Global Sensitivity Analysis Method Research Articles

Neural network driven sensitivity analysis of diffraction-based overlay metrology performance to target defect features

Abstract Overlay (OVL) is one significant performance indicator for the lithography process control in semiconductor manufacturing. The accuracy of the OVL metrology is extremely critical for guarantee the lithography quality. Currently, diffraction-based overlay (DBO) is one of the mainstream OVL metrology techniques. Unfortunately, the accuracy of the DBO metrology is largely affected by the defect features of the overlay target. Therefore, there is a strong need to investigate the impacts of these target defects on the DBO metrology performance. However, efficiently investigating the statistical and interactive impacts of various DBO target defects remains challenging. This study aims to address this issue through proposing an intelligent sensitivity analysis approach. A cumulative distribution based Global Sensitivity Analysis (GSA) method is utilized to assess the nonlinear influences of multiple defects in the OVL target on the DBO inaccuracy. The scenarios with both known and unknow distributions of the OVL target defects are considered. For the former, a neural network driven forward model is constructed for fast calculating the optical diffraction responses to accelerate the GSA process. For the latter, another neural network based inverse model are built for efficiently estimating the distribution of the target defects. Finally, a series of simulation experiments are conduct for typical DBO targets with multiple common defect features. The results demonstrate the effectiveness and robustness of the proposed approach as well as give valuable insights into the DBO defect analysis. Our study provides a strong tool to assist the practitioners in achieving intelligent and efficient DBO analysis and thus in enhancing OVL metrology performance.

APPLICATION OF MACHINE LEARNING MODELS AND GSA METHOD FOR DESIGNING STUD CONNECTORS

The design of stud connectors is aided by determining the relationship between shear strength and the input variables (number, diameter, height, tensile strength and elastic modulus of the studs, and compressive strength and elastic modulus of the concrete) that influence strength. Since strength is nonlinearly related to the influencing variables, which makes the predictions of the relevant empirical equations unreliable, the use of machine learning (ML) models is preferred. The prediction results of eight machine learning models were evaluated, including linear regression (LR1), ridge regression (RR), lasso regression (LR2), back-propagation artificial neural network (BP ANN), genetic algorithm optimized BP ANN (GA-BP ANN), extreme learning machines (ELM), random forests (RF), and support vector machines (SVM). The results show that the GA-BP ANN model is the most accurate model for prediction with a mean absolute percentage error (MAPE) of 6.17% and an R2 of 0.9599. Based on the GA-BP ANN model and the global sensitivity analysis (GSA) method, a new parameter importance analysis method was developed to compare the magnitude of the effect of different input variables on strength. It was found that stud diameter had the greatest effect on shear strength.

A SENSITIVITY ANALYSIS METHOD COMBINING DEMPSTER-SHAFER THEORY AND MACHINE LEARNING FOR ENERGY-SAVING EVALUATION OF BUILDING OCCUPANT BEHAVIOR

ABSTRACT For a very long time, the research of the sensitivity analysis of occupant behavior to energy assessment has been in the spotlight. The key element of the research is determining the exact probability of occupant behavior uncertainty. However, due to the specificity of occupant behavior, data on occupant behavior from different independent sources of information can differ significantly. This paper explores the use of Dempster-Shafer theory to the sensitivity analysis of energy evaluation of occupant behavior in buildings. The Dempster-Shafer theory is an imprecise probability theory that allows the system to create assumed confidence intervals based on interval values probability combined with knowledge of uncertainty factors from many different sources of information. The findings show that the data processing approach based on Dempster-Shafer theory provides effective and reliable information for evaluating energy related to human behavior in buildings. To begin with, the sensitivity analysis process might be accelerated by applying machine learning to process the data. Then, in order to ensure the accuracy of the simulation results, multiple learning methods can be used. Finally, in this paper, model parameters were chosen based on the specific circumstances as soon as the model had been built in order to effectively reduce costs related to operation and increase model accuracy. To establish the final results, the model is evaluated using global sensitivity analysis methods.

