Variance-based Importance Measures Research Articles

Variable importance analysis of wind turbine extreme responses with Shapley value explanation

As a leading form of renewable energy, wind power is gaining prominence with its decreasing levelized cost of electricity. The wind turbine extreme analysis is crucial for the structural reliability and integrity of wind turbines. Variable importance analysis provides a quantitative assessment of how significantly each variable influences model predictions. Variance-based importance measure quantifies each input variable’s contribution to the overall variance of a model or system. Such analyses are critical for interpreting the model and selecting features, thereby facilitating informed decision-making. This study uses Shapley value explanations to conduct a variable importance analysis of three wind turbine extreme responses, blade tip-tower clearance, blade root bending moment, and tower base bending moment. Random forest-based regression models and an improved Shapley effect evaluation algorithm are employed to improve the efficiency of the numerical analysis. The findings reveal the relative importance of five inputs, two wind parameters, one aerodynamic parameter, and two structural parameters, on each wind turbine extreme response variable. Additionally, the study explores how different dependency structures and varying correlation coefficients, even within the same correlation structure, affect the outcomes of variable importance analysis.

Derivative-based new upper bound of Sobol’ sensitivity measure

Global sensitivity (also called “uncertainty importance measure”) can reflect the effect of input variables on output response. The variance-based importance measure proposed by Sobol’ has highly general applicability. The Sobol’ total sensitivity index Sitotcan estimate the total contribution of input variables to the model output, including the self-influence of variable and the intercross influence of variable vectors. However, the computational load of Sitot is extremely heavy for double-loop simulation. The main sensitivity index Si is the lower bound of Sitot, and new upper bounds of Sitot based derivative are derived and proposed. New upper bounds of Sitot for different variable distribution types (such as uniform, normal, exponential, triangular, beta and gamma) are analyzed, and the process and formulas are presented comprehensively according to functional analysis and the Euler–Lagrange equation. Derivative-based upper bounds are easy to implement and evaluate numerically. Several numerical and engineering examples are adopted to verify the efficiency and applicability of the presented upper bounds, which can effectively estimate the Sitot value.

Moment-independent importance measure of correlated input variable and its state dependent parameter solution

For clearly exploring the origin of the uncertainty of structure failure probability in case the correlated input variables are involved, a new moment-independent importance measure of the input variable is firstly proposed on the failure probability. The intrinsic relationship between the proposed moment-independent importance measure and the corresponding variance-based importance measure is exposed. Then, based on the existing independent orthogonalization-based importance measures of the correlated input variables, the proposed moment-independent importance measure is further decomposed 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. Finally, combining the highly efficient state dependent parameter (SDP) method for the variance-based importance analysis of independent variables, a SDP solution is established for the moment-independent importance measure of correlated variables. Several examples are used to demonstrate that the proposed importance measures can effectively describe the effects of the input variables on the reliability of the structure, and the established solution can obtain the moment-independent importance measures efficiently for both the independent input variables and the correlated ones.

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.

Importance measure analysis with epistemic uncertainty and its moving least squares solution

For the structural systems with both epistemic and aleatory uncertainties, in order to analyze the effect of the epistemic uncertainty on the safety of the systems, a variance based importance measure of failure probability is constructed. Due to the large computational cost of the proposed measure, a novel moving least squares (MLS) based method is employed. By fitting the relationship of parameters and failure probability with moving least squares strategy, the conditional failure probability can be obtained conveniently, then the corresponding importance measure can be calculated. Compared with Sobol’s method for the variance based importance measure, the proposed method is more efficient with sufficient accuracy. The Ishigami function is used to test the efficiency of the proposed method. Then the proposed importance measure is used in two engineering applications, including a roof truss and a riveting process.

