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Measuring Uncertainty within the Theory of Evidence 2018 ed. [Kõva köide]

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  • Formaat: Hardback, 330 pages, kõrgus x laius: 235x155 mm, kaal: 688 g, 141 Illustrations, color; 13 Illustrations, black and white; XV, 330 p. 154 illus., 141 illus. in color. With online files/update., 1 Hardback
  • Sari: Springer Series in Measurement Science and Technology
  • Ilmumisaeg: 07-May-2018
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319741373
  • ISBN-13: 9783319741376
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Measuring Uncertainty within the Theory of Evidence 2018 ed.Suurem pilt
  • Formaat: Hardback, 330 pages, kõrgus x laius: 235x155 mm, kaal: 688 g, 141 Illustrations, color; 13 Illustrations, black and white; XV, 330 p. 154 illus., 141 illus. in color. With online files/update., 1 Hardback
  • Sari: Springer Series in Measurement Science and Technology
  • Ilmumisaeg: 07-May-2018
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319741373
  • ISBN-13: 9783319741376
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319741376.html
Märksõnad:
This monograph considers the evaluation and expression of measurement uncertainty within the mathematical framework of the Theory of Evidence. With a new perspective on the metrology science, the text paves the way for innovative applications in a wide range of areas. Building on Simona Salicone’s Measurement Uncertainty: An Approach via the Mathematical Theory of Evidence, the material covers further developments of the Random Fuzzy Variable (RFV) approach to uncertainty and provides a more robust mathematical and metrological background to the combination of measurement results that leads to a more effective RFV combination method.

While the first part of the book introduces measurement uncertainty, the Theory of Evidence, and fuzzy sets, the following parts bring together these concepts and derive an effective methodology for the evaluation and expression of measurement uncertainty. A supplementary downloadable program allows the readers to interact with the proposed approach by generating and combining RFVs through custom measurement functions. With numerous examples of applications, this book provides a comprehensive treatment of the RFV approach to uncertainty that is suitable for any graduate student or researcher with interests in the measurement field. 

