Algebraic analysis of the generating functional for discrete random sets and statistical inference for intensity in the discrete Boolean random-set model

1994 ◽  
Vol 4 (3) ◽  
pp. 273-290 ◽  
Author(s):  
N. D. Sidiropoulos ◽  
J. S. Baras ◽  
C. A. Berenstein
Keyword(s):  
Statistical Inference ◽  
Random Sets ◽  
Random Set ◽  
Generating Functional ◽  
Algebraic Analysis

Estimating the reduced moments of a random measure

1996 ◽  
Vol 28 (02) ◽  
pp. 335-336
Author(s):  
Kiên Kiêu ◽  
Marianne Mora
Keyword(s):  
Transformation Group ◽  
Smooth Boundary ◽  
Random Measure ◽  
Random Sets ◽  
Random Set ◽  
Counting Measure ◽  
Geometrical Properties ◽  
Measure Decomposition ◽  
Moment Measures ◽  
Curvature Measures

Random measures are commonly used to describe geometrical properties of random sets. Examples are given by the counting measure associated with a point process, and the curvature measures associated with a random set with a smooth boundary. We consider a random measure with an invariant distribution under the action of a standard transformation group (translatioris, rigid motions, translations along a given direction and so on). In the framework of the theory of invariant measure decomposition, the reduced moments of the random measure are obtained by decomposing the related moment measures.

MULTIVARIATE MODELS AND VARIABILITY INTERVALS: A LOCAL RANDOM SET APPROACH

2011 ◽  
Vol 19 (05) ◽  
pp. 799-823 ◽  
Author(s):  
THOMAS FETZ
Keyword(s):  
Confidence Level ◽  
Structural Mechanics ◽  
Random Sets ◽  
Confidence Limits ◽  
Random Set ◽  
Multivariate Functions ◽  
Multivariate Models ◽  
Multivariate Case ◽  
Fréchet Bounds

This article is devoted to the propagation of families of variability intervals through multivariate functions comprising the semantics of confidence limits. At fixed confidence level, local random sets are defined whose aggregation admits the calculation of upper probabilities of events. In the multivariate case, a number of ways of combination is highlighted to encompass independence and unknown interaction using random set independence and Fréchet bounds. For all cases we derive formulas for the corresponding upper probabilities and elaborate how they relate. An example from structural mechanics is used to exemplify the method.

scholarly journals Stationary countable dense random sets

2000 ◽  
Vol 32 (01) ◽  
pp. 86-100 ◽  
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Probability Theory ◽  
Polish Space ◽  
Measurable Subset ◽  
Locally Compact Group ◽  
Random Sets ◽  
Random Set ◽  
Locally Compact ◽  
Polish Spaces ◽  
Probability Zero ◽  
Group Of Symmetries

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-stationarity (null-events stay null under the group action). In passing, it is noted that the property of being countable does not correspond to a measurable subset of the space of subsets of an uncountably infinite Polish space.

Law of small numbers as concentration inequalities for sums of independent random setsand random set valued mappings

2012 ◽  
Author(s):  
Vladimir I. Norkin ◽  
Roger J-B Wets
Keyword(s):  
Hilbert Space ◽  
Discrete Distributions ◽  
Random Sets ◽  
Concentration Inequalities ◽  
Random Set ◽  
Infinite Dimensional ◽  
Random Functions ◽  
The Monte Carlo Method ◽  
Exponential Bounds ◽  
Set Valued Mappings

In the paper we study concentration of sample averages (Minkowski's sums) of independent bounded random sets and set valued mappings around their expectations. Sets and mappings are considered in a Hilbert space. Concentration is formulated in the form of exponential bounds on probabilities of normalized large deviations. In a sense, concentration phenomenon reflects the law of small numbers, describing non-asymptotic behavior of the sample averages. We sequentially consider concentration inequalities for bounded random variables, functions, vectors, sets and mappings, deriving next inequalities from preceding cases. Thus we derive concentration inequalities with explicit constants for random sets and mappings from the sharpest available (Talagrand type) inequalities for random functions and vectors. The most explicit inequalities are obtained in case of discrete distributions. The obtained results contribute to substantiation of the Monte Carlo method in infinite dimensional spaces.

