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Quadratic Point Estimate Method for Uncertainty Quantification with Dependent Non-Gaussian Inputs

Minhyeok Ko, Konstantinos G. Papakonstantinou

arXiv:2609.09053v1math.NAcs.NAmath.PRmath.STstat.COstat.MEstat.TH

TL;DR

Higher-order moment estimation with dependent, non-Gaussian inputs motivates extending point-estimate methods beyond Gaussian dependence assumptions. The paper integrates copulas with QPEM and reports a framework for mapping deterministic sigma points to dependent inputs; its conclusion cautions that no particular non-Gaussian dependence model is universally preferable.

  • Problem

    The paper addresses uncertainty quantification for dependent, non-Gaussian inputs, where Gaussian assumptions may not represent realistic engineering dependencies.

    Source: INTRODUCTION · p. 2

  • Method

    The framework combines QPEM with vine copulas and uses the inverse Rosenblatt transformation to map deterministic sigma points from independent standard Gaussian space to dependent inputs.

    Source: CONCLUSION · p. 31

  • Results

    The conclusion reports approximate moment-error values for MC-Fitted and a truss example, but the supplied passages do not fully identify the corresponding metrics and comparisons.

    Source: CONCLUSION · p. 32

  • Takeaways & Limitations

    The conclusion states that different joint distributions consistent with the same available information need not make one non-Gaussian dependence model universally preferable.

    Source: CONCLUSION · p. 32

    No particular non-Gaussian dependence model is universally preferable; the conclusion limits this claim to models consistent with the same available information.

    Source: CONCLUSION · p. 32

Abstract

from arXiv · show

As an extension of the Point Estimate Method (PEM) to evaluate probabilistic moments of quantities of interest (QoI) in general $n$-dimensional spaces, the Quadratic Point Estimate Method (QPEM) has been recently developed. This new method is defined to fully represent up to fifth-order input moments in the Gaussian space, providing general analytical expressions for sample locations and weights, without requiring any numerical optimization. The QPEM can significantly improve the estimation accuracy of the output QoI moments, in relation to PEM-based methods whose numbers of sigma points grow linearly with the problem dimension, while at the same time having an affordable and competitive computational cost up to a considerable number of dimensions. The QPEM is further enhanced in this work by enabling copula integration into the framework, which enables effective modeling of the joint input probability density function by estimating marginals and the dependence structure of the involved random variables. The validity and efficient performance of the copula-based QPEM are showcased against numerous other sampling methods in various examples considering two practical scenarios: (i) when the joint dependence structure can be inferred from data, and (ii) when only marginal distributions and correlation matrices are known.

INTRODUCTION

High-dimensional uncertainty quantification makes accurate higher-order moment estimation costly for many point-estimate methods. This work integrates vine copulas with QPEM to model dependent, non-Gaussian inputs while retaining computational efficiency.

INTRODUCTION

  • High-dimensional probabilistic problems challenge existing numerical techniques, while higher-order PEM moment estimates can require more sigma points and optimization.The resulting trade-off is between computational efficiency and accuracy.Source: INTRODUCTION · p. 2
  • QPEM was introduced to improve higher-order output-moment accuracy by capturing Gaussian input moments through fifth order without optimization.The introduction reports reduced higher-order output-moment errors relative to traditional PEM methods.Source: INTRODUCTION · p. 3
  • Gaussian dependence assumptions may misrepresent nonlinear or tail-dependent engineering inputs, motivating frameworks that represent broader dependence structures.The introduction distinguishes cases with data for dependence inference from those with only marginal distributions and pairwise correlations.Source: INTRODUCTION · pp. 3–4
  • QPEM integrates vine copulas to model joint input PDFs with complex, non-Gaussian dependencies while retaining the method’s computational efficiency.The framework is considered both when data support dependence inference and when only marginals and pairwise correlations are known.Source: INTRODUCTION · pp. 4–5

QUADRATIC POINT ESTIMATE METHOD

QPEM uses a deterministic, symmetric sigma-point set to represent Gaussian input moments through fifth order. Its analytical locations and weights avoid optimization and support higher-order output-moment estimation.

