Elementary Effects Research Articles

On the selection of Morris trajectories for parametric sensitivity analysis in spectroscopic ellipsometry modeling

Spectroscopic ellipsometry and some other optical metrology techniques, such as reflectometry and scatterometry, are model-based optical measurements and, therefore, require appropriate modeling to determine the geometric and material properties of substrates, thin films, and multilayer structures. Parametric sensitivity analysis (SA) provides essential assistance in the model-building process to quantify the relative importance of model parameters for model output and to identify those with high/little influence. SA can be performed in a variety of ways, and this article discusses an application of the Morris or elementary effect (EE) method, a screening type SA procedure, to spectroscopic ellipsometry modeling. The method is a global SA technique and uses a stepping of m parameters along certain so-called “trajectories” or sequences of points in parameter space, randomly constructed in order to maximally fill the volume of the m-dimensional parameter space. However, it was thought that the EE method relies greatly on a sampling strategy or a way of selecting “optimized trajectories” in the parameter space, i.e., a necessary number of trajectories chosen to be well spread over the space to properly cover the entire realistic ranges of all input factors. Here, we use two sampling methods for selecting trajectories with possibly different distributions and investigate their effects on the estimation of various sensitivity measures in spectroscopic ellipsometry data modeling. The SA results indicate that the performance of the sampling strategy should not be judged only by maximizing the trajectory spread but also include some additional convergence criteria for the sensitivity measures.

A novel equal frequency sampling of factor levels (EFSFL) method is applied to identify the dominant factor inducing the combined toxicities of 13 factors

The research framework combining global sensitivity analysis (GSA) with quantitative high-throughput screening (qHTS), called GSA-qHTS, provides a potentially feasible way to screen for important factors that induce toxicities of complex mixtures. Despite its value, the mixture samples designed using the GSA-qHTS technique still have a shortage of unequal factor levels, which leads to an asymmetry in the importance of elementary effects (EEs). In this study, we developed a novel method for mixture design that enables equal frequency sampling of factor levels (called EFSFL) by optimizing both the trajectory number and the design and expansion of the starting points for the trajectory. The EFSFL has been successfully employed to design 168 mixtures of 13 factors (12 chemicals and time) that each have three levels. By means of high-throughput microplate toxicity analysis, the toxicity change rules of the mixtures are revealed. Based on EE analysis, the important factors affecting the toxicities of the mixtures are screened. It was found that erythromycin is the dominant factor and time is an important non-chemical factor in mixture toxicities. The mixtures can be classified into types A, B, and C mixtures according to their toxicities at 12 h, and all the types B and C mixtures contain erythromycin at the maximum concentration. The toxicities of the type B mixtures increase firstly over time (0.25 ∼ 9 h) and then decrease (12 h), while those of the type C mixtures consistently increase over time. Some type A mixtures produce stimulation that increases with time. With the present new approach to mixture design, the frequency of factor levels in mixture samples is equal. Consequently, the accuracy of screening important factors is improved based on the EE method, providing a new method for the study of mixture toxicity.

New conceptualization and quantification method of first-flush in urban catchments: A modelling study

A major challenge for runoff pollution control lies in the quantification and identification of the first-flush. At present, there is a lack of reasonable theoretical methods to guide engineering practices. To remedy this deficiency, a novel method of cumulative pollutant mass vs. cumulative runoff volume (M(V)) curve simulation is proposed in this study. Subsequently, the first-flush phenomenon was redefined based on the M(V) curve simulation and demonstrate that the first-flush exists until the derivative of the simulated M(V) curve is equal to 1 (Ft′ = 1). Consequently, a mathematical model for first-flush quantification was developed. The Root-Mean-Square-Deviation (RMSD) and Pearson's Correlation Coefficient (PCC), as objective functions, were used to evaluate the performance of the model and the Elementary-Effect (EE) method was used to analyze the sensitivity of the parameters. The results indicated the satisfactory accuracy of the M(V) curve simulation and first-flush quantitative mathematical model. The NSE values exceeding 0.8 and 0.938, respectively, were obtained by analyzing 19 rainfall-runoff data for Xi'an, Shaanxi Province, China. The wash-off coefficient “r” was demonstrably the most sensitive factor influencing the model performance. Therefore, interactions between “r” and the other model parameters should be focused on to highlight the overall sensitivities. Overall, this study posits a novel paradigm shift from the traditional dimensionless definition criterion to redefine and quantify first-flush, which has significant implications for urban water environment management.

