Plenary Lectures
| Plenary Lecture 4 | An Introduction to Sliced-Normal Distributions |
| Date / Time | 25 September 2019, Wednesday / 09:00 - 09:40 hrs |
| Speaker | Luis Crespo, NASA Langley Research Center, Hampton, Virginia, USA |
Sliced-Normals (SN) enable the characterization of complex multivariate data as both a vector of possibly dependent random variables and as a semi-algebraic, tightly enclosing set. The versatility of SNs enables modeling complex parameter dependencies with minimal effort. A polynomial mapping is defined which injects the physical space into a higher dimensional (so-called) feature space. Optimization-based strategies for estimating SNs from data in both physical and feature space are presented. The formulations in physical space yield non-convex optimization programs whose solutions achieve the best performance. However, the formulations in feature space yield either an analytical solution or a convex program thereby facilitating their application to high dimensional datasets. Numerical experiments suggest that SNs are more versatile than most copula families thanks to their ability to handle multi-modal distributions with non-monotonic dependencies. Furthermore, whereas the selection of a copula structure is cumbersome process requiring extensive expertise, the selection of a SN structure only requires prescribing the degree of the polynomial mapping. Additionally, the closed and semi-algebraic form of the level sets of a SN density makes them amenable to rigorous uncertainty quantification. In this talk we use Scenario theory to rigorously bound the probability of new unseen data falling outside any of such level sets, and present a chance-constrained optimization framework for eliminating the effects of outliers in the identified uncertainty model. The strategies above can be used to mitigate the conservatism intrinsic to many methods in system identification, fault detection, robustness analysis, and reliability-based design.
