On the Measure of a Random Set

Robbins, H. E.

doi:10.1214/aoms/1177731315

Cited by 170 publications

(71 citation statements)

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“…See Halpern (2003), Ayyub and McCuen (1997), or Klir and Yuan (1995) for an introduction to the theory, and consult Dempster (1967) and Shafer (1976) for details . A parallel development under random sets theory (Matheron 1975 ;Robbins 1944;1945) is possible, leading to a random set on the real line. Several applications of these parallel theories have been made in engineering contexts (e .g., Castleton and Luo 1992 ;Luo and Castleton 1997 ;Tonon et al .…”

Section: Dempster-shafer Structures and Random Setsmentioning

confidence: 99%

Constructor:synthesizing information about uncertain variables.

Tucker

Ferson

Hajagos

et al. 2005

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Constructor is software for the Microsoft Windows microcomputer environment that facilitates the collation of empirical information and expert judgment for the specification of probability distributions, probability boxes, random sets or Dempster-Shafer structures from data, qualitative shape information, constraints on moments, order statistics, densities, and coverage probabilities about uncertain unidimensional quantities . These quantities may be real-valued, integer-valued or logical values .

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Section: Dempster-shafer Structures and Random Setsmentioning

confidence: 99%

Constructor:synthesizing information about uncertain variables.

Tucker

Ferson

Hajagos

et al. 2005

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“…In the model, avulsions are ultimately a consequence of the interplay between the rate at which available space for deposition is created, and the differential rate of vertical accretion between channel belts and distal floodplains. We have presented in previous fiscal year report an expression that can be used for computing the average fractions of channel belt deposits at a given basin cross section but we found now that a better representation of these fractions is given by the following formula derived from the theory of random saturation bombing [Robins, 1944[Robins, , 1945:…”

Section: Work Completedmentioning

confidence: 99%

Experimental and Theoretical Study of Strata Formation in Sedimentary Basins

Paola¹,

Fedele²

2005

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Grant Number N00014-04-1-0556 LONG TERM GOALSThe long-term goals of our stratigraphy project are to obtain experimental evidence and to develop quantitative models of the formation of strata on continental margins. Experiments and analytical models can greatly contribute to better understand the relationships between variation in forcing functions such as sediment supply, sea-level changes and subsidence, and the resulting strata in depositional basins. These relations will allow, in turn, for(1) inferring the history of external forcing from preserved strata (inversion problem) or (2) predicting the nature of preserved strata in areas where the forcing history is known. OBJECTIVESOur objective has been to develop analytical (or semi-analytical), large-scale, integrated stratigraphic models, i.e. models that use time-averaged versions of underlying transport equations, to predict the evolution of continental margins on geologic time scales. In this context, we have followed two approaches in our EuroSTRATAFORM research program:(1) to conduct experiments in our subsiding-floor experimental basin that can be used to test, inform, and refine stratigraphic models in general, and (2) to contribute to the development of large-scale coupled, moving-boundary stratigraphic models developed by other PIs (Steckler, Swenson).Our overall experimental objective is to assemble a group of experimental data sets from which we can measure the response of margin systems to a range of combinations of forcing parameters (e.g. sea level, sediment supply). Our overall modeling objective is to develop physically-based models that can be used to predict stratigraphic geometry, including grain size and facies, in response to imposed variation in the main forcing variables (sea level, subsidence, and sediment supply). The stratigraphic models are oriented to fulfill two basic needs of the larger margin modeling efforts: first, by providing 1

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“…Dempster-Shafer theory is intimately related, if not equivalent, to the theory of random sets (Matheron 1975;Robbins 1944;1945). Tonon et al (1999;2000b) have applied random set theory to various problems in reliability and design problems in engineering.…”

Section: Dempster-shafer Structuresmentioning

confidence: 99%

Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis.

Oberkampf¹,

Tucker

Zhang

et al. 2004

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LIBRARY VAULTThis report summarizes methods to incorporate information (or lack of information) about inter-variable dependence into risk assessments that use Dempster-Shafer theory or probability bounds analysis to address epistemic and aleatory uncertainty. The report reviews techniques for simulating correlated variates for a given correlation measure and dependence model, computation of bounds on distribution functions under a specified dependence model, formulation of parametric and empirical dependence models, and bounding approaches that can be used when information about the intervariable dependence is incomplete. The report also reviews several of the most pervasive and dangerous myths among risk analysts about dependence in probabilistic models.*The work described in this report was performed for Sandia National Laboratories under Contract No. 19094. 3 AcknowledgmentsDiscussions with Peter Walley were helpful in elucidating the problem of representing dependence in the context of imprecise probabilities and the role of probability boxes and Dempster-Shafer structures as models of imprecise probabilities. 2.5Using inequalities in risk assessments 2.6Caveat: best-possible calculations are NP-hard 2.7Numerical example: fault tree for a pressure tank Sym b o k -Ea is distributed as is an element of is a subset of plus or minus addition, subtraction, etc. under no assumption about the dependence between the operands addition, subtraction, etc. assuming independence addition, subtraction, etc. assuming perfect dependence addition, subtraction, etc. assuming opposite dependence the empty set, i.e., the set having no members probability-box specified by a left side F(x) and a right side E(x) where E(x) I F ( x ) for all x E %, consisting of all non-decreasing functions F from the reals into [O,l] such that E(x) I F(x) I fix). {(sl,ml),. . ., (s,,m,)} an enumeration of the elements of a Dempster-Shafer structure in terms of its focal elements si and their nonzero masses mi beta (v, w) a beta distribution with shape parameters v and w convolve(X, Y,r) convolution (usually addition) assuming that X and Y have correlation r convolve(X, Y,C) convolution (usually addition) assuming the copula C describes the dependence between X and Y covariance between random variables X and Y expectation (mean) of random variable X a functionfwhose domain is the set A and whose range is the set B. In other words, for any element in A, the functionfassigns a value that is in the set B the step function that is zero for all values of x

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On the Measure of a Random Set

Cited by 170 publications

References 0 publications

Constructor:synthesizing information about uncertain variables.

Constructor:synthesizing information about uncertain variables.

Experimental and Theoretical Study of Strata Formation in Sedimentary Basins

Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis.

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