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A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulas

E. Torre, S. Marelli, P. Embrechts, B. Sudret

arXiv:1709.08626v2stat.MEstat.APstat.CO

TL;DR

Engineering UQ commonly assumes independent or Gaussian-dependent inputs, although such assumptions can bias response estimates when dependence is more complex. This paper develops data-driven vine-copula input models and combines them with UQ methods. Across two truss applications, suitable vines gave more reliable estimates than Gaussian assumptions, including when inferred from limited observations.

  • Problem

    Engineering UQ commonly assumes independent or Gaussian-dependent inputs, but Gaussian dependence can bias estimates when the true input dependence differs.

    Source: §1 · p. 2

  • Method

    The framework models input marginals and dependence separately, represents dependence with vine copulas, and combines them with UQ methods using available transform algorithms.

    Source: §1 · p. 3, §3.3 · p. 13, §7 · p. 32

  • Results

    In two truss systems, failure probabilities were mis-estimated by one to two orders of magnitude under independence and Gaussian assumptions, while vines fitted to relatively few observations yielded far better estimates.

    Source: §7 · p. 33

  • Takeaways & Limitations

    The results indicate that choosing a suitable input-dependence model can matter more for accurate output statistics than choosing a superior UQ algorithm.

    Source: §7 · p. 33

    The marginal distributions for the transformed variables in the PCE applications were chosen for this study, and finding optimal choices remains open.

    Source: §4.2 · p. 16

Abstract

from arXiv · show

Systems subject to uncertain inputs produce uncertain responses. Uncertainty quantification (UQ) deals with the estimation of statistics of the system response, given a computational model of the system and a probabilistic model of its inputs. In engineering applications it is common to assume that the inputs are mutually independent or coupled by a Gaussian or elliptical dependence structure (copula). In this paper we overcome such limitations by modelling the dependence structure of multivariate inputs as vine copulas. Vine copulas are models of multivariate dependence built from simpler pair-copulas. The vine representation is flexible enough to capture complex dependencies. This paper formalises the framework needed to build vine copula models of multivariate inputs and to combine them with virtually any UQ method. The framework allows for a fully automated, data-driven inference of the probabilistic input model on available input data. The procedure is exemplified on two finite element models of truss structures, both subject to inputs with non-Gaussian dependence structures. For each case, we analyse the moments of the model response (using polynomial chaos expansions), and perform a structural reliability analysis to calculate the probability of failure of the system (using the first order reliability method and importance sampling). Reference solutions are obtained by Monte Carlo simulation. The results show that, while the Gaussian assumption yields biased statistics, the vine copula representation achieves significantly more precise estimates, even when its structure needs to be fully inferred from a limited amount of observations.

1 Introduction

Accurate UQ depends on representing input uncertainty well, yet engineering practice often assumes independence or Gaussian dependence. The paper proposes data-driven vine-copula input models that can be combined with common UQ methods, and finds that suitable vines produce more reliable estimates than Gaussian assumptions in two truss applications.

1 Introduction

  • Because all response uncertainty comes from uncertain inputs, an unsuitable input distribution can undermine estimates regardless of the chosen UQ method.When model runs are limited, alternatives such as FORM, importance sampling, and polynomial chaos expansions are used instead of large Monte Carlo simulations.Source: §1 · p. 2
  • Independence and Gaussian or elliptical dependence are common assumptions, but Gaussian dependence can bias estimates when the true dependence differs.The impact and validity of the Gaussian assumption are typically not quantified.Source: §1 · pp. 2–3
  • Vine copulas represent multivariate dependence through simpler pair-copulas, offering a flexible construction for complex input dependencies.The paper frames vines as a response to the difficulty of building suitable higher-dimensional copulas.Source: §1 · p. 3
  • The framework combines vine input models with FORM, importance sampling, and polynomial chaos expansions, and supports fitting model structure and parameters to data.Available Rosenblatt-transform algorithms allow vines to be used with methods requiring independent inputs.Source: §1 · p. 3
  • For two truss models, Gaussian copulas gave wrong failure probabilities and response moments, while suitable vines yielded reliable estimates even when inferred from available data.The bias was inherent to the input-dependence representation, not corrected by changing the UQ method.Source: §1 · p. 4

2 Copulas and vine copulas

This section develops copula and vine-copula representations for multivariate input dependence, from separating marginals and dependence to constructing and fitting flexible models from data.

