Total-order Sensitivity Indices Research Articles

Total effects with constrained features

Recent studies have emphasized the connection between machine learning feature importance measures and total order sensitivity indices (total effects, henceforth). Feature correlations and the need to avoid unrestricted permutations make the estimation of these indices challenging. Additionally, there is no established theory or approach for non-Cartesian domains. We propose four alternative strategies for computing total effects that account for both dependent and constrained features. Our first approach involves a generalized winding stairs design combined with the Knothe-Rosenblatt transformation. This approach, while applicable to a wide family of input dependencies, becomes impractical when inputs are physically constrained. Our second approach is a U-statistic that combines the Jansen estimator with a weighting factor. The U-statistic framework allows the derivation of a central limit theorem for this estimator. However, this design is computationally intensive. Then, our third approach uses derangements to significantly reduce computational burden. We prove consistency and central limit theorems for these estimators as well. Our fourth approach is based on a nearest-neighbour intuition and it further reduces computational burden. We test these estimators through a series of increasingly complex computational experiments with features constrained on compact and connected domains (circle, simplex), non-compact and non-connected domains (Sierpinski gaskets), we provide comparisons with machine learning approaches and conclude with an application to a realistic simulator.

Variance-based global sensitivity analysis and uncertainty quantification of the AquaCrop model parameters for basil (Ocimum basilicum L.) under different nitrogen fertilizer rates

Dynamic crop models are powerful tools for evaluating crop responses to alternative management options. These models consist of various input parameters that need calibration before being implemented in a new environment. Sensitivity analysis (SA) has proven to be an efficient method for identifying dominant model parameters. This study aimed to determine influential parameters of the AquaCrop model for basil and also quantify output uncertainty under different nitrogen fertilizer rates. The Sobol’ SA method was employed to compute both first-order and total-order sensitivity indices. Results indicated that normalized water productivity (WP*) and crop coefficient (KcTr,x) were the most influential parameters across all treatments. Furthermore, as fertility stress increased, the interaction effects among parameters decreased. The temporal analysis of model output (biomass) revealed that parameter sensitivities were highly time-dependent. In addition, the uncertainty of predicted biomass was determined using combined violin and box plots. Results demonstrated that the optimal nitrogen fertilizer application rate extended the distribution of model output. In summary, these findings offer valuable insights for model simplification and effective decision-making.

Study on flow and heat transfer characteristics in rectangular channels with lantern-shaped pin fin array: Part II—Global sensitivity analysis and optimization

Increasing gas turbine inlet temperatures are placing more stringent requirements on the efficiency of internal cooling. Pin fin arrays are a common tool for cooling electronic components or trailing edge cooling channels of gas turbine blades. A crucial concern for researchers is how to achieve a higher Nusselt number with constrained pressure loss to guarantee that the efficiency of the gas turbine will not be impacted by the power dissipation generated by the driving flow passing through the internal cooling loops. In Part I of this study, a lantern-shaped pin is proposed. To seek a more efficient design solution for the pin fin shape, the second part combines the surrogate model, Sobol’s method, and NSGA-II optimization method to perform the overall and spatial sensitivity analysis and optimization for the shape variables on the generatrix of different pin rows. The first-order and total-order sensitivity indices of different variables to Nu T and f are calculated using the surrogate model-based Sobol’s method. The variables that significantly affect the channel performance are identified. The overall and spatial sensitivity analysis results indicate that the variable X 3 i -1 contributes the most to Nu T and f, with total-order sensitivity indices of 70.59–64.26 and 58.16–59.09%, respectively. The overall optimization increases the Nu T and f by 6.95 and 9.91%, respectively, while the spatial optimization leads to an increase in the Nu T by 2.29% and a decrease in the f by 4.56%. Correspondingly, the two optimizations resulted in an increase in the TPI by 3.61 and 3.89%, respectively. The Pareto front solution obtained by the spatial optimization has a more concentrated distribution. Although the TPI obtained by the spatial optimization is only 0.21% higher than that of the overall optimization, it provides designers with a strategy considering the Nusselt number and pressure loss.

