A fuzzy approach for the expression of uncertainty in measurement

Possibility transformation of the sum of two symmetric unimodal independent/comonotone random variables

“…Therefore, the distribution function is a straight line. A remarkable result is that the equivalent possibility distribution around the middle of the interval is a triangular symmetric possibility distribution [2]. The convolution product of two identical uniform distributions gives a triangular probability density, and consequently a parabolic possibility distribution (see figure 2).…”

“…The first representation provides a quite rich information but it is not always easy to determine, unlike the second representation that is more simple to obtain but also generally more rough. A third uncertainty representation based on the possibility theory [1] has been proposed in the late 70's by Dubois and Prade and further developed within the fuzzy community, among ones the author who has proposed a probability-possibility transformation identifying the dispersion intervals of level 1 α − of a probability distribution to the α-cuts of a fuzzy subset seen as a possibility distribution [2]. This identification is closely related to the transformation principles proposed originally by Dubois, Prade and Sandri [3] (consistency, preference preservation, least commitment).…”

Monte Carlo vs. Fuzzy Monte Carlo Simulation for Uncertainty and Global Sensitivity Analysis

“…Fuzzy set theory introduced by Zadeh [11] is an effective approach for representing epistemic uncertainty under imperfect knowledge. In the fuzzy set theory, fuzzy numbers are defined as convex and normalized fuzzy sets over the universal set with their fuzzy set membership functions that can be represented as a degree of likelihood between 0 and 1 [5,6,[12][13][14][15]. In general, the triangular fuzzy number (TFN) has been widely used ( Figure 1).…”

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

“…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.…”