Parameter Uncertainty Quantification Research Articles

Research on the Application of Uncertainty Quantification (UQ) Method in High-Voltage (HV) Cable Fault Location

In HV cable fault location technology, line parameter uncertainty has an impact on the location criterion and affects the fault location result. Therefore, it is of great significance to study the uncertainty quantification of line parameters. In this paper, an impedance-based fault location criterion was used for an uncertainty study. Three kinds of uncertainty factors, namely the sheath resistivity per unit length, the equivalent grounding resistance on both sides, and the length of the cable section, were taken as random input variables without interaction. They were subject to random uniform distribution within a 50% amplitude variation. The relevant statistical information, such as the mean value, standard deviation and probability distribution, of the normal operation and fault state were calculated using the Monte Carlo simulation (MCS) method, the polynomial chaos expansion (PCE) method, and the univariate dimension reduction method (UDRM), respectively. Thus, the influence of uncertain factors on fault location was analyzed, and the calculation results of the three uncertainty quantification methods compared. The results indicate that: (1) UQ methods are effective for simulation analysis of fault locations, and UDRM has certain application prospects for HV fault location in practice; (2) the quantification results of the MCS, PCE, and UDRM were very close, while the mean convergence rate was significantly higher for the UDRM; (3) compared with the MCS, PCE, and UDRM, the PCE and UDRM had higher accuracy, and MCS and UDRM required less running time.

Data-driven uncertainty quantification in macroscopic traffic flow models

We propose a Bayesian approach for parameter uncertainty quantification in macroscopic traffic flow models from cross-sectional data. We consider both a simple first order model consisting in the mass conservation equation and its second order version including a speed evolution equation. A bias term is introduced and modeled as a Gaussian process to account for the traffic flow models limitations. We validate the results comparing the error in the macroscopic variables (flow, speed, density) for both models, showing that second order models globally perform better in reconstructing traffic quantities of interest.

Expedited Variable-Resolution Surrogate Modeling of Miniaturized Microwave Passives in Confined Domains

The design of miniaturized microwave components is largely based on computational models, primarily, full-wave electromagnetic (EM) simulations. The EM analysis is capable of giving an accurate account for cross-coupling effects, substrate and radiation losses, or interactions with environmental components (e.g., connectors). Unfortunately, direct execution of EM-based design tasks, such as parametric optimization or uncertainty quantification (UQ), may turn prohibitively expensive in computational terms. A workaround has been offered by surrogate-assisted procedures that capitalize on replacing expensive EM simulations by fast metamodels, notably data-driven ones. However, the construction of general-purpose metamodels is impeded by the curse of dimensionality as well as a limited capability of approximation techniques to represent highly nonlinear responses of microwave devices. This article proposes a novel technique that integrates the performance-driven modeling paradigm as well as variable-resolution EM simulations. The former focuses on the construction of the surrogate in the parameter space subset encompassing high-quality designs, which effectively addresses the dimensionality issues. The latter—realized through co-kriging—contributes to further computational savings by executing the majority of circuit evaluations at the level of coarse-discretization EM analysis. Verification experiments conducted for three microstrip components demonstrate the superiority of the proposed approach over existing performance-driven techniques, let alone conventional modeling procedures, both with respect to accuracy and computational cost of the surrogate construction.

Modflow-setup: Robust automation of groundwater model construction

In an age of both big data and increasing strain on water resources, sound management decisions often rely on numerical models. Numerical models provide a physics-based framework for assimilating and making sense of information that by itself only provides a limited description of the hydrologic system. Often, numerical models are the best option for quantifying even intuitively obvious connections between human activities and water resource impacts. However, despite many recent advances in model data assimilation and uncertainty quantification, the process of constructing numerical models remains laborious, expensive, and opaque, often precluding their use in decision making. Modflow-setup aims to provide rapid and consistent construction of MODFLOW groundwater models through robust and repeatable automation. Common model construction tasks are distilled in an open-source, online code base that is tested and extensible through collaborative version control. Input to Modflow-setup consists of a single configuration file that summarizes the workflow for building a model, including source data, construction options, and output packages. Source data providing model structure and parameter information including shapefiles, rasters, NetCDF files, tables, and other (geolocated) sources to MODFLOW models are read in and mapped to the model discretization, using Flopy and other general open-source scientific Python libraries. In a few minutes, an external array-based MODFLOW model amenable to parameter estimation and uncertainty quantification is produced. This paper describes the core functionality of Modflow-setup, including a worked example of a MODFLOW 6 model for evaluating pumping impacts to a lake in central Wisconsin, United States.

