The Construction of Joint Possibility Distributions of Random Contributions to Uncertainty

Abstract

The evaluation and expression of uncertainty in measurement is one of the fundamental issues in measurement science and challenges measurement experts especially when the combined uncertainty has to be evaluated. Recently, a new approach, within the framework of possibility theory, has been proposed to generalize the currently followed probabilistic approach. When possibility distributions are employed to represent random contribution to measurement uncertainty, their combination is still an open problem. This combination is directly related to the construction of the joint possibility distribution, generally performed by means of t-norms. The problem is that joint possibility distributions strongly depend on the considered t-norm, and, therefore, the choice of one particular t-norm over another has to be justified. The first goal of this paper is, hence, to provide support to the choice of a particular t-norm. The second goal is to discuss the construction of the joint possibility distribution when the two random contributions to uncertainty show mutual dependence, with specific reference to a particular class of possibility distributions.

Combination of measurement uncertainty contributions via the generalized Dombi operator

“…To keep the advantages of a unique approach to measurement uncertainty, capable of representing and processing all contributions to uncertainty, it is then important to define an optimal way to combine random contributions in the possibility domain. This paper considers the generalized Dombi operator [9] and shows that it can provide satisfactory approximations in many cases of interest in the measurement field, much better than the ones previously obtained by the same authors employing single-parameter t-norm families [8].…”

“…Therefore, f represents a generic measurement function. To obtain r Z , the joint PD r X,Y has to be first evaluated starting from the available metrological information [8,11]. In most cases, the available metrological information concerns the possible values of the measured variables and their relation and leads, as shown in [17], to the construction of the marginal PD r X and the conditional PD r Y |X , already defined in [18].…”

“…In fact, two different tnorms can lead to two significantly different joint PDs, and, therefore, two different PDs associated with measurand Z are obtained by means of (2). For this reason, a criterion has been identified in [8,11] to find the optimal t-norm for the combination of measurement uncertainty.…”

“…While possibility allows one to represent and combine systematic contributions in a strict way, when used to combine random contributions it can only provide approximate solutions [2,8] to the strict ones obtained in the probability domain. To keep the advantages of a unique approach to measurement uncertainty, capable of representing and processing all contributions to uncertainty, it is then important to define an optimal way to combine random contributions in the possibility domain.…”

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

“…Since the two PDs rint(x) and rran(x) represent different effects that combine in different ways, the mathematics used to combine RFVs must consider the different characteristics of these two PDs. This mathematics exploits the definition of joint PDs, given in [9] for the internal PDs and in [10] for the random PDs, and the application of the Zadeh's extension principle [11], as also shown in [12,13], to which the reader is addressed for further details.…”