Robust morris screening method (RMSM) for complex physiological models

Background and objectives: Morris screening sensitivity analysis (MSM) comes forth as the method needing the minimum number of model simulations to qualify the impact of input parameter variations on outputs of complex, nonlinear and overparametrized models. However, the reliability of MSM indices (mean and standard deviation) and the reproducibility of their results are rarely explored despite the input parameter tuning/identification needs. In fact, these models, such those used in medical applications as digital twins, often lie in this category and need efficient and robust tools to assess both sensitivity and reliability of the outputs to numerous input model parameters.Methods: In this study, a new Robust Morris Screening Method (RMSM) is proposed and based on new indices: the absolute median (χ*) and the median absolute deviation (ρ). The proposed RMSM approach is evaluated on a complex multi-scales neuromuscular electrophysiological model simulating HD-sEMG (high density surface electromyography) signals at the skin surface. The reliability and stability of new RMSM indicators are evaluated at different trajectories within the parameter space and compared to classical MSM results. For this purpose, We propose a new methodology for parameter screening based on the ratio ρ/χ* as a graphic indicator of (non)linearity and (non)monotonicity of parameter effects.Results: Firstly, the results demonstrated that the computed elementary effects (EE) of inputs are not normally distributed using MSM indices contrary to the proposed RMSM indices. Secondly, the ranking stability of RMSM indices was earlier obtained from 20 trajectories (T=20), while MSM ranking remained unstable until T = 100. Thirdly, The screening separation between influential and negligible input model parameters was more distinct and interpretable with RMSM than MSM.Conclusion: The proposed RMSM approach ensures a fast, reliable and stable ranking of parameters for complex and overparametrized models compared to classical MSM. this allows a more precise exploration of the model parameter influence space for future application in parameter tuning and identification.

Sensitivity Analysis of ReaxFF Potential: The Case of Si/O System.

Sensitivity analysis of the ReaxFF potential for two Si and SiO2 systems has been carried out using the method of Morris. The goal is to identify the most important force field parameters for selected properties. Thus, a clearer physical interpretation for some of the parameters can be obtained while the ranking of the parameters per magnitude of sensitivity for each property facilitates the development of parametrizations with improved quality. The parameters related to the σ-bond and van der Waals interactions have the greatest influence on the properties of the cubic diamond Si phase. Counterintuitively, parameters which have an impact on mechanical properties, such as the ones for ππ-bonds, do not influence pressure and vice versa. For the β-cristobalite SiO2 phase, the Si-O cross-interaction parameters have stronger effect on all properties than the Si-Si and O-O elemental ones. This dependence is attributed to the tetrahedral structure of the SiO2 phase. Regarding the sensitivity measures, the mean of the absolute values of the elementary effect (EE) distribution correlates with the standard deviation of the EE distribution in all cases; it quantifies the sensitivity of a property on a parameter and the cross-correlation of the parameter with the other ones. The mean of the EE distribution has a poor performance and its use should be avoided. It is also argued that performing a sensitivity analysis before force field optimization might greatly enhance the efficiency of the optimization methods by considering a two-step procedure, where the most relevant parameters are optimized first.

A mechanistic effect modeling approach to the prioritization of hidden drivers in chemical cocktails

