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# 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.

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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

... 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. ...

... 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 COVAriance) 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.

... There are two directions in the literature how to deal with the correlations during SA; (i) decomposition of the traditional sensitivity indicies into correlated and uncorrelated parts [10,11] and (ii) introducing new sets of indices which contain all correlations and indices which are reduced by the contributions due to the correlation [12,13,14]. ...

... A new set of the indices for correlated inputs was introduced by Mara and Tarantola [12,13,14]. Two distinct indices represent correlated and uncorrelated contributions of a given variable. ...

... SA with correlated parameters is studied in this work. The decorrelation approach is based on transformation of the input parameter space, such that the SA is performed using the independent distributions, following the approach (ii) and the work of Mara and Tarantola [12,13,14]. The contributions are the following: ...

Sensitivity analysis is an important tool used in many domains of computational science to either gain insight into the mathematical model and interaction of its parameters or study the uncertainty propagation through the input-output interactions. In many applications, the inputs are stochastically dependent, which violates one of the essential assumptions in the state-of-the-art sensitivity analysis methods. Consequently, the results obtained ignoring the correlations provide values which do not reflect the true contributions of the input parameters. This study proposes an approach to address the parameter correlations using a polynomial chaos expansion method and Rosenblatt and Cholesky transformations to reflect the parameter dependencies. Treatment of the correlated variables is discussed in context of variance and derivative-based sensitivity analysis. We demonstrate that the sensitivity of the correlated parameters can not only differ in magnitude, but even the sign of the derivative-based index can be inverted, thus significantly altering the model behavior compared to the prediction of the analysis disregarding the correlations. Numerous experiments are conducted using workflow automation tools within the VECMA toolkit.

... The calculation of Sobol's sensitivity indices could refer to Refs. [57,[62][63][64][65][66]. Among them, the first-order sensitivity index S i is used to describe the part of the variance in the output parameter due to the input parameter X i , known as the first-order effect of X i . ...

... PCE represents a random variable by a series of polynomial chaos basis, which allows the statistical moments of the system output to be estimated. Proposed by Ghanem and Spanos [35], the Hermite PCE has been applied to a variety of engineering fields [41][42][43][44][45]. In the most probable point-based method, first-order reliability method [46] and second-order reliability method [47] are the two most popular approaches. ...

... μ MCS =1/nΣi = 1 n μ MCSi (42) and the unbiased estimator of is written as: (43) The confidence interval (CI) is obtained as: (44) where represents the confidence level value, for 99% CI, equals 2.576. represents the standard deviation of the samples. ...

Quantification and propagation of aleatoric uncertainties distributed in complex topological structures remain a challenge. Existing uncertainty quantification and propagation approaches can only handle parametric uncertainties or high dimensional random quantities distributed in a simply connected spatial domain. There lacks a systematic method that captures the topological characteristics of the structural domain in uncertainty analysis. Therefore, this paper presents a new methodology that quantifies and propagates aleatoric uncertainties, such as the spatially varying local material properties and defects, distributed in a topological spatial domain. We propose a new random field-based uncertainty representation approach that captures the topological characteristics using the shortest interior path distance. Parameterization methods like PPCA and β-Variational Autoencoder (βVAE) are employed to convert the random field representation of uncertainty to a small set of independent random variables. Then non-intrusive uncertainties propagation methods such as polynomial chaos expansion and univariate dimension reduction are employed to propagate the parametric uncertainties to the output of the problem. The effectiveness of the proposed methodology is demonstrated by engineering case studies. The accuracy and computational efficiency of the proposed method is confirmed by comparing with the reference values of Monte Carlo simulations with a sufficiently large number of samples.

... PCE are commonly used in uncertainty quantification and sensitivity analysis. For deterministic models, the following works can be mentioned: Sudret (2008); Le Gratiet et al. (2017); Mara and Becker (2021). In the framework of stochastic models, it has been mainly studied for sensitivity analysis of models based on stochastic differential equations (SDE) (see, e.g., Le Maître and Knio (2015); Jimenez et al. (2017); Étoré et al. (2020)). ...