Uncertainties in ocean biogeochemical simulations: Application of ensemble data assimilation to a one-dimensional model

Marine biogeochemical (BGC) models are highly uncertain in their parameterization. The value of the BGC parameters are poorly known and lead to large uncertainties in the model outputs. This study focuses on the uncertainty quantification of model fields and parameters within a one-dimensional (1-D) ocean BGC model applying ensemble data assimilation. We applied an ensemble Kalman filter provided by the Parallel Data Assimilation Framework (PDAF) into a 1-D vertical configuration of the BGC model Regulated Ecosystem Model 2 (REcoM2) at two BGC time-series stations: the Bermuda Atlantic Time-series Study (BATS) and the Dynamique des Flux Atmosphériques en Méditerranée (DYFAMED). We assimilated 5-day satellite chlorophyll-a (chl-a) concentration and monthly in situ net primary production (NPP) data for 3 years to jointly estimate 10 preselected key BGC parameters and the model state. The estimated set of parameters resulted in improvements in the model prediction up to 66% for the surface chl-a and 56% for NPP. Results show that assimilating satellite chl-a concentration data alone degraded the prediction of NPP. Simultaneous assimilation of the satellite chl-a data and in situ NPP data improved both surface chl-a and NPP simulations. We found that correlations between parameters preclude estimating parameters independently. Co-dependencies between parameters also indicate that there is not a unique set of optimal parameters. Incorporation of proper uncertainty estimation in BGC predictions, therefore, requires ensemble simulations with varying parameter values.

Uncertainty Quantification and Sensitivity Analysis for the Electrical Impedance Spectroscopy of Changes to Intercellular Junctions Induced by Cold Atmospheric Plasma.

The influence of pertinent parameters of a Cole-Cole model in the impedimetric assessment of cell-monolayers was investigated with respect to the significance of their individual contribution. The analysis enables conclusions on characteristics, such as intercellular junctions. Especially cold atmospheric plasma (CAP) has been proven to influence intercellular junctions which may become a key factor in CAP-related biological effects. Therefore, the response of rat liver epithelial cells (WB-F344) and their malignant counterpart (WB-ras) was studied by electrical impedance spectroscopy (EIS). Cell monolayers before and after CAP treatment were analyzed. An uncertainty quantification (UQ) of Cole parameters revealed the frequency cut-off point between low and high frequency resistances. A sensitivity analysis (SA) showed that the Cole parameters, R0 and α were the most sensitive, while Rinf and τ were the least sensitive. The temporal development of major Cole parameters indicates that CAP induced reversible changes in intercellular junctions, but not significant changes in membrane permeability. Sustained changes of τ suggested that long-lived ROS, such as H2O2, might play an important role. The proposed analysis confirms that an inherent advantage of EIS is the real time observation for CAP-induced changes on intercellular junctions, with a label-free and in situ method manner.

