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9780387306551

Measurement Uncertainty

by
  • ISBN13:

    9780387306551

  • ISBN10:

    0387306552

  • Format: Hardcover
  • Copyright: 2006-12-01
  • Publisher: Springer Verlag

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Summary

The expression of uncertainty in measurement is a challenging aspect for researchers and engineers working in instrumentation and measurement because it involves physical, mathematical and philosophical issues. This problem is intensified by the limitations of the probabilistic approach used by the current standard (GUM). This text is the first to make full use of the mathematical theory of evidence to express the uncertainty in measurements. It gives an overview of the current standard, then pinpoints and constructively resolves its limitations through its unique approach. The text presents various tools for evaluating uncertainty, beginning with the probabilistic approach and concluding with the expression of uncertainty using random-fuzzy variables. The exposition is driven by numerous examples. The book is designed for immediate use and application in research and laboratory work. Apart from a classroom setting, this book can be used by practitioners in a variety of fields (including applied mathematics, applied probability, electrical and computer engineering, and experimental physics), and by such institutions as the IEEE, ISA, and National Institute of Standards and Technology.

Table of Contents

Prefacep. ix
Uncertainty in Measurementp. 1
Introductionp. 1
The Theory of Errorp. 2
The Theory of Uncertaintyp. 4
Toward a more modern and comprehensive approachp. 10
Fuzzy Variables and Measurement Uncertaintyp. 15
Definition of fuzzy variablesp. 17
Mathematics of fuzzy variablesp. 21
A simple example of application of the fuzzy variables to represent measurement resultsp. 26
Conclusionsp. 28
The Theory of Evidencep. 31
Basic definitionsp. 37
Rules of combinationp. 48
Possibility theoryp. 49
Necessity and possibility functionsp. 50
The possibility distribution functionp. 54
Fuzzy variables and possibility theoryp. 57
Probability theoryp. 64
Comparison between possibility and probability theoriesp. 67
Random-Fuzzy Variablesp. 73
Definition of fuzzy variables of type 2p. 77
The random-fuzzy variablesp. 81
Construction of Random-Fuzzy Variablesp. 87
Probability-possibility transformationsp. 87
A specific probability-possibility transformation for the construction of RFVsp. 90
Construction of RFVsp. 95
The internal membership functionp. 96
The external membership functionp. 97
Combination of the two membership functionsp. 97
Fuzzy Operatorsp. 99
Aggregation operationsp. 99
t-normp. 101
t-conormp. 106
Averaging operationsp. 111
Discussionp. 115
Fuzzy intersection area and fuzzy union areap. 117
Hamming distancep. 118
Greatest upper set and greatest lower setp. 119
Fuzzy-max and fuzzy-minp. 121
Yager areap. 123
The Mathematics of Random-Fuzzy Variablesp. 125
Combination of the random contributionsp. 126
Supportsp. 131
Distributionsp. 136
Mathematics for the random parts of RFVsp. 151
Sump. 152
Differencep. 160
Productp. 163
Divisionp. 167
The complete mathematicsp. 169
One or both RFVs show only nonrandom contributionsp. 171
The internal membership function of both RFVs is rectangularp. 174
General situationp. 193
Representation of Random-Fuzzy Variablesp. 195
Decision-Making Rules with Random-Fuzzy Variablesp. 197
The available methodsp. 200
Yager methodp. 200
Kerre methodp. 201
Nakamura methodp. 202
A specific methodp. 205
Definition of the credibility coefficientsp. 205
Evaluation of the credibility coefficientsp. 210
Definition of the decision rulesp. 212
Properties of the credibility coefficientsp. 216
Analysis of the credibility coefficientsp. 218
List of Symbolsp. 223
Referencesp. 225
Indexp. 227
Table of Contents provided by Ingram. All Rights Reserved.

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