Development of empirical possibility distributions in risk analysis

The goal of this dissertation is to develop methods that can be used to systematically, consistently and logically generate uncertainty distributions that do not assume randomness in the system. Uncertainty is pervasive in all systems; even more so in systems such as risk assessment where information is accumulated from varied sources that are inherently complex and where the parameters of the system cannot be assumed to be random due to data insufficiencies. On the other hand, though non-probabilistic methods have been shown to be excellent tools to capture non-random uncertainty their application has been limited by their inability to offer methods for deriving distributions from empirical data. The motivation behind this dissertation is to find new models to overcome the difficulties inherent in the ability of probability theory to model non-random uncertainty, and to adapt these new models to capture both random and non-random uncertainty as empirical possibility distributions. Two novel methods based on possibility theory are proposed to represent uncertainty in systems.; The property of consonance in data, where evidence leads an expert/model to inductively make judgments that converge to one possible outcome, is assumed in the development contained herein. Crisp notions of classical logic with a truth-value of either 0 or 1 (assuming complete evidence) on disjoint sets/outcomes (A_i ∩ A_j = Ø) are replaced by softer notions where truth-values that range between 0 and 1 are defined over overlapping sets/outcomes (A_i ∩ A_i ≠ Ø). The new methods exploit set-based mechanisms such as interval analysis and cluster analysis to accomplish the task of deriving possibility distributions and to quantitatively represent imprecise knowledge. Relaxation of the axiom of additivity with the proper assumption of sub-additivity, allows the developments made here to represent inexact, imprecise, incomplete, and incoherent information in a more realistic and consistent manner. Two novel approaches for deriving possibility distributions are developed in this dissertation: (i) Method I is used for non-consistent and non-disjoint data intervals; and (ii) Method II is used for point estimates and disjoint data intervals. The new methods are illustrated through two case studies in human health risk assessment of radon gas exposure.