Uncertainty: Only One Mathematical Approach to Its Evaluation and Expression?

Abstract

This paper considers the well-known concept of uncertainty of a measurement result and discusses the mathematical approach adopted for its evaluation and expression by the present standard documents and, in particular, the Guide to the Expression of Uncertainty in Measurement. The practical limitations introduced by the assumptions of this mathematical approach are discussed, and the requirements of a new mathematical approach capable of overcoming the present limitations are drawn. A different mathematical approach, based on the theory of evidence and originally developed by the authors in recent years, is presented and discussed, with some new findings and observations

Possibility and probability: application examples and comparison of two different approaches to uncertainty evaluation

“…Without entering too many mathematical details, for which the reader is addressed to [6,7], a possibility distribution (PD) is defined as a function r over a support X r :…”

“…can be considered as a generalization of the probabilistic concept of confidence intervals and a credibility that an element x belongs to them can be associated to each interval, in the range [0,1] [6,8]. In particular, the credibility value is given, for each α-cut at level α, by 1 − α [7].…”

“…Since the final aim of uncertainty evaluation is "to provide an interval about the result of a measurement within which the values that could reasonably be attributed to the measurand may be expected to lie with a high level of confidence" [1], expressing a measurement result in terms of a PD is in complete agreement with the requirements of the GUM. It has been proved [6] that a PD can effectively represent the effect of any kind of contributions to uncertainty. However, different effects (random and nonrandom) give different contributions to uncertainty and combine in different ways.…”

“…In order to represent such effects and their combination with other random and non-random effects in a more correct and effective way, a new mathematical framework has been proposed in the recent years [3][4][5][6], based on the theories of evidence and possibility, that generalize probability and allow also non-random effects to be handled.While the theoretical framework has been well 1 Corresponding author: alessandro.ferrero@polimi.it developed [6], few practical applications have been considered, up to now. This paper, after having briefly recalled the very fundamentals of uncertainty expression in terms of possibility distributions, is aimed at showing the effectiveness of this new approach in two simple, though significant cases, where unknown and uncompensated systematic effects may cause the GUM approach to provide incorrect results.…”

Uncertainty evaluation by means of joint possibility distributions

“…While the uncertainty concept has been universally accepted, the purely probabilistic approach suggested by the GUM has been considered too restrictive, since it implies the identification and compensation of all significant systematic effects. Especially in industrial applications, this may not be cost effective, and therefore a more general mathematical framework should be considered to evaluate the contributions of uncompensated systematic effects and their combination with the random ones on measurement uncertainty [2].…”