QUADRATIC POINT ESTIMATE METHOD

  • QPEM captures up to fifth-order moments of an n-dimensional standard Gaussian using 2n^2 + 1 sigma-point evaluations.Its symmetric points automatically satisfy odd-order zero moments, and analytical locations and weights eliminate optimization.Source: QUADRATIC POINT ESTIMATE METHOD · p. 6
  • The sigma-point set comprises an origin point, 2n axis points, and 2n(n−1) points formed from permutations and sign variations of two nonzero coordinates.The respective point groups have weights w0, w1, and w2.Source: QUADRATIC POINT ESTIMATE METHOD · pp. 6–7
  • The framework can adjust scaling parameters to capture input moments beyond fifth order without additional computational cost.The parameters modify the origin weight to tune output skewness and kurtosis estimation.Source: QUADRATIC POINT ESTIMATE METHOD · p. 7
  • QPEM places sigma points in standard normal space, transforms them to the original input space, evaluates the model, and estimates response moments.The displayed central-moment orders are 2, 3, and 4.Source: QUADRATIC POINT ESTIMATE METHOD · pp. 7–8

Copula and Sklar’s theorem

Copulas separate a joint distribution’s dependence structure from its marginal distributions. The section also describes Kendall’s tau as a rank-based measure linked to the copula.

Copula and Sklar’s theorem

  • The copula-density formulation separates dependence structure from the marginal PDFs.The joint density combines the copula density with the individual marginal densities.Source: Copula and Sklar’s theorem · p. 8
  • Kendall’s tau is favored for capturing nonlinear relationships independently of marginal distributions.Unlike linear correlation, it is invariant under strictly monotonic marginal transformations and is explicitly linked to the copula function.Source: Copula and Sklar’s theorem · p. 9

Elliptical copulas and Nataf transformation

The Nataf transformation connects correlated physical-space variables to a standard normal space through marginal transformations. Its Gaussian copula represents dependence through a correlation matrix, limiting it to linear relationships.

Elliptical copulas and Nataf transformation

  • The Gaussian copula is defined using a multivariate standard normal distribution and a correlation matrix.The section links this formulation to the Nataf transformation used in structural reliability frameworks.Source: Elliptical copulas and Nataf… · p. 9
  • The Nataf transformation maps correlated variables to a standard normal space using their marginal transformations.The joint density in this framework uses an n-dimensional standard normal density characterized by a correlation matrix and a univariate standard normal density.Source: Elliptical copulas and Nataf… · p. 9
  • Because the Gaussian copula is defined by a linear correlation matrix, it is restricted to capturing linear relationships between variables.The section notes that real-world phenomena may have complex nonlinear dependencies that linear correlation may not represent adequately.Source: Elliptical copulas and Nataf… · p. 10

Vine copulas

Pair-Copula Construction (PCC) builds multivariate copula densities from bivariate pair-copulas, while regular vines organize these dependencies through nested trees. These approaches address challenges in representing complex dependencies as dimension increases.

Vine copulas

  • Multivariate copulas can struggle to capture complex dependencies, and representing dependence becomes more challenging as input dimension increases.For high-dimensional problems, the number of potential pair-copula constructions expands combinatorially.Source: Vine copulas · pp. 10, 12
  • PCC constructs an n-dimensional copula density from products of bivariate pair-copulas.The construction uses conditional densities to factorize the joint density into marginal distributions and pair-copula terms.Source: Vine copulas · pp. 10–11
  • Regular vines organize pair-copulas in nested trees, with edges in one tree becoming nodes in the next and a proximity condition ensuring consistent conditioning.The framework offers multiple valid decompositions of the joint density; D-vines and C-vines are two predominant classes.Source: Vine copulas · p. 12
  • In the four-dimensional C-vine example, the joint PDF is the product of four marginal densities and pair-copula terms linking variable 1 to variables 2, 3, and 4.Source: Vine copulas · pp. 12–13

Rosenblatt transformation supported by vine copulas

The inverse Rosenblatt transformation maps independent Gaussian inputs to dependent inputs, and vine copulas provide a way to construct the joint density it requires. In QPEM, this mapping transforms sigma points into the original input space for moment estimation.