Confidence intervals of the Kling-Gupta efficiency

• The confidence intervals of the KG efficiency do not have a closed-form description. • The uncertainty of KG efficiency is attributed to measurement errors of training data. • Statistical inference with an informal goodness-of-fit metric is coined diagnostic regression. • Parameter confidence regions of KG efficiency in agreement with linear regression theory. • Marginal parameter distributions of KG efficiency are approximately Gaussian. The Kling-Gupta efficiency, hereafter referred to as KG efficiency rather than its common abbreviation KGE, proposed by Gupta et al. (2009) has become a widely used metric for evaluating the goodness-of-fit of n -vectors of observations, y ̃ = [ y ̃ 1 y ̃ 2 … y ̃ n ] ⊤ , and corresponding model simulations, y θ = y 1 θ y 2 θ … y n θ ⊤ . This metric rectifies some of the shortcomings of the coefficient of determination, R 2 , also known to hydrologists as the efficiency of Nash and Sutcliffe (1970) , by using a Euclidean-distance based weighting of the correlation, bias and temporal variability of the observed, y ̃ , and simulated, y θ , data. But as the KG efficiency is not borne out of assumptions with respect to the statistical distribution of the residuals, e θ = y ̃ - y θ , we cannot formally characterize its uncertainty. The NS efficiency suffers a similar problem, yet, statistical theory postulates that its confidence intervals should follow a beta distribution in certain special cases. Without a formal description of the confidence intervals of the KG efficiency, we cannot (amongst others) quantify parameter uncertainty, compute confidence and prediction limits on simulated model responses, inform decision makers about critical modeling uncertainties, evaluate model adequacy and assess the information content of calibration data. More fundamentally, without confidence intervals we cannot establish whether the KG efficiency is a consistent, efficient and unbiased estimator. In this paper we present an empirical description of the confidence intervals of the KG efficiency. We relate the unknown probability distribution of the KG efficiency to the measurement errors of the training data record, y ̃ , and use the bootstrap method to carry out statistical inference. We illustrate our method by application to a simple linear regression function for which the least squares parameter confidence regions are exactly known and two hydrologic models of contrasting complexity. The empirical parameter confidence regions and/or intervals of the KG efficiency are compared to those derived from generalized least squares, objective function contouring and Bayesian analysis using Markov chain Monte Carlo simulation. The marginal parameter distributions of the KG efficiency are generally well described by a normal distribution. Results further confirm that the distribution of the KG efficiency is a complex function of data length and the magnitude, distribution and structure of the measurement errors. This prohibits an analytic description of the empirical confidence regions and/or intervals of the KG efficiency and reiterates the need for the bootstrap method.

Uncertainty Quantification of Input Parameters in a 2-D Finite-Element Model for Broken Rotor Bar in an Induction Machine

In this article, a forward uncertainty propagation method is presented for a 2-D finite-element (FE) model in an induction machine. This method is applied to quantify the uncertainty of input parameters, for example, dimensions and material properties, and demonstrate their variability effect on harmonics related to the broken rotor bar (BRB) faults. To show the most influential input parameters in the case of BRB harmonics, a global sensitivity analysis is performed from the polynomial chaos expansion (PCE) approximation of the FE model. The results of this study indicate that BRB harmonics are highly sensitive to stator inner diameter, rotor outer diameter, rotor bar conductivity, and core materials. Moreover, the combined variability of these sensitive input parameters can attenuate the amplitude of the BRB harmonics 30%–90% compared to the simulation results at nominal values of input parameters and closely match with measurement results.

Reducing uncertainty of karst aquifer modeling with complementary hydrological observations for the sustainable management of groundwater resources

• We propose a lumped model to simulate a strategic karst aquifer under management. • Parameter estimation is conducted on multiple surface and groundwater data types using a Monte-Carlo procedure. • Parametric uncertainty is greatly reduced when hydrological surface discharge observations are introduced in the parameter estimation process. • We estimate a tipping point between sustainable and unsustainable management as a decision key for stakeholders. A quantitative estimation of the sustainability of groundwater resources is a challenge for water supply in most regions of the world. Sustainability can be estimated through numerical simulation, but uncertainty of models somehow limits the reliability of models’ predictions. To overcome this limitation, we explore how parametric and predictive uncertainties is reduced by adding complementary groundwater and surface water dynamics data to a lumped parameter model. This is illustrated with the example of a Mediterranean karst hydrosystem pumped at high flow rate (∼1 m 3 s −1 ) to supply water to about 350,000 habitants. A lumped parameter model was set up to simulate spring discharge, water table levels and surface water discharge measured within this karst hydrosystem over a 40 year period. Then, a parameter estimation and uncertainty quantification, based on groundwater discharge, water table levels and surface water discharge observations, was conducted with a Bayesian approach. Our study illustrates how the consideration of complementary hydrological data allows reducing parametric and predictive uncertainty and improving model performance. The model is then applied to assess the impact of an increasing groundwater abstraction according to different prospective scenarios, with a focus on the storage and dynamic of flux between the different compartments of the karst hydrosystem. Based on simulation results and related predictive uncertainty, a groundwater abstraction threshold, which must not be exceeded for a sustainable management of the groundwater resource, is determined.

