Measurement Uncertainty: An Approach via the Mathematical Theory of Evidence

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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

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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.

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