Measuring Uncertainty within the Theory of Evidence

Abstract

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. Intro; Preface; Contents; 1 Introduction; Part I The Background of Measurement Uncertainty; 2 Measurements; 2.1 The Theory of Error; 2.2 The Theory of Uncertainty; 3 Mathematical Methods to Handle Measurement Uncertainty; 3.1 Handling Measurement Uncertainty Within the Probability Theory; 3.1.1 Fundamental Concepts; 3.1.2 The Recommendations of the GUM; 3.1.3 The Recommendations of the Supplement to the GUM; 3.1.4 The Dispute About the Random and the Systematic Contributions to Uncertainty; 3.2 Handling Measurement Uncertainty Within the Theory of Evidence; 3.2.1 Fundamental Concepts. 3.2.2 The RFV Approach3.3 Final Discussion; 4 A First, Preliminary Example; 4.1 School Work A: Characterization of the Measurement Tapes; 4.2 School Work B: Representation of the Measurement Results; 4.2.1 Case 1B; Solution Given by the GUM Approach; Solution Given by the MC Approach; Solution Given by the RFV Approach; Comparison and Discussion; 4.2.2 Case 2B; Solution Given by the GUM Approach; Solution Given by the MC Approach; Solution Given by the RFV Approach; Comparison and Discussion; Further Considerations; 4.2.3 Case 3B; Solution Given by the GUM and MC Approaches. Solution Given by the RFV ApproachComparison and Discussion; 4.3 School Work C: Combination of the Measurement Results; 4.3.1 Case 1C; Solution Given by the GUM Approach; Solution Given by the MC Approach; Solution Given by the RFV Approach; Comparison and Discussion; 4.3.2 Case 2C; Solution Given by the GUM Approach; Solution Given by the MC Approach; Solution Given by the RFV Approach; Comparison and Discussion; 4.3.3 Case 3C; Solution Given by the GUM Approach; Solution Given by the MC Approach; Solution Given by the RFV Approach; Comparison and Discussion; 4.3.4 Case 4C. Solution Given by the GUM and MC ApproachesSolution Given by the RFV Approach; Comparison and Discussion; 4.3.5 Case 5C; Solution Given by the GUM and MC Approaches; Solution Given by the RFV Approach; Comparison and Discussion; 4.4 Conclusions; 4.5 Mathematical Derivations; 4.5.1 Example of Evaluation of the Convolution Product; 4.5.2 Example of Evaluation of the Coverage Intervals; Part II The Mathematical Theory of Evidence; 5 Introduction: Probability and Belief Functions; 6 Basic Definitions of the Theory of Evidence; 6.1 Mathematical Derivations; 6.1.1 Proof of Theorem 6.1. 6.1.2 Proof of Theorem 6.26.1.3 Proof of Theorem 6.3; 6.1.4 Proof of Theorem 6.4; 6.1.5 Proof of Theorem 6.5; 7 Particular Cases of the Theory of Evidence; 7.1 The Probability Theory; 7.1.1 The Probability Functions; 7.1.2 The Probability Distribution Functions; 7.1.3 The Representation of Knowledge in the Probability Theory; 7.2 The Possibility Theory; 7.2.1 Necessity and Possibility Functions; 7.2.2 The Possibility Distribution Function; 7.2.3 The Representation of Knowledge in the Possibility Theory; 7.3 Comparison Between the Probability and the Possibility Theories.