Sensitivity Analysis and Parametric Uncertainty Quantification of a Modular Multilevel Converter

Abstract This paper presents the numerical approaches for sensitivity analysis (SA) and its application in the modeling effort of a modular multilevel converter (MMC). A review of the state-of-the-art techniques in the sensitivity analysis is provided, with a special focus on the numerical approaches, followed by the sensitivity analysis of an MMC. To further reduce the computational cost per model evaluation in parametric uncertainty quantification (P-UQ) of the MMC, this paper also proposes a simplified model with a minimum number of power modules for P-UQ analysis without introducing any further uncertainties in the modeling and simulation.

Verification, validation, and parameter study of a computational model for corrosion pit growth adopting the level-set method. Part II: Stress corrosion

Structural components in corrosive environments such as pipelines, bridges, aircrafts, and turbines are imposed to stress corrosion. A stress corrosion model for pit growth should a) accurately consider the electrochemistry of the corrosion process, b) properly deal with the moving interface between solid and electrolyte, and c) effectively incorporates the synergism between corrosion and mechanical field at the interface. In Part II, the influence of mechanical loading is added to the approach described in Part I. Part II investigates the model’s capabilities of simulating stress corrosion via a set of numerical examples of corrosion pitting which include experimental validation and uncertainty quantification of model parameters and properties.

Model Verification and Validation of a Cable-Stayed Bridge: Interval-Based Uncertainty Quantification of the Model Parameters

Model Verification and Validation of a Cable-Stayed Bridge: Interval-Based Uncertainty Quantification of the Model Parameters

Sparse Bayesian deep learning for dynamic system identification

This paper proposes a sparse Bayesian treatment of deep neural networks (DNNs) for system identification. Although DNNs show impressive approximation ability in various fields, several challenges still exist for system identification problems. First, DNNs are known to be too complex that they can easily overfit the training data. Second, the selection of the input regressors for system identification is nontrivial. Third, uncertainty quantification of the model parameters and predictions are necessary. The proposed Bayesian approach offers a principled way to alleviate the above challenges by marginal likelihood/model evidence approximation and structured group sparsity-inducing priors construction. The identification algorithm is derived as an iterative regularised optimisation procedure that can be solved as efficiently as training typical DNNs. Remarkably, an efficient and recursive Hessian calculation method for each layer of DNNs is developed, turning the intractable training/optimisation process into a tractable one. Furthermore, a practical calculation approach based on the Monte-Carlo integration method is derived to quantify the uncertainty of the parameters and predictions. The effectiveness of the proposed Bayesian approach is demonstrated on several linear and nonlinear system identification benchmarks by achieving good and competitive simulation accuracy. The code to reproduce the experimental results is open-sourced and available online.

