Theory of Random Sets

By Ilya S. Molchanov. 2005

999 citations recorded by the source corpus.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2f8fc777fc34b045a8eeda876a6fcca9425d7c28

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In Management Science and Operations Research

In Statistics and Probability

Citations in the corpus

Imprecise probabilities in engineering analyses (2013, positive): “Imprecise probabilities have a close relationship to the theory of random sets [56, 57] and cover, for example, the concept of upper and lower probabilities [58], sets of probability measures [59], distribution envelopes [60], probability bounds analysis using p-boxes [61], interval probabilities [6”
50 years sets with positive reach -- a survey. (2008, background): “As usual, a subbasis of the topology of F is generated by {FG : G ∈ G} ∪ {F : K ∈ K}, where FG = {F ∈ F : F ∩ G 6= ∅} and FK = {F ∈ F : F ∩ K 6= ∅} (see for example [12] or [14]).”
Asymptotic theory for density ridges (2014, background): “Limiting distributions in geometric statistics often involve the Hausdorff distance [Molchanov (2005)].”
A discussion on mean excess plots (2009, background): “[19,20] are good references for the theory of random sets.”
Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems (2015, background): “Since the topology generated by the Hausdorff metric is independent of the concrete chosen metrization of the underlying space (see [26])) we can, w.”
COMPUTATION OF THE PERIMETER OF MEASURABLE SETS VIA THEIR COVARIOGRAM. APPLICATIONS TO RANDOM SETS (2011, background): “The covariogram is of particular importance in stochastic geometry when dealing with random closed sets (RACS) (Matheron, 1975; Stoyan et al., 1995; Molchanov, 2005; Schneider and Weil, 2008).”
Validation of imprecise probability models (2009, methodology): “The Wasserstein distance is a metric for any distributions for which the infimum is finite ( Dobrushin 1970 ).”
The topology of probability distributions on manifolds (2013, methodology): “The stochastic process we study in this paper is an example of a random set model – stochastic models that place probability distributions on regions or geometric objects [34, 41].”
Imprecise probabilities based on generalised intervals for system reliability assessment (2010, background): “A random set (Molchanov (2005)) is a multi-valued mapping from the probability space to the value space.”
Convergence properties of the expected improvement algorithm with fixed mean and covariance functions (2007, positive): “General measurable selection theorems (see, e.g., Molchanov, 2005) ensure that such an algorithm does exist.”
Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case (2009, background): “3 in [17], but this assumption on F is definitively too strong in our context since it would exclude the model examples of periodic and quasi–periodic Hamiltonians, see Remark 4.”
Convex geometry of max-stable distributions (2006, background): “If X is a random compact convex set [22] such that ‖X‖ = sup{‖x‖ : x ∈ X} is integrable, then the selection expectation (also called the Aumann expectation) of X is the set of expectations of E ξ for all random vectors ξ such that ξ ∈ X a.”