2.1 Copulas and Sklar’s theorem

  • A joint distribution can be built by combining marginal CDFs with a copula, which describes dependence separately from marginal behavior.For continuous marginals, the copula is unique.Source: §2.1 · pp. 4–5

2.2 Copula-based measures of dependence

  • Copula-based dependence measures describe dependence independently of the marginal distributions, while tail dependence captures the persistence of joint extreme values.The text gives Spearman’s rho and Kendall’s tau as examples of copula-based concordance measures.Source: §2.2 · pp. 5–6

2.3 Copula examples

  • The section presents independence, Gaussian, and Gumbel-Hougaard copulas as examples for representing different dependence patterns.The Gaussian copula assigns negligible probability to joint extremes, whereas the Gumbel-Hougaard copula accommodates upper-tail dependence.Source: §2.3 · pp. 6–7

2.4 Vine copulas

  • Vine copulas factor a multivariate copula into simpler unconditional or conditional pair-copulas, easing construction as input dimension grows.An R-vine organizes the non-unique factorization as a sequence of trees, with pair-copulas associated with edges and conditioning variables.Source: §2.4 · pp. 7–8
  • C-vines and D-vines are special R-vines with different tree structures and associated conditioning arrangements.Figure 1 illustrates five-dimensional examples of both structures.Source: §2.4 · p. 9

2.5 Vine inference in practice

  • Data-driven vine inference selects a structure, chooses pair-copula families, and estimates their parameters, commonly using dependence ordering and AIC-based fitting.The simplified-vine assumption is commonly used for inference but is exact only in particular cases.Source: §2.5 · pp. 9–10

3 Vine representations for UQ methods assuming independent inputs

The section explains how dependent inputs can be transformed for UQ methods that require independence. Vine copulas support the needed Rosenblatt transforms and resampling, while generalized Nataf transforms can introduce errors outside their appropriate settings.

3.1 Compositional models for dependent inputs

  • A known invertible isoprobabilistic transform lets a UQ method for independent inputs operate on a compositional model with the same response statistics as the original system.The composition applies the inverse transform before the original computational model.Source: §3.1 · p. 12

3.2 Isoprobabilistic transforms and copulas

  • The Rosenblatt transform maps inputs to independent components, but generalized Nataf transforms may cause non-negligible estimation errors for non-elliptical inputs or when independence is required.The generalized Nataf transform for elliptical copulas yields uncorrelated components that are not mutually independent except in the Gaussian case.Source: §3.2 · p. 13

3.3 Rosenblatt transform and resampling for R-vines

  • Algorithms compute Rosenblatt and inverse Rosenblatt transforms for vine models, enabling independent-space methods and resampling from the fitted input model.The cited algorithms support C-vines, D-vines, and general R-vines.Source: §3.3 · pp. 13–14

4 UQ for mutually dependent inputs

The framework combines vine input models with established UQ methods for coupled inputs, including methods that benefit from mapping inputs to independent variables. Its subsections describe Monte Carlo estimator convergence, polynomial chaos expansion, first-order reliability analysis, and importance sampling.

