environmental statisticians. It is written for readers who are familiar with elementary stochastic models and basic statistical methods.

Statistical analysis methods have to be adapted for the analysis of fuzzy data. In this book, the foundations of the description of fuzzy data are explained, including methods on how to obtain the characterizing function of fuzzy measurement results. Furthermore, statistical methods are then generalized to the analysis of fuzzy data and fuzzy a-priori information.

Key Features:

• Provides basic methods for the mathematical description of fuzzy data, as well as statistical methods that can be used to analyze fuzzy data.
• Describes methods of increasing importance with applications in areas such as environmental statistics and social science.
• Complements the theory with exercises and solutions and is illustrated throughout with diagrams and examples.
• Explores areas such quantitative description of data uncertainty and mathematical description of fuzzy data.

This work is aimed at statisticians working with fuzzy logic, engineering statisticians, finance researchers, and environmental statisticians. It is written for readers who are familiar with elementary stochastic models and basic statistical methods.

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Statistical Methods for Fuzzy Data

By Reinhard Viertl

John Wiley & Sons

Copyright © 2010 John Wiley & Sons, Ltd
All right reserved.
ISBN: 978-0-470-69945-4

Chapter One

Fuzzy data

All kinds of data which cannot be presented as precise numbers or cannot be precisely classified are called nonprecise or fuzzy. Examples are data in the form of linguistic descriptions like high temperature, low flexibility and high blood pressure. Also, precision measurement results of continuous variables are not precise numbers but always more or less fuzzy.

1.1 One-dimensional fuzzy data

Measurement results of one-dimensional continuous quantities are frequently idealized to be numbers times a measurement unit. However, real measurement results of continuous quantities are never precise numbers but always connected with uncertainty. Usually this uncertainty is considered to be statistical in nature, but this is not suitable since statistical models are suitable to describe variability. For a single measurement result there is no variability, therefore another method to model the measurement uncertainty of individual measurement results is necessary. The best up-to-date mathematical model for that are so-called fuzzy numbers which are described in Section 2.1 [cf. Viertl (2002)].

Examples of one-dimensional fuzzy data are lifetimes of biological units, length measurements, volume measurements, height of a tree, water levels in lakes and rivers, speed measurements, mass measurements, concentrations of dangerous substances in environmental media, and so on.

A special kind of one-dimensional fuzzy data are data in the form of intervals [a; b] [??] R. Such data are generated by digital measurement equipment, because they have only a finite number of digits.

1.2 Vector-valued fuzzy data

Many statistical data are multivariate, i.e. ideally the corresponding measurement results are real vectors (x₁, ..., x_k) [element of] R^k. In applications such data are frequently not precise vectors but to some degree fuzzy. A mathematical model for this kind of data is so-called fuzzy vectors which are formalized in Section 2.2.

Examples of vector valued fuzzy data are locations of objects in space like positions of ships on radar screens, spacetime data, multivariate nonprecise data in the form of vectors ([x.sup.*.sub.l], ..., [x.sup.*.sub.n]) of fuzzy numbers [x.sup.*.sub.i].

1.3 Fuzziness and variability

In statistics frequently so-called stochastic quantities (also called random variables) are observed, where the observed results are fuzzy. In this situation two kinds of uncertainty are present: Variability, which can be modeled by probability distributions, also called stochastic models, and fuzziness, which can be modeled by fuzzy numbers and fuzzy vectors, respectively. It is important to note that these are two different kinds of uncertainty. Moreover it is necessary to describe fuzziness of data in order to obtain realistic results from statistical analysis. In Figure 1.1 the situation is graphically outlined.

Real data are also subject to a third kind of uncertainty: errors. These are the subject of Section 1.4.

1.4 Fuzziness and errors

In standard statistics errors are modeled in the following way. The observation y of a stochastic quantity is not its true value x, but superimposed by a quantity e, called error, i.e.

y = x + e.

The error is considered as the realization of another stochastic quantity. These kinds of errors are denoted as random errors.

For one-dimensional quantities, all three quantities x, y, and e are, after the experiment, real numbers. But this is not suitable for continuous variables because the observed values y are fuzzy.

It is important to note that all three kinds of uncertainty are present in real data. Therefore it is necessary to generalize the mathematical operations for real numbers to the situation of fuzzy numbers.

1.5 Problems

(a) Find examples of fuzzy numerical data which are not given in Section 1.1 and Section 1.2.

(b) Work out the difference between stochastic uncertainty and fuzziness of individual observations.

(c) Make clear how data in the form of intervals are obtained by digital measurement devices.

(d) What do X-ray pictures and data from satellite photographs have in common?

(Continues...)

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Details zur Ausgabe

• Verlag: Wiley
• Erscheinungsdatum: 2011
• Sprache: Englisch
• ISBN 10: 0470699450
• ISBN 13: 9780470699454
• Einband: Gebundene Ausgabe
• Auflage: 1
• Anzahl der Seiten: 272
• Kontakt zum Hersteller: Wiley-VCH GmbH; product_safety@wiley.com; ; Boschstrasse 12; Weinheim; 69469; Deutschland