A new method for evaluating Borgonovo moment-independent importance measure with its application in an aircraft structure

doi:10.1016/j.ress.2014.07.011

Reliability Engineering & System Safety

Volume 132, December 2014, Pages 163-175

https://doi.org/10.1016/j.ress.2014.07.011 Get rights and content

Highlights

•
FM-based MaxEnt is applied to estimate the PDF of model output.
•
Improved HDRM and Gaussian integral are used to compute the FMs.
•
Nataf transformation is used to obtain the joint PDF of model input and output.
•
The number of function evaluations of the proposed method is very small.
•
The proposed method has good potential of engineering application in GSA.

Abstract

The moment-independent importance measure proposed by Borgonovo, which is defined as the average shift between the unconditional and conditional probability density functions (PDFs) of model output, is widely used to evaluate the influence of input uncertainty on the entire output distribution. And how to exactly and efficiently estimate the PDFs remains a crucial and challenging problem. In this paper, a novel PDF estimation based method is proposed to efficiently evaluate the moment-independent index. Firstly, the PDF of the model output is obtained based on the concepts of maximum entropy, fractional moment and high dimensional model representation. Secondly, the Nataf transformation is utilized to estimate the joint PDF of the output and input variable. Finally, the index can be easily computed using the generated correlated standard normal samples. Thus the importance measure can be calculated with high efficiency and accuracy using this proposed composited method. Several examples are employed to demonstrate the advantages of the proposed method. Meanwhile, the importance analysis of a stiffening rib of the wing leading edge in a certain aircraft also verifies its good engineering applicability.

Introduction

Nowadays, sensitivity analysis (SA) is becoming more and more important in structural safety analysis, Saltelli [1] defined SA as the study of “how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input factors”. Traditionally, SA can be classified into two main branches [2], [3]: local SA and global SA (GSA). Local sensitivity indices are defined as the partial derivatives of the model output with respect to the input parameters, while they are only informative at the nominal points where they are calculated and they can not provide for an exploration of the rest of the input spaces. In contrast, GSA takes account of the variation effect of the input variables, thus is more informative and robust than estimating derivatives at a nominal point of the input space.

GSA has a great potential in engineering applications, and GSA indicators are called uncertainty importance measures [4], [5] which has attracted abundant research interests in the literature. The uncertainty importance measures generally include nonparametric measures [6], [7], variance-based measures [8], [2], [9], and moment-independent measures [10], [11], [4], [12], [13], [14]. The nonparametric techniques are insufficient to capture the influence of inputs on the output variability for nonlinear models and also when interactions among inputs emerge [15]. Satelli [1] pointed out that an importance measure should satisfy the requirements: “global, quantitative, and model free”. The variance-based importance measures are widely studied and have been proved to be useful for their acknowledged features [16], which can directly reflect the contributions of the input variables to the output response. However, Borgonovo [12] illustrated that variance is not always sufficient to describe uncertainty and indicated that a sensitivity indicator should refer to the entire output distribution instead of a particular moment. With this respect, Borgonovo [4] extended Saltelli׳s three requirements by adding “moment independent”, and proposed the moment-independent importance measure (MIIM) which could assess the influence of input uncertainty on the entire output distribution without referring to a specific moment of the output, and this index is attracting increasing attention due to some attractive properties [17].

In the process of calculating Borgonovo׳s MIIM, the most crucial and challenging problem is how to estimate the unconditional and conditional PDFs of the model output efficiently and accurately. The PDF-based method adopted by Borgonovo [4] leads to considerably large computational burden as double loop samplings are involved. The CDF-based method proposed by Liu and Homma [18], which transformed calculations of the importance measures into calculations of the cumulative distribution functions (CDF) of the model output. However, as maintained by Liu and Homma, “for a computationally intensive model, when the total computational time is mainly due to the time of running the model, the improvement of the computational efficiency by the CDF-based method can be negligible”. Plischke et al. [19] presented a new method for estimating global sensitivity measures, which was a good strategy for GSA from given data. Based on the recently developed probability density evolution method (PDEM), Cui et al. [20] proposed the PDEM-based method for calculating the MIIM, while the number of function evaluations is closely related with the dimensionality of the model input. Recently, Wei et al. [21] proposed double-loop and single-loop Monte Carlo simulation (MCS) methods adopting the kernel density estimator (KDE) technique, which could avoid the problem of “curse of dimensionality”. However, large numbers of function evaluations are needed to estimate the unconditional and conditional PDFs of the model output, or the joint PDF, thus they may not be competent for the models with implicit output functions.

Interest in the principle of maximum entropy (MaxEnt) has been heightened with the emergence of the fractional moment (FM) [22], [23]. Since a FM contains information about a large number of integer moments, the FM-based MaxEnt can capture the heavy distribution tail precisely [22], [23]. Thus FM-based MaxEnt method is utilized to estimate the PDF of the model output in this paper. Besides, computations of the FMs of a multivariate output function are based on the improved high-dimensional model representation (HDMR) and the Gaussian integration [23]. Nataf transformation can be used to well estimate the joint PDF of correlated variables, which was proposed by Liu and der Kiureghian [24], [25], and developed by Li et al. [26]. The latter is applied in this paper to estimate the joint PDF of the input variable and the model output. The correlation coefficients within the correlated standard normal space are computed by solving nonlinear equations using the Gaussian–Hermite integration, thus it is not limited to the distribution types given by Liu and der Kiureghian. Finally, based on the equivalent probability transformation algorithm, the MIIM is transformed into the correlated standard normal space and can be computed easily. Note that by inheriting advantages of these existing approaches, the new PDF estimation based composited method in this paper decreases the number of function evaluations to a great extent compared with the existing methods, and it is quite fit for dealing with the complicated models with implicit output functions.