Efficient global sensitivity analysis method for dynamic models in high dimensions

AbstractDynamic models generating time‐dependent model predictions are typically associated with high‐dimensional input spaces and high‐dimensional output spaces, in particular if time is discretized. It is computationally prohibitive to apply traditional global sensitivity analysis (SA) separately on each time output, as is common in the literature on multivariate SA. As an alternative, we propose a novel method for efficient global SA of dynamic models with high‐dimensional inputs by combining a new polynomial chaos expansion (PCE)‐driven partial least squares (PLS) algorithm with the analysis of variance. PLS is used to simultaneously reduce the dimensionality of the input and output variables spaces, by identifying the input and output latent variables that account for most of their joint variability. PCE is incorporated into the PLS algorithm to capture the non‐linear behavior of the physical system. We derive the sensitivity indices associated with each output latent variable, based on which we propose generalized sensitivity indices that synthesize the influence of each input on the variance of entire output time series. All sensitivities can be computed analytically by post‐processing the coefficients of the PLS‐PCE representation. Hence, the computational cost of global SA for dynamic models essentially reduces to the cost for estimating these coefficients. We numerically compare the proposed method with existing methods by several dynamic models with high‐dimensional inputs. The results show that the PLS‐PCE method can obtain accurate sensitivity indices at low computational cost, even for models with strong interaction among the inputs.

A streamlined approach to uncertainty and sensitivity analysis for models with spatial outputs with an example from geodiversity assessment

The paper proposes a joint use of two global sensitivity analysis methods to evaluate the influence of model inputs on spatially explicit outputs. The analysis is conducted on spatial outputs of geodiversity assessment. First, the screening method is used to identify inputs with low importance and interactions, which can be discarded in later analysis. A simplified model is compared with a complete model using the mean and variance calculated on spatial outputs to evaluate outcome uncertainty. Finally, the variance decomposition method computes two sensitivity index maps per input. The results are mapped using bivariate classification, allowing for the simultaneous identification of regions with higher geodiversity scores and their corresponding variability. The research reveals that the streamlined approach preserves the overall patterns of outcome uncertainty and sensitivity observed in an original model while simultaneously reducing the number of outcome maps to interpret. Additionally, bivariate mapping proves efficient in concurrently depicting spatial sensitivity and interaction effects. The analytic efficiencies afforded by the presented approach and bivariate mapping enhance the operational knowledge of spatially explicit uncertainty and sensitivity analysis by simplifying the computational procedure and offering an alternative way of visualizing and interpreting input factors driving the sensitivity of model output.

An Improved Sobol Sensitivity Analysis Method

The Sobol method is a variance-based global sensitivity analysis method that evaluates single-input and multiple-input interaction effects by calculating the contribution of a single input to the output variance and the contribution of multiple inputs to the output variance. The Sobol method requires each input to obey a uniform distribution of U [0,1], but it is difficult to meet the requirements in practice. Taking the sum function as an example, this paper analyzes the inapplicability of the existing Sobol method when the input does not obey the uniform distribution U [0,1]. To solve the inapplicability of the Sobol method and broaden the application scope, an improved Sobol sensitivity analysis method is proposed. First, the effect of the joint probability density function not 1 on sensitivity calculation is studied; second, the input parameters are changed to uniform distribution U [0,1] through variable substitution; finally, a complete algorithm model is presented and logical sensitivity analysis results are obtained. Application verification shows that the improved Sobol method is more scientific, applicable and practical.

A Computationally Efficient Method for Parameter Sensitivity Analysis of Microbially Explicit Biogeochemical Models Accounting for Long‐Term Behavior

AbstractMicrobial ecological models become increasingly complex owing to the incorporation of many parameters and multiple biotic and abiotic processes. However, little attention has been paid to the variations in the parameter sensitivity during long‐term versus short‐term simulations. Here, we developed a Multi‐Objective Parameter Sensitivity Analysis (MOPSA) method to efficiently identify the important parameters in complex microbial ecological models with multiple response variables of interest in terms of short‐ and long‐term model simulations. We found that MOPSA was more computationally efficient for complex microbial ecological models than Sobol's method because of MOPSA's reliability and low computational sample size. In addition, we address the increased significance of microbial physiology in mediating long‐term than short‐term soil C‐N cycling, indicating that experiment‐model integration practices should examine model behaviors beyond the conventional short‐term experimental period. The outcomes of this study provide an efficient global sensitivity analysis method for parameterization and the scientific foundation for microbial physiology in mediating long‐term microbial ecological processes.

