Uncertainty Quantification Using Random Set Based on IODM

Uncertainty quantification and propagation with probability boxes

In this thesis, we concentrate on the relationship of probability intervals and random sets. Furthermore, we are interested in solving uncertain linear programming problems with probability interval and random set parameters. We discover the conditions to verify when a given probability interval obtains the same information as a random set information. If these conditions are satisfied, we can transform a problem that contains both types of uncertainty into a problem which has only the random set information. In addition, we use an idea from decision making theory with random sets for solving this problem. If a probability interval does not satisfy these conditions, we can solve the problem for finding the optimistic and pessimistic expected recourse values. Finally, we present an algorithm for checking these conditions and constructing appropriate distributions for each of the optimistic and pessimistic approaches.

We describe the application of a validation and uncertainty quantification methodology to multiphase computational fluid dynamics modeling, demonstrating the methodology with simulations of a pilot-scale circulating fluidized bed. The overall pressure drop is used as the quantity of interest (QoI); the solids circulation rate and the superficial gas velocity are chosen as the uncertain input quantities. The uncertainty in the QoI, caused by uncertainties in input parameters, surrogate model, spatial discretization, and time averaging, is calculated, and the model form uncertainty is estimated by comparing simulation results with experimental data. The spatial discretization error was determined to be the most dominant source of uncertainty, but the applicability of the method used to calculate that uncertainty needs to be further investigated. The results of the analysis are expressed as a probability box (p-box) plot. A p-box similarly constructed for predictive simulations will give the design engineer information about the confidence in the predicted values.

A Monte Carlo-based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type

This paper is one in a sequence of presentations that consider the problem of uncertainty quantification (UQ) in model verification and validation. The particular sub-topic considered here is uncertainty quantification in modeling. The objectives of the paper are to enumerate some of the various sources of uncertainty in models, as-built systems and experimental data, to define uncertainty and errors and differences between them, to introduce different types of comparative difference metrics that can be used for uncertainty quantification, and to present examples illustrating their uses. Particular emphasis is placed on the ability to quantify modeling uncertainty that includes model form uncertainty, i.e. an epistemic uncertainty that results from an incomplete understanding of the physics of the phenomenon being modeled, and/or undetected modeling error.

Abstract. The work package “Uncertainty Management multi-Actor Network”, UMAN, within EURAD, the European Joint Programme on Radioactive Waste Management, was dedicated to the management of uncertainties potentially relevant to the safety of different radioactive waste management stages and programmes. One important goal there was to compile, review, compare and refine strategies, approaches and tools for the management of uncertainties in the safety assessment that are being used, planned to be used or being developed in different countries. This paper presents major findings from the UMAN Deliverable 10.3 “Uncertainty identification, classification and quantification” (Brendler and Pospiech, 2024) and reflects on the outcome of that work. It addresses approaches to identify and categorize uncertainties that might be of relevance in the various stages of radioactive waste management as well as in the quantification of numerical uncertainties. After an introduction, bottom-up and top-down strategies are compared. Sources used for the paper as input were expert elicitation (here primarily based on a respective questionnaire send out to UMAN participants) and literature surveys. This paper then advises on how uncertainties can be structured, rendering it suitable for a comprehensive assessment of numerical uncertainties: fishbone diagrams and tables for uncertainty characteristics. Results support the identification of uncertainties with high relevance for radioactive waste management. Nine potentially useful categories are identified; the uncertainties are then grouped (including representative examples utilizing fishbone diagrams and tables) according to the occurrence by system phenomena, following the Themes and Sub-themes of the EURAD Roadmap. The last part deals with the evaluation as well as quantification of uncertainties. The paper closes with recommendations aimed at future research directions for parameter uncertainties. Finally, a glossary provides definitions for some terms frequently used (uncertainty in general, parameter uncertainty, uncertainty models and aleatory vs. epistemic uncertainties).

Uncertainty quantification in a resonant nonlinear MEMS structure

Reliable and trustworthy artificial intelligence (AI), particularly in high-stake medical diagnoses, necessitates effective uncertainty quantification (UQ). Existing UQ methods using model ensembles often introduce invalid variability or computational complexity, rendering them impractical and ineffective in clinical workflow. We propose a UQ approach based on deep neuroevolution (DNE), a data-efficient optimization strategy. Our goal is to replicate trends observed in expert-based UQ. We focused on language lateralization maps from resting-state functional MRI (rs-fMRI). Fifty rs-fMRI maps were divided into training/testing (30:20) sets, representing two labels: "left-dominant" and "co-dominant." DNE facilitated acquiring an ensemble of 100 models with high training and testing set accuracy. Model uncertainty was derived from distribution entropies over the 100 model predictions. Expert reviewers provided user-based uncertainties for comparison. Model (epistemic) and user-based (aleatoric) uncertainties were consistent in the independently and identically distributed (IID) testing set, mainly indicating low uncertainty. In a mostly out-of-distribution (OOD) holdout set, both model and user-based entropies correlated but displayed a bimodal distribution, with one peak representing low and another high uncertainty. We also found a statistically significant positive correlation between epistemic and aleatoric uncertainties. DNE-based UQ effectively mirrored user-based uncertainties, particularly highlighting increased uncertainty in OOD images. We conclude thatDNE-based UQ correlates with expert assessments, making it reliable for our use case and potentially for other radiology applications.