Importance measures for imprecise probability distributions and their sparse grid solutions

For the imprecise probability distribution of structural system, the variance based importance measures (IMs) of the inputs are investigated, and three IMs are defined on the conditions of random distribution parameters, interval distribution parameters and the mixture of those two types of distribution parameters. The defined IMs can reflect the influence of the inputs on the output of the structural system with imprecise distribution parameters, respectively. Due to the large computational cost of the variance based IMs, sparse grid method is employed in this work to compute the variance based IMs at each reference point of distribution parameters. For the three imprecise distribution parameter cases, the sparse grid method and the combination of sparse grid method with genetic algorithm are used to compute the defined IMs. Numerical and engineering examples are employed to demonstrate the rationality of the defined IMs and the efficiency of the applied methods.

Point estimate-based importance analysis for structural models with correlated variables

Importance analysis is conducted to find the contributions of the inputs to the output uncertainty. In this work, a point estimate-based importance analysis algorithm is established for models involving correlated input variables, and the variance contribution by an individual correlated input variable is decomposed into correlated contribution and uncorrelated contribution. In the established algorithm, the correlated variables are orthogonalised to generate corresponding independent variables, and the performance function is reconstructed in the independence space. Then, the point estimate is employed to compute the variance-based importance measures in the independence space, by which the variance contribution of the original correlated variables, including the correlated part and uncorrelated part, can be obtained. Different point estimate methods can be employed in the proposed algorithm; thus, the algorithm is adaptable and improvable. The proposed algorithm avoids the sampling procedure, which usually consumes a heavy computational cost. Discussion of numerical and engineering examples in this work has demonstrated that the proposed algorithm provides an effective tool to deal with uncertainty analysis involving correlated inputs.

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

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.

Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests

The estimation of variance-based importance measures (called Sobol’ indices) of the input variables of a numerical model can require a large number of model evaluations. It turns to be unacceptable for high-dimensional model involving a large number of input variables (typically more than ten). Recently, Sobol and Kucherenko have proposed the derivative-based global sensitivity measures (DGSM), defined as the integral of the squared derivatives of the model output, showing that it can help to solve the problem of dimensionality in some cases. We provide a general inequality link between DGSM and total Sobol’ indices for input variables belonging to the class of Boltzmann probability measures, thus extending the previous results of Sobol and Kucherenko for uniform and normal measures. The special case of log-concave measures is also described. This link provides a DGSM-based maximal bound for the total Sobol indices. Numerical tests show the performance of the bound and its usefulness in practice.

A new algorithm for variance-based importance measures and importance kernel sensitivity

Variance-based importance measure has proven itself as an effective tool to reflect the effects of input variables on the output. Owing to the desirable properties, researchers have paid lots of attention to improving efficiency in computing a variance-based importance measure. Based on the theory of point estimate, this article proposes a new algorithm, decomposing the importance measure into inner and outer parts, and computing each part with a point estimate method. In order to discuss the impacts on the variance-based importance measure from distribution parameters of input variables, a new concept of kernel sensitivity of the variance-based importance measure is put forward, with solving algorithms respectively, based on numerical simulation and point estimate established as well. For cases where the performance function with independent and normally distributed input variables is expressed by a linear or quadratic polynomial without cross-terms, analytical results of the variance-based importance measure and the kernel sensitivity are derived. Numerical and engineering examples have been employed to illustrate the applicability of the proposed concept and algorithm.

A modified variance-based importance measure and its solution by state dependent parameter

To overcome the disadvantage of traditional variance-based importance measures, i.e. the effects of different realizations of input variables on output response may mutually counteract each other, a modified variance-based importance measure is presented for importance analysis of the input variables. The proposed measure analyses the importance of the input variables comprehensively in terms of the expectation and variance of the output response. Compared with the traditional variance-based importance analysis method, the modified importance measure indices not only reflect the old one, but also provide a very useful supplement for it. Furthermore, combined with the advantages of the state dependent parameter model, a solution to the proposed measure indices is provided. Several examples are introduced to show that the modified importance measure is more comprehensive and reasonable, and the solution based on the state dependent parameter method can improve computational efficiency considerably with acceptable precision.