4 UQ for mutually dependent inputs

  • The framework extends PCE, FORM, and importance sampling to finite-input UQ problems with coupled inputs, including methods that benefit from independent-variable mappings.The authors do not advocate for these three methods over other UQ techniques.Source: §4 · p. 14

4.1 Convergence of MC estimates

  • Monte Carlo estimates of the mean, standard deviation, and failure probability have coefficients of variation that decay proportionally to 1/√n; a 10% CoV is a common engineering target.The standard-deviation estimator's CoV expression is only exact when Y is normally distributed.Source: §4.1 · pp. 14–15

4.2 Polynomial Chaos Expansion

  • PCE represents a finite-variance model with orthogonal polynomials, and an R-vine Rosenblatt transform maps dependent inputs to independent variables for constructing the expansion.The study uses adaptive sparse PCE; optimal choices for the independent variables' marginals remain an open question.Source: §4.2 · pp. 15–16

4.3 First order reliability method

  • For non-elliptical inputs, an R-vine Rosenblatt transform maps inputs to standard normal space, where classical FORM can search for the design point.Reliability analysis evaluates the probability mass in the failure domain, which is typically known only implicitly.Source: §4.3 · p. 17

4.4 Importance sampling

  • Importance sampling combines FORM's convergence speed with Monte Carlo robustness by sampling from a suitable density, with sample size increased until the estimated CoV meets a target.The importance density may be centered at the design point found by FORM.Source: §4.4 · p. 18

5 Results on a horizontal truss model

The horizontal-truss study compares response-moment and failure-probability estimates under independent, Gaussian, known-vine, and data-inferred-vine load couplings. Tail-dependent vine coupling substantially raises estimated failure probability, while a vine inferred from 300 observations gives good deflection-statistic estimates.

5.1 Computational model

  • The horizontal truss has 23 bars and six loaded upper nodes; its largest response is center deflection, with bar properties held fixed.Source: §5.1 · pp. 18–19, §5.2 · p. 19

5.2 Probabilistic input model

  • Gaussian and C-vine loads share Spearman correlation 0.135 with P1, but the vine adds upper-tail dependence λu=0.122.The loads use fixed Gumbel marginals; the C-vine imposes conditional independence among the other loads given P1.Source: §5.2 · pp. 19–20
  • A C-vine learned from 300 draws differed from the true structure but recovered most independence pair-copulas and gave very good deflection-statistic estimates.The inferred model had two Gumbel-Hougaard pairs in its first tree and one non-independence conditional pair-copula.Source: §5.2 · p. 20

5.3 Analysis of the moments for different load couplings

  • σ(∆) rises almost 10% from independence to the vine, while independence-based moment estimates are biased and Gaussian or fitted C-vine estimates are more accurate.Source: §5.3 · pp. 20–21
  • At equal sample counts, PCE error is significantly smaller than MCS; reference-error curves are reliable only to about 10^-4 for means and 10^-3 for standard deviations.Figure 3 compares error estimates for µ(∆) and σ(∆) across the four load couplings.Source: §5.3 · pp. 21–22, Figure 3 · p. 22

5.4 Reliability analysis for different load couplings

  • With identical marginals and matched Spearman correlations, C(V) yields failure probability over 33× independence and 14× Gaussian; data-inferred vines keep the error small.The study also reports that FORM with C(V) outperforms MCS with C(N) using 10^7 samples.Source: §5.4 · pp. 23–24, Figure 4 · p. 24

6 Results on a dome truss model under asymmetric loads

The dome study applies the vine-based UQ framework to asymmetric loads on a 120-bar truss, comparing independence, Gaussian, true-vine, and data-inferred-vine models. Its results show that the assumed input dependence strongly affects estimated moments and failure probability.

6.1 Computational model

  • A three-dimensional 120-bar dome truss demonstrated the copula-based UQ framework on a more realistic case; its ∼15 seconds/run made large MCS unaffordable.The structure has 49 nodes; nodes 1 to 37 are unsupported and, under vertical loading, may displace from their original positions in possibly all directions.Source: §6 · p. 24, §6.1 · p. 25

6.2 Probabilistic input model

  • Nine sector loads use Gumbel marginals and are coupled by independence, a Gaussian copula, or a C-vine with strong positive dependence and upper-tail dependence.The C-vine links sector A to the other sectors with Gumbel-Hougaard pair copulas; the inferred vine from 300 samples recovered the same structure and pair-copula families.Source: §6.2 · pp. 27–28

