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

Regional importance effect analysis of the input variables on failure probability and its state dependent parameter estimation

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.

Copula-based decomposition approach for the derivative-based sensitivity of variance contributions with dependent variables

Extended State Dependent Parameter modelling with a Data-Based Mechanistic approach to nonlinear model structure identification

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

Importance measure can be used to identify the most vulnerable components with respect to system functionality or failure. Traditional importance measures may not be effective to evaluate the contribution of an individual component if its reliability value does not fall in the full range between 0 and 1. Based on the cost-reliability relation, this paper proposes a generalized Birnbaum importance measure (GBIM) to quantify the contribution of individual components to system reliability improvement by considering reliability range, manufacturing complexity, and technology feasibility. Since GBIM possesses several unique features in terms of guiding system reliability optimization, in this paper, we further develop a GBIM-based genetic algorithm to solve a type of optimal reliability allocation problem. The numerical studies show that both the computational efficiency and the near global optimality based on GBIM outperforms the methods using the traditional importance measures.

Analyzing sensitivity measures using moment-matching technique

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.

Importance measures in reliability systems are used to identify weak components in contributing to proper functioning of the system. Traditional importance measures mainly concern the change of the system reliability as the change of the reliability of one component and seldom consider the expected number of repairs of the objective component in unit time. This paper proposes an improvement potential rate importance (IPR) to verify the effectiveness of the improvement in system reliability for multi-state repairable k-out-of-n: G systems. Then the comparisons between IPR and Birnbaum importance are discussed. Finally, a case study is given to demonstrate the proposed IPR.

In the non-probabilistic uncertainty structural analysis, the input uncertain variables of it, such as loads and material properties, will be propagated to the output responses, which include the displacement, stress and compliance, etc. To measure the effect of these non-probabilistic input variables on the output response, two new uncertainty importance measures on the non-probabilistic reliability index are discussed. For the linear limit state function, the analytical solutions of the importance measures are derived. To reduce computational effort, the discretization method and the surrogate model method are presented to calculate the two importance measures in case of the non-linear limit state. Finally, four examples demonstrate that the proposed importance measures can effectively describe the effect of the input variables on the reliability of the structure system, and the established methods can effectively obtain the two importance measures.

ABSTRACTFor the structural systems with both the uncertainties of input variables and their distribution parameters, three sensitivity indices are proposed to measure the influence of input variables, distribution parameters and their interactive effects. With those sensitivity indices, analysts can make a decision that whether it is worth to accumulate data of one distribution parameter to reduce its uncertainty. Due to the large computational cost, the analytical solutions are derived for quadratic polynomial output responses. Whereas for the complex models, state dependent parameter (SDP) method is utilized to solve the proposed sensitivity indices efficiently.

Variable importance analysis: A comprehensive review

Decision-making using machine learning requires a deep understanding of the model under analysis. Variable importance analysis provides the tools to assess the importance of input variables when dealing with complex interactions, making the machine learning model more interpretable and computationally more efficient. In classification problems with imbalanced datasets, this task is even more challenging. In this article, we present two variable importance techniques, a nonparametric solution, called mh-χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , and a parametric method based on Global Sensitivity Analysis. The mh-χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> employs a multivariate continuous response framework to deal with the multiclass classification problem. Based on the permutation importance framework, the proposed mh-χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> algorithm captures the dissimilarities between the distribution of misclassification errors generated by the base learner, Conditional Inference Tree, before and after permuting the values of the input variable under analysis. The GSA solution is based on the Covariance decomposition methodology for multivariate output models. Both solutions will be assessed in a comparative study of several Random Forest-based techniques with emphasis in the multiclass classification problem with different imbalanced scenarios. We apply the proposed techniques in two real application cases in order first, to quantify the importance of the 35 companies listed in the Spanish market index IBEX35 on the economic, political and social uncertainties reflected in economic newspapers in Spain during the first quadrimester of 2020 due to the COVID-19 pandemic and second, to assess the impact of energy factors on the occurrence of spike prices on the Spanish electricity market.

Traditional reliability importance measures have been successfully extended from binary-state models to multi-state models. The calculation of these measures typically relies on the true reliabilities of components. In reality, however, the true values of component reliabilities are usually unknown, and they are generally approximated by their estimates generated from testing or field failure data. The accuracy of the estimates is limited by the available data. Research on uncertainty importance measures (UIMs) has emerged on this account to rank components based on their potentials to reduce the uncertainty about the estimated system reliability. The UIMs of components for binary-state models are well studied, but there is a lack of studies dedicated to multi-state models. In this paper, the reliability estimator and the corresponding uncertainty (characterized by the variance estimator) are derived for multi-state systems with structures such as serial, parallel, bridge, and their more complex combinations. The derivation process utilizes multinomial reliability testing and the universal generating function method. With the help of the derived estimators, we extend uncertainty importance research to multi-state models through a newly defined variance-based measure. Examples are provided to demonstrate the proposed ideas.

The uncertainty of parameters will have a significant impact on slope stability, where sensitivity analysis is a commonly used method in uncertainty research. However, traditional sensitivity analysis method costs much computation time. When calculating the sensitivity index of one parameter, all other parameters are taken as fixed values, and the uncertainty of all parameters cannot be considered simultaneously. Therefore, the variance-based and the moment-independent global sensitivity analysis (GSA) methods are both introduced to determine the influence of geotechnical parameters on slope stability in this study. To solve the importance index of GSA, the least angle regression algorithm, the kernel density estimation, and orthogonal polynomial estimation methods are developed to obtain variance-based importance index and the moment-independent importance index, respectively. The proposed methods allow all variables to change simultaneously within their variation range and have high computational efficiency. The results are in good at with those obtained by the variance-based Monte Carlo simulation method, which is considered as the exact solution forobtaining the importance index. The influence of the correlation between the shear strength parameters (c and φ) on the importance index is also studied, which indicates that the negative correlation will have a great impact on the importance index, which in turn affects the safety assessment of slope. Three engineering cases have been studied for engineering application, and the compared results indicate that the impact of the geotechnical parameters uncertainty on the safety factor (Fs) and failure probability (pf) are different. Therefore, the approaches based on GSA which can integrate the Fs with pf will be a promising approach for slope stability evaluation.