Fig 1 - available via license: Creative Commons Attribution 4.0 International
Content may be subject to copyright.
First-order (left) and total-order (right) sensitivity indices versus the correlation coefficient (í µí¼ 13 ) between í µí±¥ 1 and í µí±¥ 3 in the Ishigami function. See text for comments and explanations.
Source publication
In this paper, we discuss the sensitivity analysis of model response when the uncertain model inputs are not independent of one other. In this case, two different kinds of sensitivity indices can be evaluated: (i) the sensitivity indices that account for the dependence/correlation of an input or group of inputs with the remainder and (ii) the sensi...
Citations
... The PCE is the orthogonal expansion of an M-dimensional function Y(ξ), expressed as follows (Cheng and Lu 2018;Mara and Becker 2021): ...
The probability and evidence theories are frequently used tool to deal with the mixture of aleatory and epistemic uncertainties. Due to the double-loop procedure for the mixed uncertainty quantification (UQ), the computational cost is daunting. Therefore, this study proposes a new polynomial chaos expansion (PCE) method for UQ problems with random variables and evidence variables. To enhance the computational accuracy and efficiency of the PCE model, a new low-discrepancy sequence sampling method is proposed, and the sample weights are redefined according to the Christoffel prior. Then, a weighted sparse Bayesian learning method is developed to construct the PCE model with a small sample size. Finally, the proposed method is verified through two numerical examples and one practical engineering problem and compared with three common surrogate methods. Results illustrate the proposed method has obvious advantages in computational accuracy and efficiency over the compared methods, and is powerful for the UQ problems with aleatory and epistemic uncertainties.
... The exact permutation algorithm is expensive, since it traverses the exponential space of permutations between the inputs, whereas the random permutation algorithm is approximative in that it is based of randomly sampling some permutations of the inputs. Experimental results with these estimators for Shapley effects are reported in Mara and Becker (2021). The characterization of the Shapley-Owen value in terms of its Möbius inverse (Appendix J) still requires an exponential number of summands Plischke et al. (2021). ...
We argue that relative importance and its equitable attribution in terms of Shapley-Owen effects is an appropriate one, and, if we accept a small number of reasonable imperatives for equitable attribution, the only way to measure fairness. On the other hand, the computation of Shapley-Owen effects can be very demanding. Our main technical result is a spectral decomposition of the Shapley-Owen effects, which decomposes the computation of these indices into a model-specific and a model-independent part. The model-independent part is precomputed once and for all, and the model-specific computation of Shapley-Owen effects is expressed analytically in terms of the coefficients of the model's \emph{polynomial chaos expansion} (PCE), which can now be reused to compute different Shapley-Owen effects. We also propose an algorithm for computing precise and sparse truncations of the PCE of the model and the spectral decomposition of the Shapley-Owen effects, together with upper bounds on the accumulated approximation errors. The approximations of both the PCE and the Shapley-Owen effects converge to their true values.
... One such method called polynomial chaos expansion (PCE) can be used to calculate sensitivity indices for the individual and combined effect of large groups of parameters (Sudret, 2008). These Sobol sensitivity indices are analytically calculated from the coefficients of the meta-model (Mara & Becker, 2021). As a result, this method only requires samples for fitting the meta-model. ...
... To mitigate these issues previous studies have performed uncertainty analyses using a meta-model based approach to attribute the sources of uncertainty to a large number of parameters and the interactions between parameters (Blatman & Sudret, 2011). This was done using a polynomial chaos expansion (PCE) based meta-model, where the Sobol sensitivity indices of each parameter were analytically calculated from the coefficients of the meta-model (Mara & Becker, 2021;Sudret, 2008). ...
... Polynomial Chaos Expansion sensitivity analysis is a variance-based approach [64] for estimating non-linear responses [65], [66] with an A.E. Gkikakis et al. analytical model, and thus, providing more accurate results. Fig. 4 presents the total contribution of all terms containing a variable of interest, including its interaction effects. ...
This study presents a comprehensive design optimization approach for building vibration control systems,
demonstrated through a case study on the KDamper implemented as a vertical seismic absorber (VSA). The
methodology encompasses a series of steps to identify the most suitable VSA design while maximizing the
utilization of available resources and considering model and reality imperfections. The outcomes of this
investigation include a Pareto front featuring optimal structures and a set of design guidelines specifically
tailored for the development of a VSA system. Moreover, optimization analyses conducted on simplified systems
have revealed their potential effectiveness even without certain components. Furthermore, a bi-objective
robust optimization has demonstrated the system’s resilience to manufacturing uncertainties prevalent in real-world scenarios. Finally, this approach has yielded more consistent system performance than conventional
optimization approaches. The proposed methodology holds broad applicability to diverse systems, offering
valuable support in achieving widespread utilization and facilitating the design of optimal structures, such as
the vibration-absorption system examined in this study.
