Luyi Li

A new interpretation and validation of variance based importance measures for models with correlated inputs

Abstract In order to explore the contributions by correlated input variables to the variance of the output, a novel interpretation framework of importance measure indices is proposed for a model with correlated inputs, which includes the indices of the total correlated contribution and the total uncorrelated contribution. The proposed indices accurately describe the connotations of the contributions by the correlated input to the variance of output, and they can be viewed as the complement and correction of the interpretation about the contributions by the correlated inputs presented in “Estimation of global sensitivity indices for models with dependent variables, Computer Physics Communications, 183 (2012) 937–946”. Both of them contain the independent contribution by an individual input. Taking the general form of quadratic polynomial as an illustration, the total correlated contribution and the independent contribution by an individual input are derived analytically, from which the components and their origins of both contributions of correlated input can be clarified without any ambiguity. In the special case that no square term is included in the quadratic polynomial model, the total correlated contribution by the input can be further decomposed into the variance contribution related to the correlation of the input with other inputs and the independent contribution by the input itself, and the total uncorrelated contribution can be further decomposed into the independent part by interaction between the input and others and the independent part by the input itself. Numerical examples are employed and their results demonstrate that the derived analytical expressions of the variance-based importance measure are correct, and the clarification of the correlated input contribution to model output by the analytical derivation is very important for expanding the theory and solutions of uncorrelated input to those of the correlated one.

Importance analysis for models with correlated input variables by the state dependent parameters method

For clearly exploring the origin of the variance of the output response in case the correlated input variables are involved, a novel method on the state dependent parameters (SDP) approach is proposed to decompose the contribution by correlated input variables to the variance of output response into two parts: the uncorrelated contribution due to the unique variations of a variable and the correlated one due to the variations of a variable correlated with other variables. The correlated contribution is composed by the components of the individual input variable correlated with each of the other input variables. An effective and simple SDP method in concept is further proposed to decompose the correlated contribution into the components, on which a second order importance matrix can be solved for explicitly exposing the contribution components of the correlated input variable to the variance of the output response. Compared with the existing regression-based method for decomposing the contribution by correlated input variables to the variance of the output response, the proposed method is not only applicable for linear response functions, but is also suitable for nonlinear response functions. It has advantages both in efficiency and accuracy, which are demonstrated by several numerical and engineering examples.

A new kind of sensitivity index for multivariate output

Mathematical and computational models with correlated multivariate output are commonly used for risk assessment and decision support in engineering. Traditional methods for sensitivity analysis of the model with scalar output fail to provide satisfactory results for this multivariate case. In this work, we introduce a new sensitivity index which looks at the influence of input uncertainty on the entire distribution of the multivariate output without reference to a specific moment of the output. The definition of the new index is based on the multivariate probability integral transformation (PIT), which can take into account both of the uncertainties and the correlations among multivariate output. The mathematical properties of the proposed sensitivity index are discussed and its differences with the sensitivity indices previously introduced in the literature are highlighted. Two numerical examples and a rotating shaft model of an aircraft wing are employed to illustrate the validity and potential benefits of the new sensitivity index.

Importance analysis on the failure probability of the fuzzy and random system and its state dependent parameter solution

For structure systems with fuzzy input uncertainty as well as random one, the effects of the two kinds of uncertainties on the failure probability of the structure are studied, and an importance analysis model is established to quantitatively evaluate these effects. Based on the fact that the fuzziness of the output is determined by that of the input, the importance measure is defined to evaluate the effect of the fuzzy input variable. For the random input variable, the established model analyzes its importance from two aspects: (1) its effect on the most plausible value of the failure probability, (2) its effect on the imprecision of the failure probability. In the process of computing the importance measures, the conditional failure probability and unconditional one need to be evaluated, which is time demanding for practical engineering problems. For efficiently performing importance analysis in the presence of the fuzzy input variables and the random ones, the importance sampling (IS) method combined with the state dependent parameter (SDP) method is presented. Several examples show that the established importance analysis model can reflect the effects of the two kinds of input variables on the safety of the structure comprehensively and reasonably, and the presented solution can improve computational efficiency considerably with acceptable precision.

Importance analysis for model with mixed uncertainties

In engineering, sparse or imprecise data often leads to epistemic uncertainty about the distribution parameters of the aleatory input variables. Separating and estimating the individual contributions of the aleatory variability and epistemic parameter uncertainties to the uncertainty in the model output can assist in resource allocation for data collection, as natural variability is irreducible whereas parameter uncertainty is reducible. For structural systems involving aleatory inputs with epistemic distribution parameters, a new kind of importance measure is proposed to distinguish and quantify the individual contributions of these two kinds of uncertainties, in which probability distribution is used to describe the aleatory uncertainty, and fuzzy membership function is employed to represent the epistemic distribution parameter. The mathematical properties of the proposed importance measures are discussed and proved. The defined importance measures are easy to apprehend, and can evaluate the contributions of the aleatory variability and epistemic parameter uncertainties even when the information of the epistemic parameters is very sparse. Thus, they can measure the importance of the two kinds of uncertainties more reasonably. For efficiently estimating the proposed importance measures, a monotonicity analysis is conducted for model function and the probability transformation process. The results indicate that the extreme values of model function at each membership level can generally be obtained by combining the bounds of membership intervals of the fuzzy distribution parameters. Based on the monotonicity analysis, an efficient algorithm is then formulated to compute the proposed importance measures. Several examples demonstrate the rationality and effectiveness of the proposed importance measures and the efficiency of the presented algorithm.

