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

1 | |

2 | |

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 |

225 | |

227 | |

### Other editions - View all

Measurement Uncertainty: An Approach via the Mathematical Theory of Evidence Simona Salicone No preview available - 2006 |

Measurement Uncertainty: An Approach via the Mathematical Theory of Evidence Simona Salicone No preview available - 2010 |

### Common terms and phrases

a-cut applied associated available information averaging operations basic probability assignment Bel(A belief function Central Limit Theorem Ceq(A Chapter confidence interval confidence of type credibility coefficients credibility factor defined definition degree of belief example fact focal elements fuzzy variable given Hamming distance Hence hypothesis of total initial distributions internal membership function intersection area interval of confidence Let us consider level of confidence mathematical mean value measurand measurement uncertainty min{a normal distribution obtained histogram OWA operator possibility distribution function possibility functions possibility theory probability density function probability distribution function probability theory random contributions random variables random–fuzzy variable rectangular represent respectively RFVs shown in Fig standard deviation standard fuzzy intersection standard fuzzy union subsets systematic contributions t-conorm t-norm Theorem Theory of Evidence total ignorance total negative correlation total positive correlation total uncorrelation triangular norm universal set variable of type width XD m(B zero

### 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.