Quantification and Decomposition of Uncertainty Using Sliced-Normal Distribution: With Applications to NASA Data

2026-08-14

AI Summary

This study addresses the challenges of modeling nonlinear dependencies and quantifying uncertainty in multivariate distributions by reformulating parameter estimation within the sliced normal distribution framework as a semidefinite convex optimization problem. Theoretically, it establishes a universal approximation foundation for polynomial log-densities, while methodologically introducing high-dimensional block-wise fitting and cross-block completion strategies to enhance scalability. Validation on NASA loss-of-control flight data demonstrates that the proposed approach effectively captures complex nonlinear dependency patterns, significantly improving both the accuracy of high-dimensional uncertainty quantification and model reliability. Collectively, this work offers a novel paradigm for achieving analytical structural tractability in high-dimensional density estimation.

Abstract

Modeling multivariate distributions with nonlinear dependence, multimodality, and tractable analytical structure for downstream applications is a central challenge in uncertainty quantification. Sliced Normal (SN) distributions were introduced in prior works at the National Aeronautics and Space Administration (NASA) to address this need by representing densities through polynomial feature maps. This construction provides a compact algebraic alternative to more opaque generative models, while retaining the ability to capture nonlinear parameter dependencies and multi-modal behavior. In this paper, we build on the SN framework and develop several improvements that make the approach more reliable and scalable. First, we reformulate SN parameter estimation as a convex optimization problem over a positive semidefinite matrix, replacing the original nonconvex likelihood search with a formulation amenable to standard optimization tools. Second, we clarify the expressive power of the SN class by connecting polynomial log-density modeling to a Stone--Weierstrass-type universal approximation argument on compact domains. Third, we propose a high-dimensional fitting procedure that partitions variables into approximately independent groups, fits SN models within each subgroup, and then assembles the subgroup models through a cross-block completion step to recover residual dependence. We demonstrate the resulting SN modeling pipeline on NASA loss-of-control flight data, where the method captures nonlinear dependence patterns in both low-dimensional slices and a higher-dimensional block-assembled model.

Problem

• Uncertainty Quantification
• Multivariate Distribution Modeling
• Nonlinear Dependence
• Multimodality

Innovation

• Sliced-Normal Distribution
• Convex Optimization
• Universal Approximation
• High-dimensional Fitting
• Uncertainty Quantification