Uncertainty quantification of structural flexibility identified from input–output measurement data for reliability analysis

AbstractIn contrast to the traditional uncertainty quantification of the modal parameters identified from the output‐only measurement data, this study develops an uncertainty propagation and quantification method for structural flexibility identified from input–output measurement data and then uses the derived flexibility matrix for structural reliability analysis. First, a novel procedure for variance estimations of the complete system matrix of the state‐space model is derived based on first‐order perturbation theory. Then, the mode scaling factors and integrated modal flexibility are subsequently derived and quantified. Finally, the predicted displacements and their uncertainty intervals are used to implement the reliability analysis in terms of the structural displacement serviceability limit state (SLS). The accuracies of the derived expressions are verified by a numerical example through the comparison of variance values estimated from Monte Carlo simulations. The simulation example also indicates that we can expect to identify structural parameters with higher reliability if the input force to the structure is measured, and the identified modal scaling and corresponding flexibility matrix undergo more uncertainty than those of the modal parameters. Finally, we apply the proposed uncertainty quantification method to a scaled three‐story frame structure to demonstrate its practicability. The proposed method bridges a gap between the identified structural flexibility and structural reliability analysis directly by the probability category.

Reinforcement learning and approximate Bayesian computation for model selection and parameter calibration applied to a nonlinear dynamical system

In the context of digital twins and integration of physics-based models with machine learning tools, this paper proposes a new methodology for model selection and parameter identification. It combines (i) reinforcement learning (RL) for model selection through a Thompson-like sampling with (ii) approximate Bayesian computation (ABC) for parameter identification and uncertainty quantification. These two methods are applied together to a nonlinear mechanical oscillator with periodic forcing. Experimental data are used in the analysis and two different nonlinear models are tested. The initial Beta distribution that represents the likelihood of the model is updated depending on how successful the model is at reproducing the reference data (reinforcement learning strategy). At the same time, the prior distribution of the model parameters is updated using a likelihood-free strategy (ABC). In the end, the rewards and the posterior distribution of the parameters of each model are obtained. The results show that the combined methodology (RL-ABC) is promising for model selection from bifurcation diagrams. Prior parameter distribution was successfully updated, correlations between parameters were found, probabilistic envelopes of the posterior model are consistent with the available data, the most rewarded model was selected, and the reinforcing strategy allows to speed up the selection process.

A Methodology for the Efficient Quantification of Parameter and Model Uncertainty

Abstract Complex structural systems often entail computationally intensive models that require efficient methods for statistical model calibration due to the high number of required model evaluations. In this paper, we present a Bayesian inference-based methodology for efficient statistical model calibration that builds upon the combination of the speed in the computation of a low-fidelity model with the accuracy of the computationally intensive high-fidelity model. The proposed two-stage method incorporates the adaptive Metropolis algorithm and a Gaussian process (GP)-based adaptive surrogate model as a low-fidelity model. In order to account for model uncertainty, we incorporate a GP-based discrepancy function into the model calibration. By calibrating the hyperparameters of the discrepancy function alongside the model parameters, we prevent the results of the model calibration to be biased. The methodology is illustrated by the statistical model calibration of a damping parameter in the modular active spring-damper system, a structural system developed within the collaborative research center SFB 805 at the Technical University of Darmstadt. The reduction of parameter and model uncertainty achieved by the application of our methodology is quantified and illustrated by assessing the predictive capability of the mathematical model of the modular active spring-damper system.

Hierarchical Bayesian learning framework for multi-level modeling using multi-level data

A hierarchical Bayesian learning framework is proposed to account for multi-level modeling in structural dynamics. In multi-level modeling the system is considered as a hierarchy of lower-level models, starting at the lowest material level, progressing to the component level, then the subsystem level, before ending up to the system level. Bayesian modeling and uncertainty quantification techniques based on measurements that rely on data collected only at the system level cover a quite limited number of component/subsystem operating conditions that are far from representing the full spectrum of system operating conditions. In addition, the large number of models and parameters involved from the lower to higher modeling levels of the system, constitutes this approach inappropriate for simultaneously and reliably quantifying the uncertainties at the different modeling levels. In this work, comprehensive hierarchical Bayesian learning tools are proposed to account for uncertainties through the multi-level modeling process. The uncertainty is embedded within the structural model parameters by introducing a probability model for these parameters that depend on hyperparameters. An important issue that has to be accounted for is that parameters of models at lower levels are shared at the subsystem and system levels. This necessitates a parameter inference process that takes into account data from different modeling levels. Accurate and insightful asymptotic approximations are developed, substantially reducing the computational effort required in the parameter uncertainty quantification procedure. The uncertainties inferred based on datasets available from the different levels of model hierarchy are propagated through the different levels of the system to predict uncertainties and confidence levels of output quantities of interest. A simple dynamical system consisting of components and subsystems is employed to demonstrate the effectiveness of the proposed method.