Global Sensitivity Analysis of Factors Influencing the Surface Temperature of Mold during Autoclave Processing.

During the process of forming carbon fiber reinforced plastics (CFRP) in an autoclave, deeply understanding the global sensitivity of factors influencing mold surface temperature is of paramount importance for optimizing large frame-type mold thermally and enhancing curing quality. In this study, the convective heat transfer coefficient (CHTC), the thickness of composite laminates (TCL), the thickness of mold facesheet (TMF), the mold material type (MMT), and the thickness of the auxiliary materials layer (TAL) have been quantitatively assessed for the effects on the mold surface temperature. This assessment was conducted by building the thermal-chemical curing model of composite laminates and utilizing the Sobol global sensitivity analysis (GSA) method. Additionally, the interactions among these factors were investigated to gain a comprehensive understanding of their combined effects. The results show that the sensitivity order of these factors is as follows: CHTC > MMT > TMF > TCL > TAL. Moreover, CHTC, MMT, and TMF are the main factors influencing mold surface temperature, as the sum of their first-order sensitivity indices accounts for over 97.3%. The influence of a single factor is more significant than that of the interaction between factors since the sum of the first-order sensitivity indices of the factors is more than 78.1%. This study will support the development of science-based guidelines for the thermal design of molds and associated heating equipment design.

Prediction of bridge structure deformation and strain based on dynamic testing and intelligent algorithms

The field load test is a direct and effective method for evaluating the performance of bridge structures. However, existing bridge field static load tests are costly and inefficient; moreover, they obstruct traffic and cause unavoidable damage to the bridge structure. As an alternative the the static load test, a random model update method based on bridge dynamic load tests and the Bayesian inference is proposed in this paper. The bridge static load test results were predicted with a high accuracy. To speed up the Bayesian method to infer the posterior probability density of the updated parameters, the Gaussian process was used in place of the finite element model, and the Bayesian inference used the Markov chain Monte Carlo method based on the delayed rejection adaptive Metropolis algorithm. First, the parameters to be modified for the bridge structure analysis model were determined based on the global sensitivity analysis method. Second, a uniform design sampling method was used to establish the Gaussian process optimization model to update the random model of the bridge structure. Finally, a reinforced concrete truss arch bridge was used to verify the correctness of the static load results of the bridge predicted by the random model update method based on dynamic load testing and Bayesian inference. The research results reveal that the prediction results of the bridge static load test based on the dynamic load test and Bayesian inference method agree with the actual test results.

Fatigue reliability and sensitivity analysis of aero‐disk considering correlation

AbstractAero‐disk is a key component of aero‐engine. Due to its complex working conditions, aero‐disk is prone to structural failures. Therefore, it is essential to analyze the reliability and importance input variables of aero‐disk. In this paper, a framework of aero‐disk reliability analysis and global sensitivity analysis (GSA) was established. First, the hazardous regions of aero‐disk were determined by finite element analysis, and the limit state function of aero‐disk was defined by the life interference model. Then D‐Vine model was utilized to establish the correlation model for aero‐disk hazardous regions to evaluate the reliability of aero‐disk. In addition, two GSA methods based on Copula and space partition were proposed to identify important input random variables considering underlying correlation, and the accuracy of the proposed method was verified by two examples. Finally, the proposed method was applied to the GSA of aero‐disk. The results show that the established framework fills in the gap of uncertainty analysis of aero‐disk, which can be extended to other engineering fields. The proposed GSA methods have both high efficiency and accuracy and can realize multi‐dimensional GSA when the correlation of input variables is considered.