Rosenblatt transformation supported by vine copulas

  • QPEM sigma points are generated in standard normal space and transformed into the original input space using the inverse Rosenblatt transformation.This mapping enables sigma points to be used with complex dependencies.Source: Rosenblatt transformation supported by… · pp. 13, 15
  • Vine copulas can construct the joint PDF when the full joint PDF or sequence of conditional PDFs required by the Rosenblatt transformation is unavailable.The framework is presented for high-dimensional and complex dependency settings.Source: Rosenblatt transformation supported by… · p. 13
  • The inverse Rosenblatt transformation maps independent Gaussian variables back to dependent variables in the original x-space.The forward transformation maps dependent variables to components that are independent.Source: Rosenblatt transformation supported by… · pp. 13–14
  • Selecting the vine structure and pair-copulas from available data matters because multiple valid joint-density decompositions are generally possible.The transformation is expressed using pair-copulas, with a similar formulation available for D-vine structures.Source: Rosenblatt transformation supported by… · pp. 14–15

Construction of input PDF using vine copula

This section describes vine-copula construction of joint input PDFs when dependence is inferred from data or when only marginal distributions and a correlation matrix are specified. It covers choices of vine structure, pair-copula families, and parameters, including alternative dependence models for sensitivity analysis of QPEM moment estimates when probability information is incomplete.

Construction of input PDF using vine copula

  • Vine-copula construction specifies an R-vine structure, pair-copula families, and their parameters, with feasibility and accuracy shaped by available input data.The section distinguishes direct PDF estimation from data and partial specification using marginal distributions and a covariance matrix.Source: Construction of input PDF using vine… · p. 15

Case (1): Input PDF can be estimated from available data

  • The explosive growth in possible R-vine structures makes finding globally best-fitted vine copulas computationally prohibitive; with n=10 variables, more than 5 × 10^14 R-vine tree structures are possible.Source: Case (1): Input PDF can be estimated… · pp. 15–16
  • The fitted vine’s inverse Rosenblatt transformation maps QPEM sigma points from independent standard normal space into the original dependent stochastic space.The algorithm and candidate pair-copula families are summarized in Tables 2 and 3.Source: Case (1): Input PDF can be estimated… · p. 16

Case (2): Input PDF is only partially known through marginals and correlation matrix

  • When only marginals and correlations are known, the dependence structure remains indeterminate, so the study examines QPEM moment sensitivity to flexible vine copulas versus a Gaussian copula.C-vines and D-vines are considered because the passages report no definitive evidence favoring either when the true structure is unknown.Source: Case (2): Input PDF is only partially… · pp. 17–18
  • Higher-level pair-copula parameters generally cannot be calculated directly from prescribed correlations, so the study calibrates them sequentially by bisection without evaluating the response model.For a fixed vine and pair-copula specification, induced correlation is typically monotonic in the corresponding parameter, allowing one-dimensional root finding when a feasible solution exists.Source: Case (2): Input PDF is only partially… · p. 18
  • Sigma points are transformed using the inverse Rosenblatt transformation based on the current vine copula, without evaluating the computational model.Source: Case (2): Input PDF is only partially… · p. 19
  • When the target correlation is attainable, pair-copula parameters are iteratively updated by bisection until convergence yields a vine model consistent with the prescribed correlation matrix within numerical tolerance.Source: Case (2): Input PDF is only partially… · p. 19
  • Alternative vine-copula dependence structures can preserve the prescribed marginals and correlation matrix, allowing sensitivity analysis of QPEM moment estimates under incomplete probability information.This tests how estimates depend on the assumed dependence model.Source: Case (2): Input PDF is only partially… · p. 19

NUMERICAL EXAMPLES

The numerical examples assess copula-based QPEM for estimating response moments across engineering applications and dependence-information settings, benchmarking it against Monte Carlo and other sampling methods. They include cases with dependence inferred from data and with only marginals and correlations available.

NUMERICAL EXAMPLES

  • Benchmarks use MC with 106 samples, cost-matched LHS and QMC with 2n^2+1 samples, SGH3 with 2n^2+2n+1, and HPEM with 2n+1.MC serves as the reference solution; QPEM uses 2n^2+1 samples.Source: NUMERICAL EXAMPLES · pp. 19–20

Case (1): Input joint distribution inferred from raw data

  • In Case (1), a vine copula is learned from 500 synthetic realizations, and MC-GT versus MC-Fitted references separate finite-sample inference effects from propagation-method error.The inferred model is used within QPEM; the two references are generated from the ground-truth and fitted input models, respectively.Source: Case (1): Input joint distribution… · p. 20

Example 1: Rock slope safety

  • For the rock-slope example, MC-GT and MC-Fitted differ by approximately 4% or less across all four response moments, indicating reasonable finite-sample representation of the ground truth.Relative errors for the four Case (1) examples are consolidated in Figure 4.Source: Case (1): Input joint distribution… · p. 20, Example 1: Rock slope safety · p. 22
  • QPEM’s relative errors were approximately 5.3% for skewness and 4.0% for kurtosis, whereas all competing methods showed considerably larger relative errors on these higher-order moments.LHS and QMC used the same 51 model evaluations as QPEM; SGH3 and HPEM also did not achieve comparable accuracy.Source: Example 1: Rock slope safety · p. 22

