L-moments and Chebyshev inequality driven convex model for uncertainty quantification

Abstract

In the design of real-world complex engineering systems and infrastructure, one typically deals with scarce data. When only scarce data are available, convexity approaches are widely used to construct the uncertainty model in terms of bounds. Convex models estimate the uncertainty bounds using different bounding geometries that seek minimum volume with the available points. Chebyshev inequality in conjunction with geometries such as convex hull allows accounting for unobserved future data points by inflating the initially constructed geometry. Since the Chebyshev inequality is independent of underlying distribution, the inflated convex geometry is often highly conservative. To address the conservativeness, we propose constructing a convex hull with a modified inflation coefficient that uses information on input dimension and spread of data. Since conventional metrics for spread of data such as kurtosis is volatile when sample is scarce, we use L-moments which are known to be robust to sample size. The initial convex hull built using samples from a distribution characterized by L-kurtosis, is inflated incrementally until allowable number of points corresponding to target probability are violated. The corresponding inflation coefficient is considered the modified coefficient. This exercise is repeated for different L-kurtosis and dimensions, and the modified inflation coefficient are obtained for different target probability as well to construct an empirical relationship between L-kurtosis, number of dimensions, and modified inflation coefficient. The proposed approach is demonstrated on three design examples, and the results reveal that the design from the proposed approach is less conservative compared to the classical approach while solving the probabilistic formulation deterministically with scarce samples. The performance also indicates that the proposed approach has good generalization and prediction capability.

Appendices

Appendix 1

1.1 C-moments and L-moments comparison

L-moments are robust than conventional moments (C-moments) in the presence of extremes and less sensitive to small sample size. Even if there are no extremes L-moments perform as good as C-moments if not better. A comparison between the C-moments and L-moments estimated from a sample size of 10, for medium tail is presented here. The Fig. 11 present the comparisons of ratios of population estimate to sample estimate of the respective quantity. It is clear that the variation of L-moments is low compared to that of the C-moments. Though the presented results are only for medium tail, other types of tails also exhibit similar performance.

Fig. 11

C-moments and L-moments comparison for medium tail

Appendix 2

1.1 Optimization flowchart

The optimization framework followed in this work is presented in Fig. 12. In the presented optimization formulations, the objective function is a function of only design variables and the constraints are a function of design variable and random variables. We construct the convex hull in random variable space and estimate the extrema of performance for a given design variable combination and, are used in the constraint evaluation. That is for instance, in cantilever beam example, maximum stress is computed and used in the constraint as $S_{\text{max}} < S_{\text{allowable}}$.

Fig. 12

Optimization framework for cantilever design example