Probabilistic Framework for Uncertainty Propagation With Both Probabilistic and Interval Variables

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Kais Zaman ◽  
Mark McDonald ◽  
Sankaran Mahadevan
Keyword(s):  
System Analysis ◽  
Epistemic Uncertainty ◽  
Probability Distributions ◽  
Uncertainty Propagation ◽  
Probabilistic Approach ◽  
Interval Data ◽  
Cumulative Distribution ◽  
System Response ◽  
Computationally Efficient ◽  
Probabilistic Uncertainty

This paper develops and illustrates a probabilistic approach for uncertainty representation and propagation in system analysis, when the information on the uncertain input variables and/or their distribution parameters may be available as either probability distributions or simply intervals (single or multiple). A unique aggregation technique is used to combine multiple interval data and to compute rigorous bounds on the system response cumulative distribution function. The uncertainty described by interval data is represented through a flexible family of probability distributions. Conversion of interval data to a probabilistic format enables the use of computationally efficient methods for probabilistic uncertainty propagation. Two methods are explored for the implementation of the proposed approach, based on (1) sampling and (2) optimization. The sampling-based strategy is more expensive and tends to underestimate the output bounds. The optimization-based methodology improves both aspects. The proposed methods are used to develop new solutions to challenge problems posed by the Sandia epistemic uncertainty workshop (Oberkampf et al., 2004, “Challenge Problems: Uncertainty in System Response Given Uncertain Parameters,” Reliab. Eng. Syst. Saf., 85, pp. 11–19). Results for the challenge problems are compared with earlier solutions.

scholarly journals A Computationally-Efficient Probabilistic Approach to Model-Based Damage Diagnosis

2020 ◽  
Vol 8 (2) ◽  
Author(s):  
James E. Warner ◽  
Geoffrey F. Bomarito ◽  
Jacob D. Hochhalter ◽  
William P. Leser ◽  
Patrick E. Leser ◽  
...  
Keyword(s):  
Probability Distributions ◽  
Probabilistic Approach ◽  
Measurement Data ◽  
Image Correlation ◽  
Sensor Data ◽  
Fe Model ◽  
Computationally Efficient ◽  
Damage Diagnosis ◽  
Full Field ◽  
Model Based

This work presents a computationally-efficient, probabilistic approach to model-based damage diagnosis. Given measurement data, probability distributions of unknown damage parameters are estimated using Bayesian inference and Markov chain Monte Carlo (MCMC) sampling. Substantial computational speedup is obtained by replacing a three-dimensional finite element (FE) model with an efficient surrogate model. While the formulation is general for arbitrary component geometry, damage type, and sensor data, it is applied to the problem of strain-based crack characterization and experimentally validated using full-field strain data from digital image correlation (DIC). Access to full-field DIC data facilitates the study of the effectiveness of strain-based diagnosis as the distance between the location of damage and strain measurements is varied. The ability of the framework to accurately estimate the crack parameters and effectively capture the uncertainty due to measurement proximity and experimental error is demonstrated. Furthermore, surrogate modeling is shown to enable diagnoses on the order of seconds and minutes rather than several days required with the FE model.

scholarly journals EMPIRICAL COMPARISON OF METHODS FOR THE HIERARCHICAL PROPAGATION OF HYBRID UNCERTAINTY IN RISK ASSESSMENT, IN PRESENCE OF DEPENDENCES

2012 ◽  
Vol 20 (04) ◽  
pp. 509-557 ◽  
Author(s):  
NICOLA PEDRONI ◽  
ENRICO ZIO
Keyword(s):  
Risk Analysis ◽  
Epistemic Uncertainty ◽  
Probability Distributions ◽  
Uncertainty Propagation ◽  
Random Sets ◽  
Uncertain Parameters ◽  
Random Events ◽  
Possibility Distributions ◽  
Hybrid Uncertainty ◽  
Analysis Models

Risk analysis models describing aleatory (i.e., random) events contain parameters (e.g., probabilities, failure rates, …) that are epistemically-uncertain, i.e., known with poor precision. Whereas aleatory uncertainty is always described by probability distributions, epistemic uncertainty may be represented in different ways (e.g., probabilistic or possibilistic), depending on the information and data available. The work presented in this paper addresses the issue of accounting for (in)dependence relationships between epistemically-uncertain parameters. When a probabilistic representation of epistemic uncertainty is considered, uncertainty propagation is carried out by a two-dimensional (or double) Monte Carlo (MC) simulation approach; instead, when possibility distributions are used, two approaches are undertaken: the hybrid MC and Fuzzy Interval Analysis (FIA) method and the MC-based Dempster-Shafer (DS) approach employing Independent Random Sets (IRSs). The objectives are: i) studying the effects of (in)dependence between the epistemically-uncertain parameters of the aleatory probability distributions (when a probabilistic/possibilistic representation of epistemic uncertainty is adopted) and ii) studying the effect of the probabilistic/possibilistic representation of epistemic uncertainty (when the state of dependence between the epistemic parameters is defined). The Dependency Bound Convolution (DBC) approach is then undertaken within a hierarchical setting of hybrid (probabilistic and possibilistic) uncertainty propagation, in order to account for all kinds of (possibly unknown) dependences between the random variables. The analyses are carried out with reference to two toy examples, built in such a way to allow performing a fair quantitative comparison between the methods, and evaluating their rationale and appropriateness in relation to risk analysis.

