# Measurement Uncertainty: An Approach via the Mathematical Theory of Evidence

Springer Science & Business Media, Jun 4, 2007 - Mathematics - 228 pages
It is widely recognized, by the scienti?c and technical community that m- surements are the bridge between the empiric world and that of the abstract concepts and knowledge. In fact, measurements provide us the quantitative knowledge about things and phenomena. It is also widely recognized that the measurement result is capable of p- viding only incomplete information about the actual value of the measurand, that is, the quantity being measured. Therefore, a measurement result - comes useful, in any practicalsituation, only if a way is de?ned for estimating how incomplete is this information. The more recentdevelopment of measurement science has identi?ed in the uncertainty concept the most suitable way to quantify how incomplete is the information provided by a measurement result. However, the problem of how torepresentameasurementresulttogetherwithitsuncertaintyandpropagate measurementuncertaintyisstillanopentopicinthe?eldofmetrology,despite many contributions that have been published in the literature over the years. Many problems are in fact still unsolved, starting from the identi?cation of the best mathematical approach for representing incomplete knowledge. Currently, measurement uncertainty is treated in a purely probabilistic way, because the Theory of Probability has been considered the only available mathematical theory capable of handling incomplete information. However, this approach has the main drawback of requiring full compensation of any systematic e?ect that a?ects the measurement process. However, especially in many practical application, the identi?cation and compensation of all s- tematic e?ects is not always possible or cost e?ective.

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### Contents

 Uncertainty in Measurement 1 12 The Theory of Error 2 13 The Theory of Uncertainty 4 14 Toward a more modern and comprehensive approach 10 Fuzzy Variables and Measurement Uncertainty 15 21 Definition of fuzzy variables 17 22 Mathematics of fuzzy variables 21 23 A simple example of application of the fuzzy variables to represent measurement results 26
 53 Construction of RFVs 95 Fuzzy Operators 99 62 Fuzzy intersection area and fuzzy union area 117 63 Hamming distance 118 64 Greatest upper set and greatest lower set 119 65 Fuzzymax and fuzzymin 121 66 Yager area 123 The Mathematics of RandomFuzzy Variables 125

 24 Conclusions 28 The Theory of Evidence 30 31 Basic definitions 37 32 Rules of combination 48 33 Possibility theory 49 34 Fuzzy variables and possibility theory 57 35 Probability theory 64 RandomFuzzy Variables 73 41 Deﬁnition of fuzzy variables of type 2 77 Construction of RandomFuzzy Variables 86 52 A specific probabilitypossibility transformation for the construction of RFVs 90
 71 Combination of the random contributions 126 72 Mathematics for the random parts of RFVs 151 73 The complete mathematics 169 Representation of RandomFuzzy Variables 195 DecisionMaking Rules with RandomFuzzy Variables 197 91 The available methods 200 92 A specific method 205 List of Symbols 223 References 225 Index 227 Copyright

### Popular passages

Page 1 - I often say that when you can measure what you are speaking about and express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.
Page 5 - B.1 8 uncertainty (of measurement) a parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand (NOTES — 1.
Page iv - Dipartimento di Elettrotecnica - Politecnico di Milano Piazza Leonardo Da Vinci, 32 - 20133 Milano - Italy Phone...
Page 4 - the uncertainty of the result of a measurement reflects the lack of exact knowledge of the value of the measurand.