Variance-based global sensitivity analysis of a multi-population, single-chamber microbial fuel cell operating in continuous flow mode at steady state

Microbial fuel cells (MFCs) are environmentally friendly devices which are used to convert chemical energy in organic wastes to electrical energy. MFCs have a strong non-linearity that requires a very sophisticated controlling system. Consequently, this makes optimization and performance study of MFCs a difficult process. For better estimation of the constants used for optimization of MFCs, global sensitivity analysis is performed. The global sensitivity method based on Sobol’s indices coupled with Monte Carlo simulations was applied on multi-population, single-chamber MFC operating in a continuous flow at steady state for the first time. In this paper, first-order and total-order sensitivity indices were used to visualize the impacts associated with six main parameters resulted from the maximization of power density using Matlab. Such parameters are maximum anodophilic-specific growth rate, half-rate constant of anodophilics, curve steepness factor, mediator half-rate constant, number of electrons transferred per mole mediator and decay rate constant of anodophilic bacteria. The results showed that the curve steepness factor has almost no impact on the power density of MFC. While all other studied, factors are sensitive parameters that impact the power density of MFC. It is worth mentioning that maximum anodophilic growth rate and the number of electrons transferred per mole of mediator are the most sensitive parameters that affecting the power density production having total indices of 0.74 and 0.624, respectively. While the half-rate constant of anodophilics, mediator half-rate constant and decay rate constant of anodophilics have almost similar impact by having total-order indices of 0.127, 0.144 and 0.192, respectively. The findings herein are critical in understanding and further model improvement of microbial fuel cells as the most impacting parameters on MFC power density can be optimized further to reduce uncertainty associated with the experimental parameters in the model.Graphical abstract

How Meaningful is the Elite Quality Index Ranking?

The Elite Quality Index (EQx) attempts to measure the propensity of elites—on aggregate—to create value, rather than to rent seek. The index has attracted worldwide media and press attention. In their articles, journalists have based their analyses primarily on their own countries’ position in the EQx ranking. But how meaningful is the EQx ranking? How do the uncertainties underlying some of the assumptions made in the index propagate to the country rankings? We conduct a global uncertainty and sensitivity analysis (UA and SA) of the EQx and compute Sobol’ first and total order sensitivity indices using state of the art estimators, in order to scrutinise the implications of index assumptions and assess the reliability of the EQx ranking. The UA suggests that the EQx ranking of 2021 (EQx2021) is largely stable for the top 50 countries, but exhibits considerable uncertainties especially for middle and lower performing countries. The SA highlights the handling of missing data, the normalisation process and the weighting scheme as most important methodological choices, while the largest potential for improvement is observed in how raw missing indicator data is handled.

Global Sensitivity Analysis of Metabolic Models for Phosphorus Accumulating Organisms in Enhanced Biological Phosphorus Removal

The aim of this study was to identify, quantify and prioritize for the first time the sources of uncertainty in a mechanistic model describing the anaerobic-aerobic metabolism of phosphorus accumulating organisms (PAO) in enhanced biological phosphorus removal (EBPR) systems. These wastewater treatment systems play an important role in preventing eutrophication and metabolic models provide an advanced tool for improving their stability via system design, monitoring and prediction. To this end, a global sensitivity analysis was conducted using standard regression coefficients and Sobol sensitivity indices, taking into account the effect of 39 input parameters on 10 output variables. Input uncertainty was characterized with data in the literature and propagated to the output using the Monte Carlo method. The low degree of linearity between input parameters and model outputs showed that model simplification by linearization can be pursued only in very well defined circumstances. Differences between first and total-order sensitivity indices showed that variance in model predictions was due to interactions between combinations of inputs, as opposed to the direct effect of individual inputs. The major sources of uncertainty affecting the prediction of liquid phase concentrations, as well as intra-cellular glycogen and poly-phosphate was due to 64% of the input parameters. In contrast, the contribution to variance in intra-cellular PHA constituents was uniformly distributed among all inputs. In addition to the intra-cellular biomass constituents, notably PHB, PH2MV and glycogen, uncertainty with respect to input parameters directly related to anaerobic propionate uptake, aerobic poly-phosphate formation, glycogen formation and temperature contributed most to the variance of all model outputs. Based on the distribution of total-order sensitivities, characterization of the influent stream and intra-cellular fractions of PHA can be expected to significantly improve model reliability. The variance of EBPR metabolic model predictions was quantified. The means to account for this variance, with respect to each quantity of interest, given knowledge of the corresponding input uncertainties, was prescribed. On this basis, possible avenues and pre-requisite requirements to simplify EBPR metabolic models for PAO, both structurally via linearization, as well as by reduction of the number of non-influential variables were outlined.