Exposure to a single chemical does not exist in reality. Mixtures, which are the ecological norm, are often characterized by numerous intrinsic driving factors with unknown combined effects. Interactions between heterogeneous chemicals, or chemical and nonchemical stressors, could alter their toxicity traits relative to single exposure. Hence, revealing the hidden environmental effects affecting multiple stressor interactions is essential to expand our knowledge about uncertainty sources in chemical risk-based decision contexts. Global sensitivity analysis (GSA) techniques involving Morris method sampling and elementary effects (EE) sensitivity analysis was applied to investigate the driving factors underlying the combined effects on Scenedesmus obliquus, and identify the mode of interaction in mixtures at environmentally-relevant concentrations. One hundred mixed-exposure formulas were generated with 9 variables (8 chemicals and temperature) via the Morris method, representing environmental perspective in the field. Subsequently, EE sensitivity analysis combined with quantitative high-throughput screening (q-HTS) was adopted to identify the most critical mixture and its primary drivers. Combined exposure exerted significantly increased effects on S. obliquus compared to the effects of individual exposure. The critical drivers were identified and validated by the control variate method. For the mode of combined action, mixture toxicity did not match the additivity relationship, and a strong interaction existed among chemicals. Collectively, the data provides evidence that a combination of specific pesticides and emerging brominated flame retardants can produce comparable, or even stronger, bionegative effects than pure chemicals due to complicated interactions. The method used offers direct comparison of multifarious factors in a unified standard scale, bridges the actual interaction scenarios in the field to toxicity simulations in the laboratory, and fill a gap in ecotoxicology.

Early-design sensitivity of radiant cooled office buildings in the tropics for building performance

The research community has worked on improving the design, performance, and operation of radiant system technologies; however, the interactions between architectural design decisions and radiant cooling systems design in the tropics – where radiant system design and operation represents a challenge due to high external heat gains and latent loads – have been left unexplored by existing work. This manuscript (1) addresses the holistic early-design of effective radiant cooling systems and the architectural environments that they serve in hot and humid tropical climates, (2) analyses how different design choices and their interaction affect buildings performance compared to conventional all-air systems, and (3) proposes preliminary early-design indications of the parameters values to adopt when designing radiant cooled office buildings in the tropics to achieve high multi-objective design goals. Morris sensitivity analysis is used to analyse the impact of 13 different building and system design parameters on comfort, system and energy performances for 3 radiant systems typologies applied to a parametric office building in the tropical climate of Singapore. The elementary effect (EE) of each parameter is analysed based on 8200 individual EnergyPlus simulation models being run, and the impact, interactions & (non) linear effects of different design factors are assessed. It is found that radiant cooling systems require different early-design phase considerations than conventional air-based cooling systems since different parameters prove to be dominant or significant. All three radiant system typologies, if coupled with efficient operational strategies and a complimentary set of design variables, ensure high multi-objective building performance, showing how the implementation of radiant cooling technologies could be beneficial for a tropical climate. The window solar heat gain coefficient and the façade window-to-wall ratio play a key role in this effort with respective values falling below or equal to 54% and 57%. The system set-point temperature has an increased importance for operational efficiency allowing high dry-bulb room temperatures ranging from 24.8 °C to 27 °C.

ELM-Based AFL–SLFN Modeling and Multiscale Model-Modification Strategy for Online Prediction

Online prediction of key parameters (e.g., process indices) is essential in many industrial processes because online measurement is not available. Data-based modeling is widely used for parameter prediction. However, model mismatch usually occurs owing to the variation of the feed properties, which changes the process dynamics. The current neural network online prediction models usually use fixed activation functions, and it is not easy to perform dynamic modification. Therefore, a few methods are proposed here. Firstly, an extreme learning machine (ELM)-based single-layer feedforward neural network with activation-function learning (AFL–SLFN) is proposed. The activation functions of the ELM are adjusted to enhance the ELM network structure and accuracy. Then, a hybrid model with adaptive weights is established by using the AFL–SLFN as a sub-model, which improves the prediction accuracy. To track the process dynamics and maintain the generalization ability of the model, a multiscale model-modification strategy is proposed. Here, small-, medium-, and large-scale modification is performed in accordance with the degree and the causes of the decrease in model accuracy. In the small-scale modification, an improved just-in-time local modeling method is used to update the parameters of the hybrid model. In the medium-scale modification, an improved elementary effect (EE)-based Morris pruning method is proposed for optimizing the sub-model structure. Remodeling is adopted in the large-scale modification. Finally, a simulation using industrial process data for tailings grade prediction in a flotation process reveals that the proposed method has better performance than some state-of-the-art methods. The proposed method can achieve rapid online training and allows optimization of the model parameters and structure for improving the model accuracy.