A new kind of regional importance measure of the input variable and its state dependent parameter solution

Abstract To further analyze the effect of different regions within input variable on the variance and mean of the model output, two new regional importance measures (RIMs) are proposed, which are the “contribution to variance of conditional mean (CVCM)” and the “contribution to mean of conditional mean (CMCM)”. The properties of the two RIMs are analyzed and their relationships with the existing contribution to sample variance (CSV) and contribution to sample mean (CSM) are derived. Based on their characteristics, the highly efficient state dependent parameter (SDP) method is introduced to estimate them. By virtue of the advantages of the SDP-based method, the same set of sample points utilized for solving CSM and CSV is enough to estimate CVCM and CMCM. Several examples demonstrate that CVCM can provide further information on the existing CSV, which can effectively instruct the engineer on how to achieve a targeted reduction of the main effect of each input variable. CMCM can act as effectively as the CSM, but the convergence and stability for estimating CMCM by numerical simulation is better than those for estimating CSM. Besides, the efficiency and accuracy of the SDP-based method are also testified by the examples.

Importance analysis for models with correlated variables and its sparse grid solution

For structural models involving correlated input variables, a novel interpretation for variance-based importance measures is proposed based on the contribution of the correlated input variables to the variance of the model output. After the novel interpretation of the variance-based importance measures is compared with the existing ones, two solutions of the variance-based importance measures of the correlated input variables are built on the sparse grid numerical integration (SGI): double-loop nested sparse grid integration (DSGI) method and single loop sparse grid integration (SSGI) method. The DSGI method solves the importance measure by decreasing the dimensionality of the input variables procedurally, while SSGI method performs importance analysis through extending the dimensionality of the inputs. Both of them can make full use of the advantages of the SGI, and are well tailored for different situations. By analyzing the results of several numerical and engineering examples, it is found that the novel proposed interpretation about the importance measures of the correlated input variables is reasonable, and the proposed methods for solving importance measures are efficient and accurate.

Global Sensitivity Analysis of Fuzzy Distribution Parameter on Failure Probability and Its Single-Loop Estimation

An extending Borgonovo’s global sensitivity analysis is proposed to measure the influence of fuzzy distribution parameters on fuzzy failure probability by averaging the shift between the membership functions (MFs) of unconditional and conditional failure probability. The presented global sensitivity indices can reasonably reflect the influence of fuzzy-valued distribution parameters on the character of the failure probability, whereas solving the MFs of unconditional and conditional failure probability is time-consuming due to the involved multiple-loop sampling and optimization operators. To overcome the large computational cost, a single-loop simulation (SLS) is introduced to estimate the global sensitivity indices. By establishing a sampling probability density, only a set of samples of input variables are essential to evaluate the MFs of unconditional and conditional failure probability in the presented SLS method. Significance of the global sensitivity indices can be verified and demonstrated through several numerical and engineering examples.

Discussion of paper by Matieyendou Lamboni, Hervé Monod, David Makowski “Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models”, Reliab. Eng. Syst. Saf. 99 (2011) 450–459

Abstract In the subject paper, a new set of sensitivity indices for multivariate outputs are defined when the uncertainty on the input factors is either discrete or continuous and when the dynamic model is either discrete or functional. Admittedly, these indices can quantify the contribution of each input factor to each principal component and the total inertia reasonably, and provide rich information on dynamic model behaviour. However, Propositions 2–4 in the subject paper which claim that the generalized sensitivity index is equal to the sensitivity index defined on the sum of the principal components are not correct. These incorrect propositions are concluded by the incorrect derivation in the Appendices B and C of the subject paper. This discussion intends to clarify this issue associated with the paper.

A new algorithm for importance analysis of the inputs with distribution parameter uncertainty

ABSTRACT Importance analysis is aimed at finding the contributions by the inputs to the uncertainty in a model output. For structural systems involving inputs with distribution parameter uncertainty, the contributions by the inputs to the output uncertainty are governed by both the variability and parameter uncertainty in their probability distributions. A natural and consistent way to arrive at importance analysis results in such cases would be a three-loop nested Monte Carlo (MC) sampling strategy, in which the parameters are sampled in the outer loop and the inputs are sampled in the inner nested double-loop. However, the computational effort of this procedure is often prohibitive for engineering problem. This paper, therefore, proposes a newly efficient algorithm for importance analysis of the inputs in the presence of parameter uncertainty. By introducing a ‘surrogate sampling probability density function (SS-PDF)’ and incorporating the single-loop MC theory into the computation, the proposed algorithm can reduce the original three-loop nested MC computation into a single-loop one in terms of model evaluation, which requires substantially less computational effort. Methods for choosing proper SS-PDF are also discussed in the paper. The efficiency and robustness of the proposed algorithm have been demonstrated by results of several examples.