Global sensitivity analysis approaches applied to parameter selection for numerical model-updating of structures

PurposeThis paper aims to measure the sensitivity of the structure’s deformation numerical model (NM) related to the various types of the design parameters, which is a suitable method for parameter selection to increase the time of model-updating.Design/methodology/approachIn this research, a variance-based sensitivity analysis (VBSA) approach is proposed to measure the sensitivity of NM of structures. In this way, the contribution of measurements of the structure (such as design parameter values and geometry) on the output of NM is studied using first-order and total-order sensitivity indices developed by Sobol’. In this way the generated data set of parameters by considering different distributions such as Gaussian or uniform distribution and different order as input along with, the resulted deformation variables of NM as output has been submitted to the Sobol’ indices estimation procedure. To the verification of VBSA results, a gradient-based sensitivity analysis (SA), which is developed as a global SA method has been developed to measure the global sensitivity of NM then implemented over the NM’s results of a tunnel.FindingsRegarding the estimated indices, it has been concluded that the derived deformation functions from the tunnel’s NM usually are non-additive. Also, some parameters have been determined as most effective on the deformation functions, which can be selected for model-updating to avoid a time-consuming process, so those may better to be considered in the group of updating parameters. In this procedure for SA of the model, also some interactions between the selected parameters with other parameters, which are beneficial to be considered in the model-updating procedure, have been detected. In this study, some parameters approximately (27 per cent of the total) with no effect over the all objective functions have been determined to be excluded from the parameter candidates for model-updating. Also, the resulted indices of implemented VBSA were approved during validation by the gradient-based indices.Practical implicationsThe introduced method has been implemented for a circular lined tunnel’s NM, which has been created by Fast Lagrangian Analysis of Continua software.Originality/valueThis paper plans to apply a statistical method, which is global on the results of the NM of a soil structure by a complex system for parameter selection to avoid the time-consuming model-updating process.

Global sensitivity analysis of n-butanol reaction kinetics using rate rules

We investigate the sensitivity of the ignition delay time, τign, of n-butanol to uncertainties in the rate-rule kinetic rate parameters, at various initial temperatures (600–1000 K), pressures (10–80 bar) and equivalence ratios (0.5–2.0). We start by considering a 30-dimensional stochastic germ in which each random variable is associated with one reaction class, and build a surrogate model for the ignition delay time using polynomial chaos expansions. The adaptive pseudo-spectral projection technique is used for this purpose. The surrogate model is used to estimate first-order and total-order sensitivity indices characterizing the dependence of the ignition delay time on the uncertain inputs. Results indicate that τign is mostly sensitive to variations in four dominant reaction classes, namely, H–atom abstraction from the fuel (reaction class 2), addition of O2 to the fuel radicals (reaction class 11), fuel radical isomerization including Waddington type reactions (reaction class 15), and concerted elimination reactions (reaction class 16). We consequently focus our attention on these four reaction classes, and consider variations within corresponding subrules. We explore two approaches to define the subrules of reaction class 2, one based on the radical abstracting from the fuel resulting in eleven subrules and another based on the abstraction site resulting in five subrules. Hence, we investigate the sensitivity of τign due to variability in the rate parameters of 26 and 20 subrules of the resulting models. In particular, the simulations indicate that in reaction class 2 H–atom abstraction by HO2 dominates the variability in τign at all initial conditions considered. Analysis of this finding reveals that correlations inherent in the rate rule construction plays an important role in the resulting sensitivity predictions, and suggests a hierarchical approach to the calibration of elementary reaction rates.

Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics

Abstract Reliable models for structural materials are a vital issue. Hierarchical multiscale models for concrete, incorporating microstructural properties of the material, have been developed to predict the mechanical behavior. The present paper aims to study the major sources of uncertainty in the framework of multiscale modeling based on continuum micromechanics. The effect of uncertainy in input parameters on the Young's modulus of cement-based materials is studied systematically by means of a probabilistic multiscale modeling approach covering three length scales. Sensitivity and uncertainty analyses are adapted to assess the stochastic predictions of the multiscale model. The total order sensitivity indices are computed using the variance-based sensitivity analysis. The total uncertainty according to the adjustment factor approach, comprising the parameter uncertainty and the model uncertainty, is quantified, aiming at the comparison of two commonly utilized hydration models. Application of the probabilistic methodology reveals that the uncertainties of the model responses increase significantly during the upscaling process. Hence, considering the stochastic variability of the predicted mechanical properties is of utmost importance for the assessment and evaluation of the model responses.

Global sensitivity analysis in a mathematical model of the renal insterstitium

The pressure in the renal interstitium is an important factor for normal kidney function. Here we develop a computational model of the rat kidney and use it to investigate the relationship between arterial blood pressure and interstitial fluid pressure. In addition, we investigate how tissue flexibility influences this relationship. Due to the complexity of the model, the large number of parameters, and the inherent uncertainty of the experimental data, we utilize Monte Carlo sampling to study the model’s behavior under a wide range of parameter values and to compute first- and total-order sensitivity indices. Characteristically, at elevated arterial blood pressure, the model predicts cases with increased or reduced interstitial pressure. The transition between the two cases is controlled mostly by the compliance of the blood vessels located before the afferent arterioles.

