On the quantification of aleatory and epistemic uncertainty using Sliced-Normal distributions

This paper proposes a means to characterize multivariate data. This characterization, given in terms of both probability distributions and data-enclosing sets, is instrumental in assessing and improving the robustness properties of system designs. To this end, we propose the Sliced-Normal (SN) class of distributions. The versatility of SNs enables characterizing complex parameter dependencies with minimal modeling effort. A polynomial mapping which injects the physical space into a higher dimensional (so-called) feature space is first defined. Optimization-based strategies for the estimation of SNs from data in both physical and feature space are proposed. The non-convex formulations in physical space yield SNs having the best performance. However, the formulations in feature space either admit an analytical solution or yield a convex program thereby facilitating their application to high-dimensional datasets. The semi-algebraic form of the superlevel sets of a SN, form which a tight data-enclosing set can be readily identified, makes them amenable to rigorous worst-case based approaches to robustness analysis and robust design. Furthermore, we propose a chance-constrained optimization framework for identifying and eliminating the effects of outliers in the prescription of such a set. In addition, the distribution-free and non-asymptotic Scenario Theory framework is used to rigorously bound the probability of unseen data falling outside the identified data-enclosing set.