scholarly journals Learning on dynamic statistical manifolds

2020 ◽  
Vol 476 (2239) ◽  
pp. 20200213 ◽  
Author(s):  
F. Boso ◽  
D. M. Tartakovsky
Keyword(s):  
Loss Function ◽  
Uncertainty Propagation ◽  
Cumulative Distribution ◽  
Computationally Efficient ◽  
Random System ◽  
Statistical Manifolds ◽  
Hyperbolic Balance Laws ◽  
Leibler Divergence ◽  
Spatio Temporal ◽  
Non Gaussian

Hyperbolic balance laws with uncertain (random) parameters and inputs are ubiquitous in science and engineering. Quantification of uncertainty in predictions derived from such laws, and reduction of predictive uncertainty via data assimilation, remain an open challenge. That is due to nonlinearity of governing equations, whose solutions are highly non-Gaussian and often discontinuous. To ameliorate these issues in a computationally efficient way, we use the method of distributions, which here takes the form of a deterministic equation for spatio-temporal evolution of the cumulative distribution function (CDF) of the random system state, as a means of forward uncertainty propagation. Uncertainty reduction is achieved by recasting the standard loss function, i.e. discrepancy between observations and model predictions, in distributional terms. This step exploits the equivalence between minimization of the square error discrepancy and the Kullback–Leibler divergence. The loss function is regularized by adding a Lagrangian constraint enforcing fulfilment of the CDF equation. Minimization is performed sequentially, progressively updating the parameters of the CDF equation as more measurements are assimilated.

scholarly journals Evidential Model Validation under Epistemic Uncertainty

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Wei Deng ◽  
Xi Lu ◽  
Yong Deng
Keyword(s):  
Computational Model ◽  
Model Validation ◽  
Bayes Factor ◽  
Epistemic Uncertainty ◽  
Probability Distributions ◽  
Uncertainty Propagation ◽  
Evidence Theory ◽  
Validation Method ◽  
Basic Probability ◽  
The Given

This paper proposes evidence theory based methods to both quantify the epistemic uncertainty and validate computational model. Three types of epistemic uncertainty concerning input model data, that is, sparse points, intervals, and probability distributions with uncertain parameters, are considered. Through the proposed methods, the given data will be described as corresponding probability distributions for uncertainty propagation in the computational model, thus, for the model validation. The proposed evidential model validation method is inspired by the idea of Bayesian hypothesis testing and Bayes factor, which compares the model predictions with the observed experimental data so as to assess the predictive capability of the model and help the decision making of model acceptance. Developed by the idea of Bayes factor, the frame of discernment of Dempster-Shafer evidence theory is constituted and the basic probability assignment (BPA) is determined. Because the proposed validation method is evidence based, the robustness of the result can be guaranteed, and the most evidence-supported hypothesis about the model testing will be favored by the BPA. The validity of proposed methods is illustrated through a numerical example.

scholarly journals A Probabilistic Approach to Aeropropulsion System Assessment

2000 ◽  
Author(s):  
Michael T. Tong
Keyword(s):  
Mechanical Design ◽  
Probability Distributions ◽  
Probabilistic Approach ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Sensitivity Analyses ◽  
Turbine Engine ◽  
Technical Performance ◽  
Design Variables ◽  
System Assessment

A probabilistic approach is described for aeropropulsion system assessment. To demonstrate this approach, the technical performance of a wave rotor-enhanced gas turbine engine (i.e. engine net thrust, specific fuel consumption, and engine weight) is assessed. The assessment accounts for the uncertainties in component efficiencies/flows and mechanical design variables, using probability distributions. The results are presented in the form of cumulative distribution functions (CDFs) and sensitivity analyses, and are compared with those from the traditional deterministic approach. The comparison shows that the probabilistic approach provides a more realistic and systematic way to assess an aeropropulsion system.

Robust optimization based on a polynomial expansion of chaos constructed with integration point rules

2008 ◽  
Vol 223 (5) ◽  
pp. 1263-1272 ◽  
Author(s):  
D L Wei ◽  
Z S Cui ◽  
J Chen
Keyword(s):  
Robust Optimization ◽  
Engineering Design ◽  
Uncertainty Propagation ◽  
Probabilistic Approach ◽  
Main Idea ◽  
Optimization Procedure ◽  
Optimization Method ◽  
Computationally Efficient ◽  
Design Problems ◽  
Engineering Design Problems