Bayesian, frequentist, and information geometric approaches to parametric uncertainty quantification of classical empirical interatomic potentials

In this paper, we consider the problem of quantifying parametric uncertainty in classical empirical interatomic potentials (IPs) using both Bayesian (Markov Chain Monte Carlo) and frequentist (profile likelihood) methods. We interface these tools with the Open Knowledgebase of Interatomic Models and study three models based on the Lennard-Jones, Morse, and Stillinger-Weber potentials. We confirm that IPs are typically sloppy, i.e., insensitive to coordinated changes in some parameter combinations. Because the inverse problem in such models is ill-conditioned, parameters are unidentifiable. This presents challenges for traditional statistical methods, as we demonstrate and interpret within both Bayesian and frequentist frameworks. We use information geometry to illuminate the underlying cause of this phenomenon and show that IPs have global properties similar to those of sloppy models from fields, such as systems biology, power systems, and critical phenomena. IPs correspond to bounded manifolds with a hierarchy of widths, leading to low effective dimensionality in the model. We show how information geometry can motivate new, natural parameterizations that improve the stability and interpretation of uncertainty quantification analysis and further suggest simplified, less-sloppy models.

An approach to quantify parameter uncertainty in early assessment of novel health technologies.

Health economic modeling of novel technology at the early stages of a product lifecycle has been used to identify technologies that are likely to be cost-effective. Such early assessments are challenging due to the potentially limited amount of data. Modelers typically conduct uncertainty analyses to evaluate their effect on decision-relevant outcomes. Current approaches, however, are limited in their scope of application and imposes an unverifiable assumption, that is, uncertainty can be precisely represented by a probability distribution. In the absence of reliable data, an approach that uses the fewest number of assumptions is desirable. This study introduces a generalized approach for quantifying parameter uncertainty, that is, probability bound analysis (PBA), that does not require a precise specification of a probability distribution in the context of early-stage health economic modeling. We introduce the concept of a probability box (p-box) as a measure of uncertainty without necessitating a precise probability distribution. We provide formulas for a p-box given data on summary statistics of a parameter. We describe an approach to propagate p-boxes into a model and provide step-by-step guidance on how to implement PBA. We conduct a case and examine the differences between the status-quo and PBA approaches and their potential implications on decision-making.

Parametric uncertainty quantification of SST turbulence model for a shock train and pseudo-shock phenomenon

Computational fluid dynamics simulations are performed to identify the inadequacy of the Menter shear-stress transport (SST) turbulence model for the prediction of the shock train and pseudo-shock phenomenon in a square duct (Mr = 1.61, Re = 3 × 107 m−1), and the uncertainty of the SST model coefficients is quantified. Due to the underprediction of Reynolds stress and failures in predicting corner flow in a square duct, both two-dimensional and three-dimensional SST simulations tended to predict weaker first shock, more secondary shocks, and larger spacing between shocks. The static pressure of the wall fluctuates in the shock train region and suffers a greater loss after the shocks. Based on the two-dimensional simulations, parametric sensitivity analysis using Sobol indices is performed, and the SST model coefficients are classified into three groups according to their sensitivity and influence regions (before and within the shock train). Bayesian inference is conducted on three-dimensional simulations to quantify the uncertainty and calibrate the coefficients that mainly influence the shock train structure. After calibration using maximum a posteriori estimates, the eddy viscosity is greatly increased, and the prediction of the shock structures and the static pressure of the wall is significantly improved, while the prediction of the friction deteriorates.