... This is because, as mentioned, the Hermite polynomials are orthogonal only to standard normal inputs. However, this limitation can be circumvented by approximating the random vector X with a given joint probability density function into a vector with a Gaussian distribution using transformations such as the Nataf transform [58], as described in (24). ...
In conventional vehicle design approaches, there is typically little understanding of the consequences of early stage design choices. This may be attributed to the conventional approach’s limitations in capturing complex interactions, which leads to increased design iterations. To overcome this, holistic multidisciplinary models have been developed. However, these models introduce the burden of complexity and costs due to their intricate nature. Additionally, it is challenging to gain meaningful insights without a deeper understanding of the model’s structure and behavior. In this article, an alternative form of model representation is proposed to address these shortcomings. This was achieved by integrating two concepts: network theory and sensitivity analysis. A detailed and robust framework is provided, which represents complex multidisciplinary models as network models, reducing their complexity and allowing insights to be extracted from them. This approach was demonstrated through a case study of a rail vehicle traction system, including a traction motor and an inverter, coupled with an operational drive cycle. Among the 246 factors identified in the traction system network model, the three most influential inputs were determined for the selected output factor of interest. Subsequently, the knock-on effects of these inputs were assessed. The results indicate a significant reduction in the network graph size compared to the complete network graph of the traction system model, highlighting a substantial decrease in the number of factors considered in the analysis. This demonstrates the capability of the proposed framework to simplify the analysis while retaining the ability to examine intricate interaction effects.
... This approach is originally shown in [6], using the coefficients of generalized PCE [8], which is extended using sparse PCE [9,10] and partial least squares-driven PCE (PLS-PCE) [7] to deal with high-dimensional problems. Furthermore, advances have been made to account for the dependence in input factors [11], generalized modeling of both aleatory and epistemic uncertainty in input factors [12], and derivative-based sensitivity measures for efficient screening of unimportant factors [13]. These studies have demonstrated PCE to be a versatile and efficient tool for sensitivity analysis. ...
... The multi-indices are determined based on (0 < < 1) [29]. For = 1, the hyperbolic truncation corresponds to the standard totaldegree truncation degree in Eq. (11), where the polynomials of maximum total degree of are retained. When < 1, the truncation penalizes high-degree terms with many interacting variables, while favoring the main effects and low-order interactions. ...
... If ( ) < , it stops the iterative process (line 9). Otherwise = + 1 (line 10), and repeat the process (line [4][5][6][7][8][9][10][11][12]. The optimal set * is eventually retained, associated with the lowest error * (line 8). ...
The use of sensitivity analysis is essential in model development for the purposes of calibration, verification, factor prioritization, and mechanism reduction. While most contributions to sensitivity methods focus on the average model response, this paper proposes a new sensitivity method focusing on the extreme response and structural limit states, which combines an extreme-oriented sensitivity method with polynomial chaos expansion. This enables engineers to perform sensitivity analysis near given limit states and visualize the relevance of input factors to different design criteria and corresponding thresholds. The polynomial chaos expansion is used to approximate the model output and alleviate the computational cost in sensitivity analysis, which features sparsity and adaptivity to enhance efficiency. The accuracy and efficiency of the method are verified in a truss structure, which is then illustrated on a dynamic train–track–bridge system. The role of the input factors in response variability is clarified, which differs in terms of the design criteria chosen for sensitivity analysis. The method incorporates multi-scenarios and can thus be useful to support decision-making in design and management of engineering structures.
... At present, the most effective method is to adopt a proxy model based on computer model responses. The proxy model is a statistical model used to simulate the input-output relationship of the original model, which can greatly reduce the calculation cost and improve the analysis efficiency [12,13]. The commonly used methods of establishing a proxy model include the regression method, Kriging method, artificial neural network, etc. [14], and the polynomial chaotic expansion method (PCE) is widely used because of its concise mathematical principle, wide applicability and rapid convergence. ...
... As a kind of the spectral approaches, PCE converts the model response into the sum of orthogonal polynomials to obtain the approximate value of the model response. In sensitivity analysis, the Parseval-Plancherel theorem can be used to analytically obtain Sobol' indices from PCE coefficients without repeatedly running PCE as a proxy model [12,15]. At present, PCE has been successfully applied to sensitivity analysis in different fields [12][13][14][15][16][17], and its techniques can be mainly divided into the invasive method and the non-invasive method [18]. ...