1 Introduction
1(8)
Part I The Background of Measurement Uncertainty
2 Measurements
9(8)
2.1 The Theory of Error
10(4)
2.2 The Theory of Uncertainty
14(3)
3 Mathematical Methods to Handle Measurement Uncertainty
17(20)
3.1 Handling Measurement Uncertainty Within the Probability Theory
18(11)
3.1.1 Fundamental Concepts
18(1)
3.1.2 The Recommendations of the GUM
19(5)
3.1.3 The Recommendations of the Supplement to the GUM
24(2)
3.1.4 The Dispute About the Random and the Systematic Contributions to Uncertainty
26(3)
3.2 Handling Measurement Uncertainty Within the Theory of Evidence
29(6)
3.2.1 Fundamental Concepts
31(2)
3.2.2 The RFV Approach
33(2)
3.3 Final Discussion
35(2)
4 A First, Preliminary Example
37(50)
4.1 School Work A: Characterization of the Measurement Tapes
38(1)
4.2 School Work B: Representation of the Measurement Results
39(19)
4.2.1 Case IB
42(4)
4.2.2 Case 2B
46(9)
4.2.3 Case 3B
55(3)
4.3 School Work C: Combination of the Measurement Results
58(21)
4.3.1 Case 1C
61(4)
4.3.2 Case 2C
65(6)
4.3.3 Case 3C
71(4)
4.3.4 Case 4C
75(2)
4.3.5 Case 5C
77(2)
4.4 Conclusions
79(1)
4.5 Mathematical Derivations
80(7)
4.5.1 Example of Evaluation of the Convolution Product
80(3)
4.5.2 Example of Evaluation of the Coverage Intervals
83(4)
Part II The Mathematical Theory of Evidence
5 Introduction: Probability and Belief Functions
87(6)
6 Basic Definitions of the Theory of Evidence
93(14)
6.1 Mathematical Derivations
97(10)
6.1.1 Proof of Theorem 6.1
97(3)
6.1.2 Proof of Theorem 6.2
100(3)
6.1.3 Proof of Theorem 6.3
103(1)
6.1.4 Proof of Theorem 6.4
103(1)
6.1.5 Proof of Theorem 6.5
103(4)
7 Particular Cases of the Theory of Evidence
107(22)
7.1 The Probability Theory
107(4)
7.1.1 The Probability Functions
108(1)
7.1.2 The Probability Distribution Functions
109(1)
7.1.3 The Representation of Knowledge in the Probability Theory
110(1)
7.2 The Possibility Theory
111(8)
7.2.1 Necessity and Possibility Functions
112(3)
7.2.2 The Possibility Distribution Function
115(2)
7.2.3 The Representation of Knowledge in the Possibility Theory
117(2)
7.3 Comparison Between the Probability and the Possibility Theories
119(4)
7.4 Mathematical Derivations
123(6)
7.4.1 Proof of Theorem 7.1
123(1)
7.4.2 Proof of Theorem 7.2
124(1)
7.4.3 Proof of Theorem 7.3
124(2)
7.4.4 Proof of Theorem 7.4
126(1)
7.4.5 Proof of Theorem 7.5
126(1)
7.4.6 Proof of Theorem 7.6
127(1)
7.4.7 Proof of Theorem 7.7
127(1)
7.4.8 Proof of Theorem 7.8
127(1)
7.4.9 Proof of Theorem 7.9
128(1)
8 Operators Between Possibility Distributions
129(24)
8.1 Aggregation Operators
129(16)
8.1.1 t-Norm
131(4)
8.1.2 t-Conorm
135(5)
8.1.3 Averaging Operators
140(5)
8.2 Other Operators
145(4)
8.2.1 Fuzzy Intersection Area and Fuzzy Union Area
145(1)
8.2.2 Hamming Distance
146(1)
8.2.3 Greatest Upper Set and Greatest Lower Set
146(1)
8.2.4 Fuzzy-Max and Fuzzy-Min
147(2)
8.3 Mathematical Derivations
149(4)
8.3.1 Proof of Theorem 8.1
149(1)
8.3.2 Proof of Theorem 8.2
150(1)
8.3.3 Proof of Theorem 8.3
151(1)
8.3.4 Proof of Theorem 8.4
151(1)
8.3.5 Proof of Theorem 8.5
151(1)
8.3.6 Proof of Theorem 8.6
151(2)
9 The Joint Possibility Distributions
153(8)
9.1 Joint PDFs
153(4)
9.1.1 Conditional PDFs
157(1)
9.2 Joint PDs
157(4)
9.2.1 Conditional PDs
159(2)
10 The Combination of the Possibility Distributions
161(2)
11 The Comparison of the Possibility Distributions
163(4)
11.1 Definition and Evaluation of the Credibility Coefficients
163(4)
12 The Probability-Possibility Transformations
167(18)
12.1 1-D Probability-Possibility Transformations
167(3)
12.2 2-D Probability-Possibility Transformations
170(9)
12.2.1 Natural Extension of the 1-D p-p Transformation
171(1)
12.2.2 Ad Hoc 2-D p-p Transformation
172(7)
12.3 Mathematical Derivations
179(6)
Part III The Fuzzy Set Theory and the Theory of Evidence
13 A Short Review of the Fuzzy Set Theory
185(10)
13.1 Basic Definitions of the Fuzzy Set Theory
186(2)
13.2 Fuzzy Numbers
188(7)
14 The Relationship Between the Fuzzy Set Theory and the Theory of Evidence
195(12)
14.1 Equivalence of the Mathematical Definitions
195(5)
14.2 A Possible Misunderstanding
200(1)
14.3 Example
201(1)
14.4 Further Considerations
202(5)
Part IV Measurement Uncertainty Within the Mathematical Framework of the Theory of Evidence
15 Introduction: Toward an Alternative Representation of the Measurement Results
207(2)
16 Random-Fuzzy Variables and Measurement Results
209(14)
16.1 Why the RFV?
209(3)
16.2 From PDs to RFVs
212(6)
16.3 Definition of the RFVs
218(1)
16.4 Construction of the RFVs from the Available Information
218(5)
16.4.1 The Internal PD rint
219(1)
16.4.2 The Random PD rran
219(1)
16.4.3 The External PD rext and the RFV
220(3)
17 The Joint Random-Fuzzy Variables
223(4)
18 The Combination of the Random-Fuzzy Variables
227(42)
18.1 Nguyen's Theorem
227(1)
18.2 Interval Arithmetics
228(2)
18.3 Random PDs Combination
230(19)
18.3.1 Random Joint PD
230(14)
18.3.2 Random Interval Arithmetic
244(5)
18.4 Internal PD Combination
249(15)
18.4.1 α-Cuts of the Internal Joint PD
250(6)
18.4.2 Internal Joint PD
256(4)
18.4.3 Internal Interval Arithmetic
260(4)
18.5 Conditional RFVs
264(2)
18.6 Conclusion
266(3)
19 The Comparison of the Random-Fuzzy Variables
269(4)
20 Measurement Uncertainty Within Fuzzy Inference Systems
273(18)
20.1 The Standard Fuzzy Inference Systems
274(5)
20.1.1 The Steps of the Standard Fuzzy Inference Systems
275(4)
20.2 The Modified Fuzzy Inference Systems
279(12)
Part V Application Examples
21 Phantom Power Measurement
291(12)
21.1 Experimental Setup
291(1)
21.2 Uncertainty Evaluation
292(6)
21.3 Results
298(5)
22 Characterization of a Resistive Voltage Divider
303(6)
22.1 Experimental Setup
303(1)
22.2 Uncertainty Evaluation
304(2)
22.3 Results
306(3)
23 Temperature Measurement Update
309(6)
24 The Inverted Pendulum
315(8)
24.1 Definition of the FIS
315(3)
24.2 Results
318(5)
25 Conclusion
323(2)
References 325(4)
Index 329
Simona Salicone is Associate Professor of electrical and electronic measurements in the Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano. Her principal research interests are the analysis of advanced mathematical methods for uncertainty representation and estimation, and she has contributed to the development and application of the mathematical Theory of Evidence to the expression and evaluation of uncertainty in measurement.  Marco Prioli is an IEEE Instrumentation and Measurement Society member. He is also a memeber of the Italian Association for Electrical and Electronic Measurements (GMEE).