A two-step Bayesian network-based process sensitivity analysis for complex nitrogen reactive transport modeling

Process-based nitrogen reactive transport models (NRTMs) are vital tools for simulating complex nitrogen reactive transport processes in soils and groundwater to reduce nutrient loss and contamination risk. To develop and improve these models, global sensitivity analysis (GSA) is used to identify key model parameters and processes. Because existing GSA approaches focus mainly on model parameters and not model processes, in this study, we developed a two-step GSA method to improve our understanding of model processes. First, we applied the Morris method to identify less influential parameters. Then, we developed a Bayesian network (BN)-based GSA method to identify the dominant processes for different model predictions. The BN-GSA method accounts for the hierarchical uncertainty in the structure of the model system and flexibly groups uncertain inputs into processes based on their characteristics. The two-step GSA method was applied to a real-world nitrogen reactive transport model with 68 uncertain parameters in three different climate scenarios. The results show that the dominant processes differed for various nitrogen and agricultural outputs; thus, different processes should be emphasized based on specific modeling purposes.

Accurate solution of the SAIL model by leaf inclination angle calculation based on laser point clouds

ABSTRACT The canopy radiative transfer model enables quantitative inversion of vegetation biophysical parameters by rapidly simulating spectral information. However, structural input parameters, such as the leaf inclination angle distribution (LAD), are often set to empirical values in classic models, leading to simulation errors. Laser point cloud technology has developed rapidly in recent years, and the 3D structure of vegetation can be detailed using laser sensors and processing software, providing the necessary conditions for accurate solutions of radiative transfer models of the canopy. However, the mechanisms for improving existing models of canopy radiative transfer by introducing laser point clouds have not yet been theoretically analyzed in detail. In this study, a precise solution was proposed by incorporating the accurate LAD obtained from laser point clouds into the intermediate function of the traditional Scattering by Arbitrarily Inclined Leaves (SAIL) model, which is a widely used multiangle radiative transfer model. The calculation of the leaf inclination angle and the derivation of the improved model were detailed in this paper. Experiments from different observation angles and bands compared the similarity between the actual measured spectra and the simulated spectra for both the traditional SAIL model and the proposed model with accurate structural parameters. The overall similarity of the proposed model with respect to the conventional SAIL model was improved, indicating that introducing laser point clouds improved the simulation accuracy. The proposed model achieved more significant improvements in the blue 450 nm and red 670 nm bands. Moreover, the similarity improvements differed between different angles. Finally, we analyzed the sensitivity of the structural parameters of the SAIL model and the proposed model using local and global sensitivity analysis methods.

An efficient data-driven global sensitivity analysis method of shale gas production through convolutional neural network

The shale gas development process is complex in terms of its flow mechanisms and the accuracy of the production forecasting is influenced by geological parameters and engineering parameters. Therefore, to quantitatively evaluate the relative importance of model parameters on the production forecasting performance, sensitivity analysis of parameters is required. The parameters are ranked according to the sensitivity coefficients for the subsequent optimization scheme design. A data-driven global sensitivity analysis (GSA) method using convolutional neural networks (CNN) is proposed to identify the influencing parameters in shale gas production. The CNN is trained on a large dataset, validated against numerical simulations, and utilized as a surrogate model for efficient sensitivity analysis. Our approach integrates CNN with the Sobol' global sensitivity analysis method, presenting three key scenarios for sensitivity analysis: analysis of the production stage as a whole, analysis by fixed time intervals, and analysis by declining rate. The findings underscore the predominant influence of reservoir thickness and well length on shale gas production. Furthermore, the temporal sensitivity analysis reveals the dynamic shifts in parameter importance across the distinct production stages.