Robust optimization is a probabilistic approach to engineering design under uncertainty. The main idea is to select designs insensitive to changes in given parameters. Robust optimization using numerical simulations for black-box problems has received increasing interest. However, when the simulation programmes are computationally expensive, robust optimization is difficult to implement due to the intensive computational demand of uncertainty propagation. Based on polynomial chaos expansion (PCE), an efficient robust optimization method is presented. The PCE is constructed with points of monomial cubature rules (MCRs) to approximate the original model. As the number of points of MCRs is small and all of the points are sampled, the robust optimization procedure is computationally efficient and stable. Two engineering design problems are employed to demonstrate the availability of the proposed method.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

scholarly journals Stochastic and epistemic uncertainty propagation in LCA

2013 ◽  
Vol 18 (7) ◽  
pp. 1393-1403 ◽  
Author(s):  
Julie Clavreul ◽  
Dominique Guyonnet ◽  
Davide Tonini ◽  
Thomas H. Christensen
Keyword(s):  
Epistemic Uncertainty ◽  
Uncertainty Propagation

Comparative Analysis of Methodologies for Uncertainty Propagation and Quantification

2017 ◽  
Author(s):  
Alessandra Cuneo ◽  
Alberto Traverso ◽  
Shahrokh Shahpar
Keyword(s):  
Engineering Design ◽  
Polynomial Chaos ◽  
Epistemic Uncertainty ◽  
Uncertainty Propagation ◽  
Computational Cost ◽  
Computational Effort ◽  
Optimization Under Uncertainty ◽  
Design Parameters ◽  
Test Case ◽  
Aleatory And Epistemic Uncertainty

In engineering design, uncertainty is inevitable and can cause a significant deviation in the performance of a system. Uncertainty in input parameters can be categorized into two groups: aleatory and epistemic uncertainty. The work presented here is focused on aleatory uncertainty, which can cause natural, unpredictable and uncontrollable variations in performance of the system under study. Such uncertainty can be quantified using statistical methods, but the main obstacle is often the computational cost, because the representative model is typically highly non-linear and complex. Therefore, it is necessary to have a robust tool that can perform the uncertainty propagation with as few evaluations as possible. In the last few years, different methodologies for uncertainty propagation and quantification have been proposed. The focus of this study is to evaluate four different methods to demonstrate strengths and weaknesses of each approach. The first method considered is Monte Carlo simulation, a sampling method that can give high accuracy but needs a relatively large computational effort. The second method is Polynomial Chaos, an approximated method where the probabilistic parameters of the response function are modelled with orthogonal polynomials. The third method considered is Mid-range Approximation Method. This approach is based on the assembly of multiple meta-models into one model to perform optimization under uncertainty. The fourth method is the application of the first two methods not directly to the model but to a response surface representing the model of the simulation, to decrease computational cost. All these methods have been applied to a set of analytical test functions and engineering test cases. Relevant aspects of the engineering design and analysis such as high number of stochastic variables and optimised design problem with and without stochastic design parameters were assessed. Polynomial Chaos emerges as the most promising methodology, and was then applied to a turbomachinery test case based on a thermal analysis of a high-pressure turbine disk.

scholarly journals Probabilistic Precipitation Forecast Postprocessing Using Quantile Mapping and Rank-Weighted Best-Member Dressing

2018 ◽  
Vol 146 (12) ◽  
pp. 4079-4098 ◽  
Author(s):  
Thomas M. Hamill ◽  
Michael Scheuerer
Keyword(s):  
Probability Distributions ◽  
The United States ◽  
Ensemble Forecast ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Precipitation Forecast ◽  
Prediction System ◽  
Quantile Mapping ◽  
State Dependent ◽  
Grid Points

Abstract Hamill et al. described a multimodel ensemble precipitation postprocessing algorithm that is used operationally by the U.S. National Weather Service (NWS). This article describes further changes that produce improved, reliable, and skillful probabilistic quantitative precipitation forecasts (PQPFs) for single or multimodel prediction systems. For multimodel systems, final probabilities are produced through the linear combination of PQPFs from the constituent models. The new methodology is applied to each prediction system. Prior to adjustment of the forecasts, parametric cumulative distribution functions (CDFs) of model and analyzed climatologies are generated using the previous 60 days’ forecasts and analyses and supplemental locations. The CDFs, which can be stored with minimal disk space, are then used for quantile mapping to correct state-dependent bias for each member. In this stage, the ensemble is also enlarged using a stencil of forecast values from the 5 × 5 surrounding grid points. Different weights and dressing distributions are assigned to the sorted, quantile-mapped members, with generally larger weights for outlying members and broader dressing distributions for members with heavier precipitation. Probability distributions are generated from the weighted sum of the dressing distributions. The NWS Global Ensemble Forecast System (GEFS), the Canadian Meteorological Centre (CMC) global ensemble, and the European Centre for Medium-Range Weather Forecasts (ECMWF) ensemble forecast data are postprocessed for April–June 2016. Single prediction system postprocessed forecasts are generally reliable and skillful. Multimodel PQPFs are roughly as skillful as the ECMWF system alone. Postprocessed guidance was generally more skillful than guidance using the Gamma distribution approach of Scheuerer and Hamill, with coefficients generated from data pooled across the United States.

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