Random variables with moment-matching staircase density functions

• Published in: Applied Mathematical Modelling Vol. 64; pp. 196 - 213
• Main Authors: Crespo, Luis G., Kenny, Sean P., Giesy, Daniel P., Stanford, Bret K.
• Format: Journal Article
• Language: English
• Published: Langley Research Center Elsevier Inc 01.12.2018; Elsevier; Elsevier BV
• Subjects: Aeroelastic stability; Aeroelasticity; Computational geometry; Convex analysis; Convexity; Datasets; Density; Dynamic stability; Empirical analysis; Flutter; Mathematical models; Moments; Numerical Analysis; Optimality criteria; Optimization; Probability; Probability distribution; Random variables; Risk; Risk analysis; Sampling error; Staircases; Uncertainty; Probability; Risk; Uncertainty; Moments; Optimization
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2018.07.029

•This paper proposes the means to model phenomena exhibiting a possibly skewed and multimodal response.•The approach is based on calculating variables having a finite range and fixed values for the first four moments.•This paper provides the means to estimate the above variables and to quantify the corresponding sampling error.•The versatility of the method is illustrated by modeling the dynamics of an aeroelastic structure subject to flutter. This paper proposes a family of random variables for uncertainty modeling. The variables of interest have a bounded support set, and prescribed values for the first four moments. We present the feasibility conditions for the existence of any of such variables, and propose a class of variables that conforms to such constraints. This class is called staircase because the density of its members is a piecewise constant function. Convex optimization is used to calculate their distributions according to several optimality criteria, including maximal entropy and maximal log-likelihood. The flexibility and efficiency of staircases enable modeling phenomena having a possibly skewed and/or multimodal response at a low computational cost. Furthermore, we provide a means to account for the uncertainty in the distribution caused by estimating staircases from data. These ideas are illustrated by generating empirical staircase predictor models. We consider the case in which the predictor matches the sample moments exactly (a setting applicable to large datasets), as well as the case in which the predictor accounts for the sampling error in such moments (a setting applicable to sparse datasets). A predictor model for the dynamics of an aeroelastic airfoil subject to flutter instability is used as an example. The resulting predictor not only describes the system's response accurately, but also enables carrying out a risk analysis for safe flight.