Global sensitivity analysis of uncertain parameters based on 2D modeling of solid oxide fuel cell

In this study, we propose an enhanced quasi–two-dimensional and nonisothermal model for solid oxide fuel cell (SOFC) parametric simulation and optimization. The dependence of effective properties on microstructural parameters is fully considered in this model. Besides, an elementary effect (EE) approach based on Monte Carlo experiments is adopted to comprehensively evaluate the sensitivity of totally 24 parameters. A two sample Kolmogorov-Smirnov (K-S) test is carried out to evaluate the ability of EE method for robust and accurate sensitivity analysis. The investigation focuses on the important microstructural parameters of the composite anode/cathode function layers (AFL/CFL). With this research, relative volume fractions of conducting materials in the AFL/CFL are the most sensitive factors among all input parameters while particle radius is found to be the least sensitive microstructural parameters. The particle size ratio of electronic particles to ionic particles is found to be much more sensitive than particle size due to its significant effect on effective conductivity. The cathodic electrochemical parameters reflect cell performance more significantly than the anodic ones. To further elucidate the role of input factors, this study provides a principle for parametric sensitivity classification as well. Besides, impacts of current density variation on parametric sensitivity are comprehensively considered. Negative or positive effects of parameters on cell performance and their influencing mechanisms are also discussed. Furthermore, global sensitivity analysis for single parameters at different positions is performed to assess the probability of structural optimization along the channel length. Then, a feasible nonuniform distribution method in allusion to function layers is proposed for further improvement of cell performance according to global SA results of single parameters along the channel direction.

Three-Objective Optimization of a Centrifugal Pump to Reduce Flow Recirculation and Cavitation

This work presents a three-objective design optimization of a centrifugal pump impeller to reduce flow recirculation and cavitation using three-dimensional (3D) Reynolds-averaged Navier–Stokes equations. A cavitation model was used to simulate the multiphase cavitating flow inside the centrifugal pump. The numerical results were validated by comparing them with experimental data for the total head coefficient and critical cavitation number. To achieve the optimization goals, blockage at 50% of the design flow rate, hydraulic efficiency at the design flow rate, and critical cavitation number for a head-drop of 3% at 125% of the design flow rate were selected as the objective functions. Based on the results of the elementary effect (EE) method, the design variables selected were the axial length of the blade, the control point for the meridional profile of the shroud, the inlet radius of the blade hub, and the incidence angle of tip of the blade. Kriging models were constructed to approximate the objective functions in the design space using the objective function values calculated at the design points selected by Latin hypercube sampling (LHS). Pareto-optimal solutions were obtained using a multi-objective genetic algorithm (MOGA). Six representative Pareto-optimal designs (POD) were analyzed to evaluate the optimization results. The PODs showed large improvements in the objective functions compared to the baseline design. Thus, both the hydraulic performance and the reliability of the centrifugal pump were improved by the optimization.

Sensitivity analysis of uncertain parameters based on an improved proton exchange membrane fuel cell analytical model

In this study, an enhanced non-isothermal, two-phase 1D analytical proton exchange membrane fuel cell (PEMFC) model is developed, which not only considers the water saturation jump, but also proposes a novel method to analytically solve the water phase changes and couple the liquid and vapor transport together. A stringent model validation procedure is used to show good agreement between the simulated results and the experimental data, taking advantage of the “three-step” and “multi-case” validation methods. It is revealed that the uncertain parameters may deteriorate model reliability and credibility, thus demonstrating the necessity to conduct sensitivity analysis. A multi-parametric screening method i.e. the elementary effect (EE) method based on Monte Carlo experiments is implemented to comprehensively analyze the total 22 uncertain parameters (including geometric, physical and electrochemical parameters), which are finally classified into very sensitive ones, rather sensitive ones and insensitive ones. The cathodic parameters are found more sensitive than the anodic ones, and the parameters of different components may have distinct sensitivity. Besides, whether the effect of each parameter is positive or negative on cell performance is also discussed. Furthermore, three cases with different groups of parameters are presented, which show almost the same polarization curve, and the two-sample Kolmogorov-Smirnov (KS) test is applied to verify the stability difference. It is concluded that those uncertain parameters not only influence the cell performance but also affect the model stability, and hence the effects of varying operating conditions should be taken into account in validation work.