Computation of Reliability with Both Aleatory and Epistemic Uncertainty Based on Random Sets Theory

2010 ◽  
Vol 29-32 ◽  
pp. 1252-1257 ◽  
Author(s):  
Hui Xin Guo ◽  
Lei Chen ◽  
Hang Min He ◽  
Dai Yong Lin
Keyword(s):  
Epistemic Uncertainty ◽  
Limit State ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Random Sets ◽  
Extension Principle ◽  
State Function ◽  
Random Set ◽  
Cumulative Distribution Functions ◽  
Aleatory And Epistemic Uncertainty

A method to handle hybrid uncertainties including aleatory and epistemic uncertainty is proposed for the computation of reliability. The aleatory uncertainty is modeled as random variable and the epistemic uncertainty is modeled with evidence theory. The two types of uncertainty are firstly transformed into random set, and the limit-state function of a product is mapped into a random set by using the extension principle of random set. Then, the belief function and the plausibility function of safety event are determined. The two functions are viewed as the lower and the upper cumulative distribution functions of reliability, respectively. The reliability of a product will be bounded by two cumulative distribution functions, and then an interval estimation of reliability can be obtained. The proposed method is demonstrated with an example.

scholarly journals COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

2014 ◽  
Vol 20 (1) ◽  
pp. 80-90 ◽  
Author(s):  
LAURENT BIENVENU ◽  
ADAM R. DAY ◽  
NOAM GREENBERG ◽  
ANTONÍN KUČERA ◽  
JOSEPH S. MILLER ◽  
...  
Keyword(s):  
Random Sets ◽  
Random Set ◽  
Halting Problem

AbstractEveryK-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.

RELATIONSHIPS BETWEEN POSSIBILITY MEASURES AND NESTED RANDOM SETS

2002 ◽  
Vol 10 (01) ◽  
pp. 1-15 ◽  
Author(s):  
ENRIQUE MIRANDA ◽  
INÉS COUSO ◽  
PEDRO GIL
Keyword(s):  
Random Sets ◽  
General Context ◽  
Random Set ◽  
Possibility Measure

Different authors have observed some relationships between consonant random sets and possibility measures, specially for finite universes. In this paper, we go deeply into this matter and propose several possible definitions for the concept of consonant random set. Three of these conditions are equivalent for finite universes. In that case, the random set considered is associated to a possibility measure if and only if any of them is satisfied. However, in a general context, none of the six definitions here proposed is sufficient for a random set to induce a possibility measure. Moreover, only one of them seems to be necessary.

AGGREGATION AND COMPLETION OF RANDOM SETS WITH DISTRIBUTIONAL FUZZY MEASURES

1996 ◽  
Vol 04 (04) ◽  
pp. 307-329 ◽  
Author(s):  
CLIFF JOSLYN
Keyword(s):  
Information Theory ◽  
Possibility Theory ◽  
Random Sets ◽  
Random Set ◽  
Topological Structures ◽  
Fuzzy Measures ◽  
New Information ◽  
Set Intersection

The two known information theories, probability and possibility theory, are based on t-conorm decomposable fuzzy measures, so that bijective mappings exist between their set-valued measures and their point-valued distributions. Further, their random set (Dempster-Shafer evidence theoretical) interpretations have simple topological structures, with bijective mappings between the subset focal elements and the point singletons. We introduce the concepts of distributional and aggregable random sets and random set completion, and first use them as a model in which to cast probability and possibility measures and distributions. Then, towards the goal of deriving new forms of information theory, general Sugeno conorm decomposable fuzzy measures and ring-like aggregable random sets with set-intersection structural aggregation are examined, but it is shown that in these two cases no new information theories are forthcoming.

On the construction of effectively random sets

2004 ◽  
Vol 69 (3) ◽  
pp. 862-878 ◽  
Author(s):  
Wolfgang Merkle ◽  
Nenad Mihailović
Keyword(s):  
Truth Table ◽  
Random Sets ◽  
Random Set ◽  
Basic Construction ◽  
Computably Enumerable

Abstract.We present a comparatively simple way to construct Martin-Löf random and rec-random sets with certain additional properties, which works by diagonalizing against appropriate martingales. Reviewing the result of Gács and Kučera, for any given set X we construct a Martin-Löf random set from which X can be decoded effectively.By a variant of the basic construction we obtain a rec-random set that is weak truth-table autoreducible and we observe that there are Martin-Löf random sets that are computably enumerable self-reducible. The two latter results complement the known facts that no rec-random set is truth-table autoreducible and that no Martin-Löf random set is Turing-autoreducible [8, 24].

scholarly journals BAYESIAN IMAGE RESTORATION, USING CONFIGURATIONS

2011 ◽  
Vol 25 (3) ◽  
pp. 129
Author(s):  
Thordis Linda Thorarinsdottir
Keyword(s):  
Image Restoration ◽  
Random Sets ◽  
Salt And Pepper Noise ◽  
Random Set ◽  
Bayesian Procedure ◽  
Prior Probabilities ◽  
Configuration Theory ◽  
Normal Measure ◽  
The Mean ◽  
Salt And Pepper

In this paper, we develop a Bayesian procedure for removing noise from images that can be viewed as noisy realisations of random sets in the plane. The procedure utilises recent advances in configuration theory for noise free random sets, where the probabilities of observing the different boundary configurations are expressed in terms of the mean normal measure of the random set. These probabilities are used as prior probabilities in a Bayesian image restoration approach. Estimation of the remaining parameters in the model is outlined for salt and pepper noise. The inference in the model is discussed in detail for 3 X 3 and 5 X 5 configurations and examples of the performance of the procedure are given.

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