A Comparative Study of Polynomial Chaos Expansion-Based Methods for Global Sensitivity Analysis in Power System Uncertainty Control

A Comparative Study of Polynomial Chaos Expansion-Based Methods for Global Sensitivity Analysis in Power System Uncertainty Control

Global sensitivity analysis for studying hot-mix asphalt dynamic modulus parameters

The dynamic modulus (E*) of hot-mix asphalt mixtures is one of the most laborious and time-consuming material parameters to measure in the laboratory. It involves expensive, specialized equipment and expertize that are not readily available in most laboratories. Consequently, several efforts have been devoted to E* prediction models. Unfortunately, developing these prediction models is complex because of the numerous contributory factors and their non-linear influence on E* values. Moreover, such models are not able to prioritize or screen the major factors influencing the E* values. This study presents a new framework for analyzing the dynamic modulus influencing factors by adopting two modeling approaches. First, deep residual neural networks (DRNNs) for non-parametric approaches are used to improve the E* prediction capabilities and derive deep insight into the contributory parameters' effect on the E* value. Second, the well-known Witczak 1–40D prediction equation is used as a representative of the classical statistical modeling approach. In the validation of the models, a comprehensive laboratory database is utilized to account for all significant contributory parameters, such as binder characteristics, volumetric properties, mixture gradation, and testing circumstances parameters. Then, the performance is assessed using typical performance metrics. Lastly, intensive global sensitivity analysis (GSA) is undertaken with the assistance of Latin Hypercube Simulation. Three distinct GSA methods are used to emphasize the influence of each contributory factor on the value of E* in actual practice while reducing the possibility for result distortion owing to correlations between contributory variables. Performance metrics of the DRNNs and the Witczak 1–40D prediction models give the GSA conclusions high credibility. The GSA reveals that, among all possible inputs, the binder content, shear modulus, voids in the mineral aggregates, and temperature are the most significant factors in determining the E* value.

Analysis of discharge characteristics of a symmetrical stepped labyrinth side weir based on global sensitivity

Abstract In this paper, the discharge coefficient prediction model for this structure in a subcritical flow regime is first established by extreme learning machine (ELM) and Bayesian network, and the model's performance is analyzed and verified in detail. In addition, the global sensitivity analysis method is introduced to the optimal prediction model to analyze the sensitivity for the dimensionless parameters affecting the discharge coefficient. The results show that the Bayesian extreme learning machine (BELM) can effectively predict the discharge coefficients of the symmetric stepped labyrinth side weir. The range of 95% confidence interval [−0.055,0.040] is also significantly smaller than that of the ELM ([−0.089,0.076]) and the Kernel extreme learning machine (KELM) ([−0.091,0.081]) at the testing stage. The dimensionless parameter ratio of upstream water depth of stepped labyrinth side weir p/y1 has the greatest effect on the discharge coefficient Cd, accounting for 55.57 and 54.17% under single action and other parameter interactions, respectively. Dimensionless step number bs/L has little effect on Cd, which can be ignored. Meanwhile, when the number of steps is less (N = 4) and the internal head angle is smaller (θ = 45°), a larger discharge coefficient value can be obtained.

Efficient sensitivity analysis for biomechanical models with correlated inputs.

In most variance-based sensitivity analysis (SA) approaches applied to biomechanical models, statistical independence of the model input is assumed. However, often the model inputs are correlated. This might alter the interpretation of the SA results, which may severely impact the guidance provided during model development and personalization. Potential reasons for the infrequent usage of SA techniques that account for input correlation are the associated high computational costs, especially for models with many parameters, and the fact that the input correlation structure is often unknown. The aim of this study was to propose an efficient correlated global sensitivity analysis method by applying a surrogate model-based approach. Furthermore, this article demonstrates how correlated SA should be interpreted and how the applied method can guide the modeler during model development and personalization, even when the correlation structure is not entirely known beforehand. The proposed methodology was applied to a typical example of a pulse wave propagation model and resulted in accurate SA results that could be obtained at a theoretically 27,000× lower computational cost compared to the correlated SA approach without employing a surrogate model. Furthermore, our results demonstrate that input correlations can significantly affect SA results, which emphasizes the need to thoroughly investigate the effect of input correlations during model development. We conclude that our proposed surrogate-based SA approach allows modelers to efficiently perform correlated SA to complex biomechanical models and allows modelers to focus on input prioritization, input fixing and model reduction, or assessing the dependency structure between parameters.