Global sensitivity analysis as a support for the generation of simplified building stock energy models

Buildings are responsible for 40% of total final energy consumptions in Europe. Numerous bottom-up models were recently developed to support local authorities in assessing the energy consumption of large building stocks and reduction potentials. However, current models rarely consider uncertainty associated to building usage and characteristics within the stock, resulting in potentially biased results.This study presents a generic model simplification approach using uncertainty propagation and stochastic sensitivity analysis to derive fast simplified (surrogate) models to estimate the current building stock energy use for improved urban planning. The methodology includes an engineering-based energy model as input to global sensitivity analysis (GSA) using the elementary effects (EE) screening and Sobol’ method for key parameters identification and regression analysis to derive simplified models for entire building stocks.The application to the housing stock of Esch-sur-Alzette (Luxembourg) showed that the parameters explaining most of the variability in final energy use for heating and domestic hot water are floor area, set-point temperature, external walls U-values, windows and heating system type. Results of the simplified models were validated against measured data and confirmed the validity of the approach for a simple yet robust assessment of the building stock energy use considering uncertainty and variability.

Parameter sensitivity analysis of a mechanistic model to simulate microalgae growth

In this paper, sensitivity analysis is applied to a mechanistic model developed to simulate microalgae growth. The Morris method of Elementary Effects (EEs) is applied to evaluate the sensitivity of model outputs with respect to a subset of key input parameters. For an easier interpretation, results were plotted as distributions of elementary effects means and standard deviations for each input parameter. The model outputs were very sensitive with respect to the maximum specific growth rate of microalgae (μalg). Results of the sensitivity analysis indicate that the transfer of ammonia (Ka,NH3) and carbon dioxide (Ka,CO2) have a non-linear relation with nitrogen uptake and carbonate concentrations, respectively. This analysis helped identify the parameters with the greatest impact on simulation outputs. The results indicated that maximum specific growth rate of microalgae (μalg) was the most critical parameter to calibrate properly.

Sensitivity analysis of machining accuracy of multi-axis machine tool based on POE screw theory and Morris method

The geometric errors have a significant effect on the machining accuracy of multi-axis machine tool. Because of their complex inter-coupling, the process to control these geometric errors and then to improve the machining accuracy on this basis is recognized as a difficult problem. This paper proposes a method based on the product of exponential (POE) screw theory and Morris approach for volumetric machining accuracy global sensitivity analysis of a machine tool. When a five-axis machine tool is chosen as an example, there are five screws to represent the six basic error components of each axis (in an original way) according to the geometric definition of the errors and screws. This type of POE model is precise and succinct enough to express the relation of each of the components as the Morris method is based on the elementary effect (EE). The method can compare incidence of these errors and be used to describe the nonlinear relationship by less calculated amount in a global system. Based on the POE modelling, the Morris method is adopted to identify the key geometric errors which have a greater influence on the machining accuracy by global sensitivity analysis. Finally, according to the results obtained from analysis, suggestions, and guidelines are provided to adjust and modify the machine tool components to improve the machining accuracy economically.

Sensitivity Analysis and Validation for Numerical Simulation of Water Infiltration into Unsaturated Soil.

A FORTRAN code for liquid water flow in unsaturated soil under the isothermal condition was developed to simulate water infiltration into Yolo light clay. The governing equation, that is, Richards' equation, was approximated by the finite-difference method. A normalized sensitivity coefficient was used in the sensitivity analysis of Richards' equation. Normalized sensitivity coefficient was calculated using one-at-a-time (OAT) method and elementary effects (EE) method based on hydraulic functions for matric suction and hydraulic conductivity. Results from EE method provided additional insight into model input parameters, such as input parameter linearity and oscillating sign effect. Boundary volumetric water content (θ L (upper bound)) and saturated volumetric water content (θ s) were consistently found to be the most sensitive parameters corresponding to positive and negative relations, as given by the hydraulic functions. In addition, although initial volumetric water content (θ L (initial cond)) and time-step size (Δt), respectively, possessed a great amount of sensitivity coefficient and uncertainty value, they did not exhibit significant influence on model output as demonstrated by spatial discretization size (Δz). The input multiplication of parameters sensitivity coefficient and uncertainty value was found to affect the outcome of model simulation, in which parameter with the highest value was found to be Δz.