Sensitivity of Wind Farm Output to Wind Conditions, Land Configuration, and Installed Capacity, Under Different Wake Models

In conventional wind farm design and optimization, analytical wake models are generally used to estimate the wake-induced power losses. Different wake models often yield significantly dissimilar estimates of wake velocity deficit and wake width. In this context, the wake behavior, as well as the subsequent wind farm power generation, can be expressed as functions of a series of key factors. A quantitative understanding of the relative impact of each of these key factors, particularly under the application of different wake models, is paramount to reliable quantification of wind farm power generation. Such an understanding is however not readily evident in the current state of the art in wind farm design. To fill this important gap, this paper develops a comprehensive sensitivity analysis (SA) of wind farm performance with respect to the key natural and design factors. Specifically, the sensitivities of the estimated wind farm power generation and maximum farm output potential are investigated with respect to the following key factors: (i) incoming wind speed, (ii) ambient turbulence, (iii) land area per MW installed, (iv) land aspect ratio, and (v) nameplate capacity. The extended Fourier amplitude sensitivity test (e-FAST), which helpfully provides a measure of both first-order and total-order sensitivity indices, is used for this purpose. The impact of using four different analytical wake models (i.e., Jensen, Frandsen, Larsen, and Ishihara models) on the wind farm SA is also explored. By applying this new SA framework, it was observed that, when the incoming wind speed is below the turbine rated speed, the impact of incoming wind speed on the wind farm power generation is dominant, irrespective of the choice of wake models. Interestingly, for array-like wind farms, the relative importance of each input parameter was found to vary significantly with the choice of wake models, i.e., appreciable differences in the sensitivity indices (of up to 70%) were observed across the different wake models. In contrast, for optimized wind farm layouts, the choice of wake models was observed to have marginal impact on the sensitivity indices.

Simulation of long‐term soil redistribution by tillage using a cellular automata model

AbstractThe process of tillage translocation is well studied and can be described adequately by different existing models. Nevertheless, in complex environments with numerous obstacles, such as olive orchards, the application of conventional tillage erosion models is not straightforward. However, such obstacles have important effects on the spatial pattern of soil redistribution and on resulting soil properties. Cellular automata could provide a valuable alternative here. This study aims at developing a cellular automata model for tillage translocation (CATT) that can take into account such obstacles, exploring its possibilities and limitations.Firstly, model outcome was tested on a traditional field with rolling topography, for which caesium‐137 (137Cs) inventories are available. The observed spatial soil redistribution patterns could be adequately represented by the CATT model. Secondly, a global sensitivity analysis was performed to explore the effect of input parameter uncertainty on several selected model outputs. The variance‐based extended Fourier Amplitude Sensitivity Test (FAST) method was used to determine first‐ and total‐order sensitivity indices. Tillage depth was identified as the input parameter that determined most of the output variance, followed respectively by tillage direction and speed. The high difference between the total‐ and first‐order sensitivity indices indicated that, in spite of the simple model structure, the model behaves non‐linearly with respect to some of the model output variables. Higher order interactions were especially important for determining the proportion of eroding and deposition cells.Finally, simulations were performed to analyse the model behaviour in complex landscapes, applying it to a field with protruding obstacles (representing olive trees). The model adequately represented some morphological features observed in actual olive orchards, such as mounds around the olive trees. The results show that cellular automata are an appropriate tool to describe long‐term tillage soil redistribution. Copyright © 2010 John Wiley & Sons, Ltd.

Making best use of model evaluations to compute sensitivity indices

This paper deals with computations of sensitivity indices in sensitivity analysis. Given a mathematical or computational model y= f( x 1, x 2,…, x k ), where the input factors x i 's are uncorrelated with one another, one can see y as the realization of a stochastic process obtained by sampling each of the x i from its marginal distribution. The sensitivity indices are related to the decomposition of the variance of y into terms either due to each x i taken singularly (first order indices), as well as into terms due to the cooperative effects of more than one x i . In this paper we assume that one has computed the full set of first order sensitivity indices as well as the full set of total-order sensitivity indices (a fairly common strategy in sensitivity analysis), and show that in this case the same set of model evaluations can be used to compute double estimates of: • the total effect of two factors taken together, for all such k 2 couples, where k is the dimensionality of the model; • the total effect of k−2 factors taken together, for all k 2 such ( k−2) ples. We further introduce a new strategy for the computation of the full sets of first plus total order sensitivity indices that is about 50% cheaper in terms of model evaluations with respect to previously published works. We discuss separately the case where the input factors x i 's are not independent from each other.