6.3 Model response to the uncertain input

  • The response is reduced to the vertical displacement at node 2, which had the largest absolute vertical displacement in all but 17 of 1,000 simulations.Excessive displacement in any direction is defined as failure; the dome analysis uses a critical displacement of −11 mm.Source: §6.3 · pp. 28–29, §6.5 · p. 30

6.4 Analysis of the moments

  • With the true vine as reference, independence and Gaussian models misestimate deflection standard deviation by 111% and 6.5%, respectively.The inferred vine with Monte Carlo estimates moments more precisely than the independence or Gaussian models, while PCE combined with the vine yields the smallest errors.Source: Figure 6 · p. 30, Table 9 · p. 30, §6.4 · p. 30

6.5 Reliability analysis

  • Failure probability falls by an order of magnitude from independence to Gaussian loads, then by another order from Gaussian to true or inferred vine loads.Under the true C-vine, independence and Gaussian assumptions overestimate failure probability; data-driven vine inference recovers the reference solution with high precision.Source: Figure 7 · p. 31, §6.5 · pp. 31–32, Table 10 · p. 32

7 Discussion

The framework combines vine-copula models of complex input dependencies with UQ methods, including methods that benefit from independent inputs. In the truss examples, independence and Gaussian assumptions biased estimates, while fitted vine models gave better estimates; vine selection and computation remain challenging in very high dimensions or on large samples.

7 Discussion

  • Vine copulas model complex, non-Gaussian, non-elliptical input dependencies as products of simpler pair-copulas, supporting a wide range of dependence structures.Separating marginals and the copula eases model construction, especially in high dimensions.Source: §7 · p. 32
  • R-vines support data-driven fitting and Rosenblatt transforms, allowing UQ methods that benefit from independent inputs to be applied to dependent inputs with continuous marginals.The discussion notes that extensions to discrete data have been proposed, but this work restricts attention to continuous marginals.Source: §7 · p. 32
  • Independence and Gaussian assumptions mis-estimated failure probabilities by one to two orders of magnitude, while fitted vine models yielded far better estimates.The comparison held regardless of the UQ method used.Source: §7 · p. 33
  • Using a suitable input-dependence model can matter more for accurate output statistics than using a superior UQ algorithm.The authors also report that vine representations can be inferred for simple Gaussian dependencies and cover a broader range of problems than the Nataf transform, including its generalized form.Source: §7 · p. 33
  • Automated vine selection may become computationally prohibitive with many dimensions or pair-copula families, and selection remains open above roughly 50 dimensions or on very large samples.Rosenblatt transforms may also be computationally demanding or numerically unstable in these cases.Source: §7 · p. 33

A Some families of pair copulas and their properties

The appendix describes the pair-copula families and properties used for vine inference, including rotated families and measures of dependence. In this study, family selection used AIC, and rotations expanded the set considered from 19 to 62.

A Some families of pair copulas and their properties

  • Nineteen parametric pair-copula families and their rotated versions were considered, for 62 families in total, with the family yielding the lowest AIC selected for each pair-copula.Table 11 lists the implemented families used for inference.Source: A Some families of pair copulas and… · p. 39, Table 11 · p. 40
  • The appendix gives formulas for the 180° and 270° rotated copulas and notes that the 90° and 270° rotations flip the copula density around the horizontal and vertical axes, respectively.The text also provides alternative formulas for the 90° and 270° rotations.Source: A Some families of pair copulas and… · p. 39
  • A 180° rotation switches lower and upper tail-dependence coefficients, while 90° and 270° rotations model negative dependence.Table 12 reports Kendall’s tau and lower and upper tail-dependence coefficients for considered pair-copula families.Source: A Some families of pair copulas and… · p. 39, Table 12 · p. 41
  • The reported family properties include analytical expressions for Kendall’s tau and tail-dependence coefficients when available, with some expressions derived by the authors.The appendix identifies the standard normal and bivariate normal distributions, and the univariate and bivariate t distributions used in the notation.Source: A Some families of pair copulas and… · pp. 39–41
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