Just as an interval bounds an uncertain real value, a probability box bounds an uncertain cumulative distribution function. It can express doubt about the shape of a probability distribution, the distribution parameters, the nature of intervariable dependence, or some other aspect of model uncertainty. Probability bounds analysis rigorously projects probability boxes through mathematical expressions. Ben-Haim's info-gap decision theory is a non-probabilistic decision theory that can address poorly characterized and even unbounded uncertainty. It bases decisions on optimizing robustness to failure rather than expected utility. Nested probability boxes can be used to define info-gap models for probability distributions, and probability bounds analysis provides a ready calculus for the calculations needed for an info-gap analysis involving probabilistic uncertainty.

We are interested in improving risk and reliability analysis of complex systems where our knowledge of system performance is provided by large simulation codes, and where moreover input parameters are known only imprecisely. Such imprecision lends itself to interval representations of parameter values, and thence to quantifying our uncertainty through Dempster-Shafer or Probability Bounds representations on the input space. In this context, the simulation code acts as a large box function f, transforming one input Dempster-Shafer structure on the line into an output random interval f(A). Our quantification of output uncertainty is then based on this output random interval.. If some properties of f are known, then some information about f(A) can be determined. But when f is a pure black box, we must resort to sampling approaches. We present the basic formalism of a Monte Carlo approach to sampling a functionally propagated general random set, as opposed to a random interval. We show that the results of straightforward formal definitions are mathematically coherent, in the sense that bounding and convergence properties are achieved.

Model and analysis of single event transient sensitivity based on uncertainty quantification

Assessment of hydrological parameter uncertainty versus climate projection spread on urban streamflow and floods

Quantifying the uncertainty in data-driven mechanistic models is fundamental in public health applications. COVID-19 is a complex disease that had a significant impact on global health and economies. Several mathematical models were used to understand the complexity of the transmission dynamics under different hypotheses to support the decision-making for disease management. This paper highlights various scenarios of a 6D epidemiological model known as SEIQRD (Susceptible-Exposed-Infected-Quarantined-Recovered-Deceased) to evaluate its effectiveness in prediction and state estimation during the spread of COVID-19 pandemic. Then we investigate the suitability of the classical 4D epidemiological model known as SIRD (Susceptible-Infected-Recovered-Deceased) in the long-term behaviour in order to make a comparison between these models. The primary aim of this paper is to establish a foundational basis for the validity and epidemiological model comparisons in long-term behaviour which may help identify the degree of model complexity that is required based on two approaches viz. the Bayesian inference employing the nested sampling algorithm and recursive state estimation utilizing the Extended Kalman Filter (EKF). Our approach acknowledges the potential imperfections and uncertainties inherent in compartmental epidemiological models. By integrating our proposed methodology, these models can consistently generate predictions closely aligned with the observed data on active cases and deaths. This framework, implemented within the EKF algorithm, offers a robust tool for addressing future, unknown pandemics. Moreover, we present a systematic methodology for time-varying parameter estimation along with uncertainty quantification using Saudi Arabia COVID-19 data and obtain the credible confidence intervals of the epidemiological nonlinear dynamical system model parameters.

Identification of linear models in view of robust control design requires the identification of a control‐relevant nominal model, and a quantification of model uncertainty. In this paper a procedure is presented to quantify the model uncertainty of any prespecified nominal model, from a sequence of measurement data of input and output signals from a plant. By employing a nonparametric empirical transfer function estimate (ETFE), we are able to split the model uncertainty into three parts: the inherent uncertainty in the data due to data imperfections, the unmodelled dynamics in the nominal model, and the uncertainty due to interpolation. A frequency‐dependent hard error bound is constructed, and results are given for tightening the bound through appropriate input design.

Material parameters related to deterministic models can have different values due to variation of experiments outcome. From a mathematical point of view, probabilistic modeling can improve this problem. It means that material parameters of constitutive models can be characterized as random variables with a probability distribution. To this end, we propose a constitutive models of rubber-like materials based on uncertainty quantification (UQ) approach. UQ reduces uncertainties in both computational and real-world applications. Constitutive models in elastomers play a crucial role in both science and industry due to their unique hyper-elastic behavior under different loading conditions (uni-axial extension, biaxial, or pure shear). Here our goal is to model the uncertainty in constitutive models of elastomers, and accordingly, identify sensitive parameters that we highly contribute to model uncertainty and error. Modern UQ models can be implemented to use the physics of the problem compared to black-box machine learning approaches that uses data only. In this research, we propagate uncertainty through the model, characterize sensitivity of material behavior to show the importance of each parameter for uncertainty reduction. To this end, we utilized Bayesian rules to develop a model considering uncertainty in the mechanical response of elastomers. As an important assumption, we believe that our measurements are around the model prediction, but it is contaminated by Gaussian noise. We can make the noise by maximizing the posterior. The uni-axial extension experimental data set is used to calibrate the model and propagate uncertainty in this research.