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Preface | p. ix |
Uncertainty in Measurement | p. 1 |
Introduction | p. 1 |
The Theory of Error | p. 2 |
The Theory of Uncertainty | p. 4 |
Toward a more modern and comprehensive approach | p. 10 |
Fuzzy Variables and Measurement Uncertainty | p. 15 |
Definition of fuzzy variables | p. 17 |
Mathematics of fuzzy variables | p. 21 |
A simple example of application of the fuzzy variables to represent measurement results | p. 26 |
Conclusions | p. 28 |
The Theory of Evidence | p. 31 |
Basic definitions | p. 37 |
Rules of combination | p. 48 |
Possibility theory | p. 49 |
Necessity and possibility functions | p. 50 |
The possibility distribution function | p. 54 |
Fuzzy variables and possibility theory | p. 57 |
Probability theory | p. 64 |
Comparison between possibility and probability theories | p. 67 |
Random-Fuzzy Variables | p. 73 |
Definition of fuzzy variables of type 2 | p. 77 |
The random-fuzzy variables | p. 81 |
Construction of Random-Fuzzy Variables | p. 87 |
Probability-possibility transformations | p. 87 |
A specific probability-possibility transformation for the construction of RFVs | p. 90 |
Construction of RFVs | p. 95 |
The internal membership function | p. 96 |
The external membership function | p. 97 |
Combination of the two membership functions | p. 97 |
Fuzzy Operators | p. 99 |
Aggregation operations | p. 99 |
t-norm | p. 101 |
t-conorm | p. 106 |
Averaging operations | p. 111 |
Discussion | p. 115 |
Fuzzy intersection area and fuzzy union area | p. 117 |
Hamming distance | p. 118 |
Greatest upper set and greatest lower set | p. 119 |
Fuzzy-max and fuzzy-min | p. 121 |
Yager area | p. 123 |
The Mathematics of Random-Fuzzy Variables | p. 125 |
Combination of the random contributions | p. 126 |
Supports | p. 131 |
Distributions | p. 136 |
Mathematics for the random parts of RFVs | p. 151 |
Sum | p. 152 |
Difference | p. 160 |
Product | p. 163 |
Division | p. 167 |
The complete mathematics | p. 169 |
One or both RFVs show only nonrandom contributions | p. 171 |
The internal membership function of both RFVs is rectangular | p. 174 |
General situation | p. 193 |
Representation of Random-Fuzzy Variables | p. 195 |
Decision-Making Rules with Random-Fuzzy Variables | p. 197 |
The available methods | p. 200 |
Yager method | p. 200 |
Kerre method | p. 201 |
Nakamura method | p. 202 |
A specific method | p. 205 |
Definition of the credibility coefficients | p. 205 |
Evaluation of the credibility coefficients | p. 210 |
Definition of the decision rules | p. 212 |
Properties of the credibility coefficients | p. 216 |
Analysis of the credibility coefficients | p. 218 |
List of Symbols | p. 223 |
References | p. 225 |
Index | p. 227 |
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