This thesis focuses on the sensitivity analysis of stochastic models. These models include uncertainties that originate mainly from two sources: the parametric uncertainty due to the lack of knowledge of parameters and the intrinsic randomness that represents the noise inherent to the model coming from the way chance intervenes in the description of the modeled phenomenon. The presence of intrinsic randomness is a challenge in sensitivity analysis because, on the one hand, it is generally hidden and therefore cannot be characterized and, on the other hand, it acts as noise when evaluating the impact of the parameters on the model output However, in epidemiology, the issues associated with the sensitivity of a model can be important in the control of epidemics because they impact the decisions made on the basis of this model. This thesis studies approaches for sensitivity analysis of stochastic models such as epidemiological models based on stochastic processes, in the framework of variance-based analysis. In a general context, we introduce a method for estimating sensitivity indices that optimizes the trade-off between the number of input parameter values of the model and the number of replications of model evaluation in each of these values. For this method, we consider the class of quantities of interest of stochastic model outputs that are in the form of conditional expectations with respect to uncertain parameters. In the context of estimation of sensitivity indices by the Monte Carlo method, we control the quadratic risk of the estimators, show its convergence and find a trade-off between the bias related to the presence of the intrinsic randomness and the variance. In the specific context of stochastic compartmental models in epidemiology, we characterize the intrinsic randomness of the stochastic processes on which these models are based. These stochastic processes can be Markovian or non-Markovian. For Markovian processes, we use Gillespie algorithms to make explicit the intrinsic randomness and to separate it from uncertain parameters. Regarding non-Markovian processes, we extend to a large class of compartmental models the Sellke construction, which was originally introduced to describe epidemic dynamics of the SIR model in a framework that is not necessarily Markovian. This extension has allowed us to develop an algorithm that generates exact trajectories in a non-Markovian framework for a large class of compartmental models but also to be able to separate intrinsic randomness from parameter uncertainty in the output of these models. Thus, for both types of processes, Markovian and non-Markovian, the separation of the two sources of uncertainty has been obtained and it allows to represent model outputs as a deterministic function of the uncertain parameters and the variables representing the intrinsic randomness. When the uncertainty on the parameters is assumed to be independent of the intrinsic randomness, this representation allows to assess the contributions of the intrinsic randomness on the model outputs, in addition to the contributions of the parameters. It is also possible to characterize different interactions. This thesis has contributed to develop an approach to estimate sensitivity indices and to evaluate the contribution of intrinsic randomness in compartmental models in epidemiology based on stochastic processes.

... The popularity of PCE stems from its fast training time, theoretically converged properties, and various implementations of sparse algorithms for non-intrusive PCE [13,14,12]. Variants of non-intrusive PCE have been widely used in several tasks including UQ and GSA [4,15,16,17,18], and reliability analysis [19,20,21]. The nonlinearity and interactions between variables can be theoretically investigated from PCE terms and coefficients. ...

Surrogate models are indispensable tools in uncertainty quantification and global sensitivity analysis. Polynomial chaos expansion (PCE) is one of the most widely used surrogate models, thanks to its faster convergence rate compared to Monte Carlo simulation. In some cases, especially for complex problems, analyzing the complexity of the random input-output relationship (e.g., nonlinearity and interactions between input variables) may reveal additional information and useful insight. To that end, this paper introduces the use of Shapley additive explanations (SHAP) to help the explanation of a PCE model. Originating from game theory and machine learning, SHAP computes the contribution of the input variables to the single prediction level. SHAP enables visual inspection of the nonlinearity and interaction between variables from a PCE model. In addition, as an alternative to Sobol indices, SHAP also quantifies the relative importance of the inputs to the output. This paper introduces a procedure to calculate SHAP values from a PCE model without explicitly building multiple PCE models. A fast and exact algorithm that enables the calculation of SHAP for high-dimensional problems is presented. The usefulness of SHAP with PCE is demonstrated on several algebraic and non-algebraic problems with varying complexities.