Organization of this paper is as follows. Section 2 gives a brief review of Borgonovo׳s MIIM. Section 3 describes the detailed algorithms for estimating the MIIM. Section 4 shows the implementation process of the proposed method and the computational effort. Section 5 introduces three examples to demonstrate the advantages of the proposed method and applies it to the importance analysis of the stiffening rib of the wing leading edge of an aircraft. Section 6 summarizes the conclusions.

Section snippets

Brief review of Borgonovo׳s MIIM

Consider a input–output model $Y = g (X)$ , where $Y$ is the model output, and $X = {(X_{1}, X_{2}, .. ., X_{n})}^{T}$ (n is the input dimension) is the vector of random input variables. Denote $f_{Y} (y)$ and $f_{Y | X_{i}} (y)$ as the unconditional PDF and conditional PDF of $Y$ , respectively. $f_{Y | X_{i}} (y)$ can be obtained by fixing the input variable $X_{i}$ at its realization. The absolute value of the difference between $f_{Y} (y)$ and $f_{Y | X_{i}} (y)$ while $y$ varies from $- \infty$ to $+ \infty$ can be estimated by the area $s (X_{i})$ shown in Fig. 1, and $s (X_{i})$ represents the

A new PDF estimation based method for estimating MIIM

In this section, a new PDF estimation based method is proposed to estimate the MIIM, and it is a composited technique by combing some existing excellent approaches.

Analogously to the derivation in [19], [21], firstly, the $δ_{i}$ index can be derived to a different expression, i.e. $δ_{i} = \frac{1}{2} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} | f_{Y} (y) - f_{Y | X_{i}} (y) | f_{X_{i}} (x_{i}) d y d x_{i} = \frac{1}{2} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} | f_{Y} (y) f_{X_{i}} (x_{i}) - f_{Y, X_{i}} (y, x_{i}) | d y d x_{i}$ where $f_{Y, X_{i}} (y, x_{i})$ is the joint PDF of the model output $Y$ and the input variable $X_{i}$ .

In this paper, we consider models with known PDFs of

Implementation steps of the proposed method

Computation of $δ_{i_{1}, i_{2}, .. ., i_{s}}$ is similar as that of $δ_{i}$ , thus in this section, we only gives the detailed steps for computing $δ_{i}$ of the proposed method as follows:

Step 1: Establish the numerical moment integration grid based on the Gaussian rules and improved Cut-HDMR as shown in Table 1, where the last column $F = X_{i} \cdot g (X, c)$ is used to calculate $E (Y \cdot X_{i})$ .
Step 2: Estimate the PDF $f_{Y} (y)$ of the model output by FM-based MaxEnt. Actually, $f_{Y} (y)$ is obtained by solving the optimization problem in Eq. (11).

Test examples and application

In this section, we employ three examples to verify the advantages of the proposed method. The first example is an analytical test case, and there are slight differences among the $δ_{i}$ indices of all the input variables. The aim of this test case is to demonstrate whether the proposed method can produce the correct importance ranking. The second example is a risk assessment model with third order interactions. The third example is a root truss structure. The latter two examples are used to

Conclusions

In this paper, we focus on establishing a highly efficient method to estimate Borgonovo׳ MIIM. A novel PDF estimation based method combining the principle of FM-based MaxEnt with the Nataf transformation is proposed. Based on the HDMR and Gaussian integration, the PDF of the model output can be precisely estimated by the FM-based MaxEnt. Then the joint PDF of the model output and the input variable can be estimated by Nataf transformation. At last, the $δ_{i}$ indices can be easily computed by the

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. NSFC 51175425), the Special Research Found for the Doctoral Program of Higher Education of China (Grant no. 20116102110003).

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A new method for evaluating Borgonovo moment-independent importance measure with its application in an aircraft structure

Highlights

Abstract

Introduction

Section snippets

Brief review of Borgonovo׳s MIIM

A new PDF estimation based method for estimating MIIM

Implementation steps of the proposed method

Test examples and application

Conclusions

Acknowledgments

Reliab Eng Syst Saf

Comput Chem Eng

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Risk Anal

Risk Anal

Eur J Oper Res

Sci China Technol Sci

Commun Stat Theor Methods

Probab Eng Mech

J Eng Mech ASCE

Risk Anal

Sensitivity analysis for importance assessment

Risk Anal

Importance Measures in Reliability, Risk, and Optimization: Principles and Applications

Sensitivity analysis for computer model projections of hurricane losses

Risk Anal

Sensitivity estimates for nonlinear mathematical models

Matorsz. Modelirovanie

Math Model Comput Exp

Sensitivity analysis as an ingredient of modelling

Stat Sci

A new approach for measuring uncertainty importance and distributional sensitivity in probabilistic safety assessment

Reliab Eng Syst Saf