A multi-criteria trajectory-based parameter sampling strategy for the screening method of elementary effects

Environmental models are inherently complex and often characterized by high dimensionality. The method of elementary effects (EE) is one of the most widely used parameter screening technique implemented to reduce burden on computational resources required for thorough model evaluation. Due to issues like inefficient screening and excessive sampling time, the development of more effective EE sampling strategies has been a recent research focus. This paper presents a new sampling strategy - Sampling for Uniformity (SU) – based on the principles of meeting close-to-theoretical parameter distributions and maximizing trajectory spread. The performance of the SU relative to existing strategies was evaluated using a number of criteria including generated parameter distributions' uniformity, time efficiency, trajectory spread, and screening efficiency. The SU performed better than some trajectory-based benchmark strategies across the evaluation criteria, underlining the effectiveness of multi-criteria based sampling and the need to focus future efforts on exploring other combinations of sampling criteria.

An Efficient Screening Method for Computer Experiments

Computer simulators of real-world processes are often computationally expensive and require many inputs. The problem of the computational expense can be handled using emulation technology; however, highly multidimensional input spaces may require more simulator runs to train and validate the emulator. We aim to reduce the dimensionality of the problem by screening the simulator’s inputs for nonlinear effects on the output rather than distinguishing between negligible and active effects. Our proposed method is built upon the elementary effects (EE) method for screening and uses a threshold value to separate the inputs with linear and nonlinear effects. The technique is simple to implement and acts in a sequential way to keep the number of simulator runs down to a minimum, while identifying the inputs that have nonlinear effects. The algorithm is applied on a set of simulated examples and a rabies disease simulator where we observe run savings ranging between 28% and 63% compared with the batch EE method. Supplementary materials for this article are available online.

An Efficient Sensitivity Analysis Approach for Computationally Expensive Microscopic Traffic Simulation Models

Microscopic traffic simulators are useful tools for designing, evaluating and optimizing transportation systems. In order for a simulator to accurately describe reality, the corresponding traffic model must be properly calibrated. However, the calibration can be rather difficult when the model is computationally expensive and has many parameters. To overcome these difficulties, Sensitivity Analysis (SA) can be applied as an essential instrument for supporting model calibration. Through SA the practitioners can obtain valuable information about the relationship between model inputs and outputs, and hence focus on the proper set of most influential parameters for the calibration. Notice, however, that many quantitative SA techniques may also fail with computationally expensive models. To address the above issues, in this paper we developed the quasi-OTEE method, an efficient and qualitative SA approach based on the Elementary Effects (EE) method. The quasi-OTEE approach is able to screen the most influential parameters of a complex model through computing and comparing their Sensitivity Indexes. With the improved sampling strategy, this approach is much more efficient than the original EE method. The approach is validated through a numerical analysis, and a case study of a small synthetic network in Aimsun. A more detailed case study of the Zurich network in VISSIM is then provided to illustrate its application. The results demonstrate that the proposed approach is an effective SA tool for the computationally expensive microscopic traffic models, as well as other complex models in the general scientific community.

Scheduling of Changes in Complex Engineering Design Process via Genetic Algorithm and Elementary Effects Method

Engineering design changes constantly occur in a complex engineering design process. Designers have to put an appropriate procedure in place to handle these changes in order to realize successful product development in a timely and cost-effective manner. When many change propagation paths are present, selection of the best change evolution paths and distribution of change results to downstream tasks become critical to the progress management of the project. In this paper, based on the available change propagation simulation algorithm, a global sensitivity analysis method known as elementary effects (EE) is employed to rank the importance of each potential propagation path with those involved design dependencies in the process. Further, an EE-based heuristic design dependency encoding method is applied to the genetic algorithm which is then adopted to schedule the change updating process. Finally, the optimal results obtained by the complete search and the heuristic dependency encoding methods are compared to illustrate the improvements and effectiveness of the latter method.