... In sensitivity analysis, the Parseval-Plancherel theorem can be used to analytically obtain Sobol' indices from PCE coefficients without repeatedly running PCE as a proxy model [12,15]. At present, PCE has been successfully applied to sensitivity analysis in different fields [12][13][14][15][16][17], and its techniques can be mainly divided into the invasive method and the non-invasive method [18]. The former is put forward under the background of the stochastic finite element method, which is used to discretize constitutive equations in physical space and random space; the latter is based on post-processing multiple analog outputs of existing numerical models [19,20]. ...
To effectively identify the key material parameters of different zones of concrete face rockfill dams and improve the efficiency of parameter optimization, a global sensitivity analysis method of parameters based on sparse polynomial chaotic expansion (sPCE) is proposed in this paper. The latin hypercube sampling method is used to select multiple groups of material parameters, and then finite element method is used to calculate the displacement of dam characteristic nodes in dam body. On this basis, the displacement is expanded by sPCE, and the polynomial basis function is reconstructed by orthogonal matching pursuit to improve the construction and analysis efficiency of the proxy model. According to the chaos coefficients, Sobol’ indices are calculated to evaluate the influence of the material parameters and their interaction on different displacements of the dam. The results show that the sPCE model can accurately simulate dam displacement and its statistical characteristics with a relatively small sample size. The sensitivity of the same parameter has spatial variability, and under the influence of parameter levels and spatial distribution of different materials, the parameter sensitivity ranking of different zones has certain differences. The proposed method provides a new reference to sensitivity analysis and uncertainty analysis for practical engineering.
... In fact, to the best of our knowledge, there have been no investigations or comparisons regarding the use of PCE-based models in estimating global sensitivity indices for correlated random inputs in power systems, which nevertheless are crucial for effective uncertainty control. In previous works, Mara et al. [10] calculated two sensitivity indices based on ANOVA (Analysis of variance) techniques for dependent inputs, which, nevertheless, may be time-consuming and impractical for highdimensional problems. Another two PCE-based methods were suggested to handle correlated random inputs in GSA based on ANCOVA (ANalysis of COVAriance). ...
... Once the PCE-based model (6) is built by one of the aforementioned two methods, ML samples (ML ≫ Mp) of Z (l) , l = 1, ..., ML can be substituted into the PCE-based model to obtain corresponding responsesŶ (l) efficiently. Then the ANCOVA indices Sj for each Zj can be estimated by (10), based on which effective control measures can be designed to reduce the variance of the system response Y in the most effective way. The detailed steps of the two PCE-based methods for ANCOVA indices estimation and uncertainty control are summarized in Method 1 and Method 2. Remark 1. ...
... Step 5. Calculate the ANCOVA indices S j by (10). Identify the critical random inputs with the highest S j values. ...
In this letter, we compare three polynomial chaos expansion (PCE)-based methods for ANCOVA (ANalysis of CO-VAriance) indices based global sensitivity analysis for correlated random inputs in two power system applications. Surprisingly, the PCE-based models built with independent inputs after decorrelation may not give the most accurate ANCOVA indices, though this approach seems to be the most correct one and was applied in [1] in the field of civil engineering. In contrast, the PCE model built using correlated random inputs directly yields the most accurate ANCOVA indices for global sensitivity analysis. Analysis and discussions about the errors of different PCE-based models will also be presented. These results provide important guidance for uncertainty management and control in power system operation and security assessment.
... In fact, to the best of our knowledge, there have been no investigations or comparisons regarding the use of PCE-based models in estimating global sensitivity indices for correlated random inputs in power systems, which nevertheless are crucial for effective uncertainty control. In previous works, Mara et al. [10] calculated two sensitivity indices based on ANOVA (Analysis of variance) techniques for dependent inputs, which, nevertheless, may be time-consuming and impractical for highdimensional problems. Another two PCE-based methods were suggested to handle correlated random inputs in GSA based on ANCOVA (ANalysis of COVAriance). ...
... Once the PCE-based model (6) is built by one of the aforementioned two methods, ML samples (ML ≫ Mp) of Z (l) , l = 1, ..., ML can be substituted into the PCE-based model to obtain corresponding responsesŶ (l) efficiently. Then the ANCOVA indices Sj for each Zj can be estimated by (10), based on which effective control measures can be designed to reduce the variance of the system response Y in the most effective way. The detailed steps of the two PCE-based methods for ANCOVA indices estimation and uncertainty control are summarized in Method 1 and Method 2. Remark 1. ...
... Step 5. Calculate the ANCOVA indices S j by (10). Identify the critical random inputs with the highest S j values. ...