Example 2: Rock tunnel excavation

  • The model involves five random variables; uniform support pressure pi is deterministic at 0.868 MPa, while random critical pressure pcr causes stochastic regime switching and a gradient discontinuity between elastic and plastic states.Source: Example 2: Rock tunnel excavation · pp. 23–24
  • The tunnel dependence model uses a C-vine rooted at Young’s modulus, with varied pair-copula families and conditional dependence terms.The five random variables are Young’s modulus, cohesion, friction angle, Poisson’s ratio, and hydrostatic pressure.Source: Example 2: Rock tunnel excavation · p. 24
  • QPEM most accurately represents all four moments of the inferred model in the rock-tunnel example, with errors below 0.6% for mean, standard deviation, and skewness and about 6.9% for kurtosis.The response is non-smooth, with stochastic elastic-plastic regime switching; its kurtosis error is substantially smaller than those of the other methods.Source: Example 2: Rock tunnel excavation · pp. 24–25

Example 3: Maximum vertical displacement of horizontal truss

  • QPEM’s relative errors versus MC-Fitted were approximately 0.2%, 1.2%, 4.7%, and 1.9% for mean, standard deviation, skewness, and kurtosis, respectively; all four moments were within approximately 3.3% of MC-GT.The estimates used only 201 model evaluations.Source: Example 3: Maximum vertical… · p. 26

Example 4: Maximum displacement of space truss structure

  • For the 15-variable space truss, QPEM’s relative errors were 0.02%, 0.07%, 5.1%, and 3.6% for mean, standard deviation, skewness, and kurtosis, respectively.Competing methods were less balanced: SGH3 had substantial skewness and kurtosis errors, while QMC’s skewness estimate was much less accurate.Source: Example 4: Maximum displacement of… · p. 27

Case (2): Only marginals and correlation matrix are known

  • When only marginals and the correlation matrix are available, the first four response moments are computed with a vine copula, without a separate MC-Fitted reference.The assumed dependence model is an explicit modeling choice under incomplete probability information; MC-GT remains the reference solution.Source: Case (2): Only marginals and… · p. 28
  • Case (2) compared copula-based QPEM and LHS/QMC at 2n^2+1 samples, SGH3 at 2n^2+2n+1, and HPEM at 2n+1.Source: Case (2): Only marginals and… · p. 28

Example 5: Maximum vertical displacement of horizontal truss

Example 6: Rock tunnel excavation

  • In the rock-tunnel example, supplying vine structure and pair-copula families reduced QPEM skewness error from approximately 7.9% to 3.8% and kurtosis error from approximately 15.1% to 8.6%.Source: Example 6: Rock tunnel excavation · p. 31
  • With an informed vine-copula model, QPEM provided the lowest relative errors across all four rock-tunnel response moments, although kurtosis remained more challenging to estimate.Source: Example 6: Rock tunnel excavation · p. 31

CONCLUSION

The conclusion presents copula-based QPEM as an efficient, flexible approach to estimating response moments under dependent non-Gaussian inputs. Across the examples, it reports accurate moment estimates while emphasizing that dependence-model assumptions affect probabilistic predictions.

CONCLUSION

  • The framework maps deterministic QPEM sigma points from independent standard Gaussian space to dependent inputs using the inverse Rosenblatt transformation, covering data-inferred vines and alternatives calibrated to marginals and correlations.Source: CONCLUSION · p. 31
  • For data-informed studies, separate MC-GT and MC-Fitted references distinguish input-distribution inference discrepancy from numerical moment-propagation error.Source: CONCLUSION · pp. 31–32
  • Across rock slope, tunnel excavation, horizontal truss, and space truss examples, QPEM generally gave the most balanced and accurate estimates, particularly for skewness and kurtosis.The proposed framework is described as providing accurate estimates of the first four response moments at relatively low computational cost.Source: CONCLUSION · pp. 31–32
  • For MC-Fitted, errors were approximately 0.2%, 1.2%, 4.7%, and 1.9% for the mean, standard deviation, skewness, and kurtosis, respectively, using 201 model evaluations.Source: CONCLUSION · p. 32
  • In the 15-dimensional space truss example, the four moment errors were approximately 0.02%, 0.07%, 5.1%, and 3.6%, using 451 model evaluations.Source: CONCLUSION · p. 32
  • With the same marginals and correlation matrix, changing the assumed dependence structure changed response moments; a Gumbel vine reduced horizontal-truss skewness and kurtosis errors from 9.1% and 16.0% to 4.4% and 10.6%.For the rock tunnel, when qualitative information about the vine structure and pair-copula families was assumed available, skewness and kurtosis errors fell from 7.9% and 15.1% under the Gaussian assumption to 3.8% and 8.6%.Source: CONCLUSION · p. 32
  • The results do not establish a universally preferable non-Gaussian dependence model; different joint distributions consistent with limited second-order information can produce different predictions, while credible additional dependence information can improve them.Source: CONCLUSION · p. 32