Comparing global sensitivity analysis methods for the joint thermal design parameters of a space manipulator based on the Sobol’ and PAWN

The Global Sensitivity Analysis method works well for screening and ranking impacting factors. It can help with thermal design optimization and thermal model calibration. This study compares the use of Sobol's variance-based method with PAWN's density-based method for 12 thermal design parameters (such as the properties of thermal control coating and the contact heat transfer coefficient between components) of a space manipulator joint thermal model. The two techniques have high consistent parameter sensitivity. The analysis's findings indicate that the junction electric box and shell's temperature variations are significantly influenced by the cooling surface's coating properties. The cooling surface's absorption rate has a bigger impact than emissivity does. Only the characteristics of the multilayer mask and the state of parcel coverage have a significant impact on the temperature of the joint shell. The outcomes offer technical backing for cooperative optimization in thermal design. The identical no-effect parameters are discovered to be recognized by Sobol' and PAWN at the same time, but they are ordered in different ways. The limitation of Sobol's technique, which uses variance as a sensitivity index, may be the cause of this outcome. Finally, the sensitivity response results are re-substituted into the finite element thermal model. The changing rule of temperature curve is used to further confirm the accuracy of the sensitivity results.

Quantifying vertical streambed fluxes and streambed thermal properties using heat as a tracer during extreme hydrologic events

As a natural tracer, heat has some remarkable advantages over competing test methods and has become a popular and widely used tool for quantifying streambed fluxes. In recent decades, extreme hydrological events have become intense and frequent, resulting in irregular or complex temperature changes at the stream-streambed interface. During extreme hydrological events, most existing analytical models with periodic changing top boundary conditions are inapplicable. In this study, we present a heat-based analytical model that accounts for the arbitrary top boundary and initial conditions to estimate time-variant vertical streambed fluxes (VSFs) and streambed effective thermal diffusivity using temperature–time series from streambed sediments during extreme hydrologic events. The particle swarm optimization is used to estimate the time-variant VSFs. The Morris global sensitivity analysis method is used to test the impacts of the input parameters on heat transport in the streambed. Results show that the output temperature data is most sensitive to VSFs, followed by the volumetric heat capacity of the streambed, thermal conductivity and thermal dispersivity. The performance of the new model is tested using a numerical model with time-variant VSFs associated with extreme hydraulic events such as hurricanes and dam-controlled river conditions. Results demonstrate that despite oscillations in the estimated VSFs and fluctuations being stronger when the VSFs begin to reverse and at the peak of the VSFs, the new model of this study captures changes in VSFs under extreme hydraulic events. In addition, the frequency domain model, LPMLEn, performs satisfactorily in estimating VSFs in hurricane conditions. Application to field data demonstrates that the temperature distribution can be effectively remodeled based on the estimated VSFs and streambed effective thermal diffusivity. Both the present model and LPMLEn model could be considered as alternatives to the traditional isotope-based method in estimating VSFs during extreme hydrologic events.

Over-limit risk assessment method of integrated energy system considering source-load correlation

In an integrated energy system, source-load multiple uncertainties and correlations lead to an over-limit risk in operating state, including voltage, temperature, and pressure over-limit. Therefore, efficient probabilistic energy flow calculation methods and risk assessment theories applicable to integrated energy systems are crucial. This study proposed a probabilistic energy flow calculation method based on polynomial chaos expansion for an electric-heat-gas integrated energy system. The method accurately and efficiently calculated the over-limit probability of the system state variables, considering the coupling conditions of electricity, heat, and gas, as well as uncertainties and correlations in renewable energy unit outputs and multiple types of loads. To further evaluate and quantify the impact of uncertainty factors on the over-limit risk, a global sensitivity analysis method for the integrated energy system based on the analysis of covariance theory is proposed. This method considered the source-load correlation and aimed to identify the key uncertainty factors that influence stable operation. Simulation results demonstrated that the proposed method achieved accuracy to that of the Monte Carlo method while significantly reducing calculation time. It effectively quantified the over-limit risk under the presence of multiple source-load uncertainties.