A new method on ANN for variance based importance measure analysis of correlated input variables

Due to the strong flexibility of artificial neural networks (ANNs), a new method on ANN is presented to analyze the variance based importance measure (VBIM) of correlated input variables. An individual input variable’s global variance contribution to the output response can be evaluated and decomposed into the contributions uncorrelated and correlated with other input variables by use of the ANN model. Furthermore, the ANN model is used to decompose the correlated contribution into components, which reflect the contributions of the individual input variable correlated with each of other input variables. Combining the uncorrelated contributions and the correlated contribution components of all input variables, an importance matrix can be obtained to explicitly expose the contribution components of the correlated input variables to the variance of the output response. Several properties of the importance matrix are discussed. One numerical example and three engineering examples are used to verify the presented new method, the results show that the new ANN-based method can evaluate the VBIM with acceptable precision, and it is suitable for the linear and nonlinear output responses.

Moment-independent importance measure of basic variable and its state dependent parameter solution

To analyze the effect of basic variable on output of the structure or system in reliability engineering, two moment-independent importance measures of the basic variable are proposed respectively on the failure probability and distribution function of the output. The importance measures proposed not only inherit the advantages of the traditional moment-independent importance measures, but also reflect the intrinsic relationship of the moment-independent measures and the corresponding variance-based importance measures. For the problem that the computational effort of the moment-independent importance measure is usually too high, the computation of the proposed moment-independent importance measures is transformed into that of the variance-based importance measures on their intrinsic relationship. And then combining the high efficient state dependent parameter (SDP) method for the calculation of the conditional moments of the model output, a SDP solution is established to solve two moment-independent importance measures. Several examples are used to demonstrate that the proposed importance measures can effectively describe the effect of the basic variable on the reliability of the structure system, and the established solution can obtain the two importance measures simultaneously with only a single set of model runs, which allows for a strong reduction of the computational cost.

Global sensitivity analysis for models with spatially dependent outputs

The global sensitivity analysis of a complex numerical model often calls for the estimation of variance-based importance measures, named Sobol' indices. Metamodel-based techniques have been developed in order to replace the CPU time-expensive computer code with an inexpensive mathematical function, which predicts the computer code output. The common metamodel-based sensitivity analysis methods are well suited for computer codes with scalar outputs. However, in the environmental domain, as in many areas of application, the numerical model outputs are often spatial maps, which may also vary with time. In this paper, we introduce an innovative method to obtain a spatial map of Sobol' indices with a minimal number of numerical model computations. It is based upon the functional decomposition of the spatial output onto a wavelet basis and the metamodeling of the wavelet coefficients by the Gaussian process. An analytical example is presented to clarify the various steps of our methodology. This technique is then applied to a real hydrogeological case: for each model input variable, a spatial map of Sobol' indices is thus obtained. Copyright © 2010 John Wiley & Sons, Ltd.

Calculations of Sobol indices for the Gaussian process metamodel

Global sensitivity analysis of complex numerical models can be performed by calculating variance-based importance measures of the input variables, such as the Sobol indices. However, these techniques, requiring a large number of model evaluations, are often unacceptable for time expensive computer codes. A well-known and widely used decision consists in replacing the computer code by a metamodel, predicting the model responses with a negligible computation time and rending straightforward the estimation of Sobol indices. In this paper, we discuss about the Gaussian process model which gives analytical expressions of Sobol indices. Two approaches are studied to compute the Sobol indices: the first based on the predictor of the Gaussian process model and the second based on the global stochastic process model. Comparisons between the two estimates, made on analytical examples, show the superiority of the second approach in terms of convergence and robustness. Moreover, the second approach allows to integrate the modeling error of the Gaussian process model by directly giving some confidence intervals on the Sobol indices. These techniques are finally applied to a real case of hydrogeological modeling.