APPENDIX I. MAXIMUM SPANNING TREE BY PRIM’S ALGORITHM

The appendix describes a greedy maximum spanning tree procedure that selects the strongest available dependencies while excluding cycle-forming edges. The resulting tree is the first tree in the R-vine structure and captures significant pairwise dependencies.

APPENDIX I. MAXIMUM SPANNING TREE BY PRIM’S ALGORITHM

  • Prim’s procedure grows a tree from an arbitrary root by repeatedly adding the highest-weight edge connecting visited and unvisited nodes.Here, edge weights are absolute Kendall’s τ.Source: Appendix I · p. 33
  • Edges that would create a cycle are excluded, even when their weight is high.The example rejects edge (3, 8) because both endpoints are already in the tree.Source: Appendix I · p. 33
  • The resulting maximum spanning tree becomes the first R-vine tree and captures the most significant pairwise dependencies among the variables.For n=8, the spanning tree contains n−1=7 edges.Source: Appendix I · p. 34
  • Because the procedure uses edge ranks, any monotone increasing transformation of the weights produces the same tree.The appendix gives squared τ as an alternative weight transformation.Source: Appendix I · p. 34

APPENDIX II. PARAMETER ESTIMATION AND MODEL SELECTION

The appendix sets out a stepwise procedure for estimating vine-copula parameters and selecting pair-copula families. It also provides copula-family formulations, explains rotations, and describes how target correlations are obtained for Case (2).

APPENDIX II. PARAMETER ESTIMATION AND MODEL SELECTION

  • Stepwise semiparametric estimation is used because global maximum likelihood can be computationally prohibitive and numerically unstable in high-dimensional engineering problems.The procedure decouples marginal estimation from dependence estimation and breaks high-dimensional copula estimation into low-dimensional steps.Source: APPENDIX II. PARAMETER ESTIMATION AND… · p. 35

Stage 1: Transformation to Pseudo-observations via Empirical Marginals

  • The empirical CDF transforms each variable’s observed values into uniform pseudo-observations using normalized empirical ranks.Dividing ranks by N+1 keeps the transformed values inside (0, 1).Source: Stage 1: Transformation to… · p. 35

Stage 2: Sequential (Stepwise) Estimation of Copula Parameters

  • Copula parameters are estimated sequentially by tree, using each tree’s pseudo-log-likelihood and conditional distributions derived from earlier trees.The SSP estimator is less efficient than global MLE due to accumulated stepwise estimation errors, but remains consistent and asymptotically normal.Source: Stage 2: Sequential (Stepwise)… · pp. 35–36

Model Selection Criterion

  • For each vine edge, candidate pair-copula families are compared using AIC to balance fit and parsimony.The candidates include Gaussian, Student-t, Clayton, Gumbel, and Frank families.Source: Model Selection Criterion · pp. 36–37

APPENDIX III. DEFINITIONS OF PAIR-COPULA FAMILIES

Rotated Copulas

  • Rotations of copulas are used to represent negative dependence or different tail behavior; for asymmetric families, rotations provide distinct dependence structures.A 90°-rotated Clayton copula, for example, captures negative dependence in the lower-right tail.Source: Rotated Copulas · pp. 39–40

APPENDIX IV. TARGET CORRELATION MATRICES FOR CASE (2)

  • For Case (2), target correlation matrices specify second-order dependence, and their correlations are evaluated numerically from the ground-truth input distributions.These calculations use input-distribution sampling without evaluating the computational response models.Source: APPENDIX IV. TARGET CORRELATION… · p. 41

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