Evidence Theory for Engineering Applications

Abstract

Computational analysis of the performance, reliability, and safety of engineered systems is spreading rapidly in industry and government. To many managers, decision makers, and politicians not trained in computational simulation, computer simulations can appear most convincing. Terminology such as "virtual prototyping," "virtual testing," "full physics simulation," and "modeling and simulation-based acquisition" are extremely appealing when budgets are highly constrained; competitors are taking market share; or when political constraints do not allow testing of certain systems. To assess the accuracy and usefulness of computational simulations, three key aspects are needed in the analysis and experimental process: computer code and solution verification; experimental validation of most, if not all, of the mathematical models of the engineered system being simulated; and estimation of the uncertainty associated with analysis inputs, physics models, possible scenarios experienced by the system, and the outputs of interest in the simulation. The topics of verification and validation are not addressed here, but these are covered at length in the literature (see, for example, [1-6]). A number of fields have contributed to the development of uncertainty estimation techniques and procedures, such as nuclear reactor safety, underground storage of radioactive and toxic wastes, and structural dynamics (see, for example, [7][8][9][10][11][12][13][14][15][16][17][18]). only thought to be within specified intervals; for example, the parameters are estimated from expert opinion, not measurements. Assume each of these parameters is treated as a random variable and assigned the least informative distribution (i.e., a uniform distribution). If extreme system responses correspond to extreme values of these parameters (i.e., values near the ends of the uniform distribution), then their probabilistic combination could predict a very low probability for such extreme system responses. Given that the parameters are only known to occur within intervals, however, this conclusion is grossly inappropriate. Improved Models for Epistemic UncertaintyDuring the past two decades, the information theory and expert systems communities have made significant progress in developing a number of new theories that can be pursued for modeling epistemic uncertainty. Examples of the newer theories include fuzzy set theory [17,[42][43][44][45][46], interval analysis [47,48], evidence (Dempster-Shafer) theory [49][50][51][52][53][54][55], possibility theory [56,57], and theory of upper and lower previsions [58]. Some of these theories only deal with epistemic uncertainty; most deal with both epistemic and aleatory uncertainty; and some deal with other varieties of uncertainty (e.g., nonclassical logics appropriate for artificial intelligence and data fusion systems [59]).A recent article summarizes how these theories of uncertainty are related to one another from a hierarchical viewpoint [60]. The article shows that evidence theory is a generalization of cl...

DAKOTA : a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis. Version 5.0, user's reference manual.

“…Belief represents the smallest possible probability that is consistent with the evidence, while plausibility represents the largest possible probability that is consistent with the evidence. For more information about the Dempster-Shafer theory of evidence, see Helton, 2003 andOberkampf, 2004. Similar to the interval approaches, one may use global or local methods to determine plausbility and belief measures for the outputs. Note that to calculate the plausibility and belief cumulative distribution functions, one has to look at all combinations of intervals for the uncertain variables.…”

Sensitivity in risk analyses with uncertain numbers.

“…Dempster-Shafer evidence theory (Oberkampf et al 2005;Klir and Yuan 1995;Shafer 1976) has been widely studied in computer science and artificial intelligence, although it has never achieved wide acceptance among probabilists and traditional statisticians. In a discrete probability distribution on the real line, a nonzero probability mass is associated with each of the possible points of the distribution.…”

“…There have been several applications of the new methods to a wide variety of problems in risk analysis and uncertainty projection, including engineering applications (Oberkampf and Helton 2005), reliability studies for dikeworks (Hall and Lawry 2001), competing failure calculations for strong/weak link switches (Helton et al 2004a;2005b;; automotive design (Rekuc et al 2006), aircraft reliability (Tonon et al 1999); global warming (Kriegler and Held 2005), geological engineering (Tonon et al 2000a,b), slope stability and landslide analysis (Rubio et al 2004), human health risk assessments (MacDonald et al 2002;EA 2002EA -2005, and endangered species viability analyses .…”

Epistemic Uncertainty Analysis: An Approach Using Expert Judgment and Evidential Credibility

“…"Epistemic uncertainty as a source of nondeterministic behavior derives from lack of knowledge of the system or the environment" Oberkampf et al [6]. Oberkampf and Helton [5] Epistemic uncertainty often becomes an issue when expert opinion is required to solve a problem. In trying to determine the likelihood of a terrorist attack on a given building, a decision maker may solicit many expert opinions due to a lack of sufficient knowledge about the problem.…”

“…"Aleatory uncertainty is also referred to as variability, irreducible uncertainty, inherent uncertainty, stochastic uncertainty, and uncertainty due to chance. Epistemic uncertainty is also referred to as reducible uncertainty, subjective uncertainty, and uncertainty due to lack of knowledge" ( [5], p. 10-2). Aleatory uncertainty refers to variation which is inherent to a given system, typically as a result of the random nature of model inputs.…”