Limit theorems for convex hulls of random sets

1993 ◽  
Vol 25 (02) ◽  
pp. 395-414 ◽  
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Convex Hulls ◽  
Stable Sets ◽  
Limiting Distributions ◽  
Closed Set ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set ◽  
Simple Limit

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn , and independent copies A 1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

Limit theorems for convex hulls of random sets

1993 ◽  
Vol 25 (2) ◽  
pp. 395-414 ◽  
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Convex Hulls ◽  
Stable Sets ◽  
Limiting Distributions ◽  
Closed Set ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn, and independent copies A1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

scholarly journals Spectral theory for random closed sets and estimating the covariance via frequency space

2003 ◽  
Vol 35 (03) ◽  
pp. 603-613 ◽  
Author(s):  
Karsten Koch ◽  
Joachim Ohser ◽  
Katja Schladitz
Keyword(s):  
Power Spectrum ◽  
Spectral Theory ◽  
Real World ◽  
Second Order ◽  
Mathematical Basis ◽  
Closed Set ◽  
Frequency Space ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set

A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. The definition and a proof of the existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khinchin theorem for the power spectrum are used to two ends. First, well-known second-order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second-order characteristics in frequency space. Examples show that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real-world counterparts.

scholarly journals ON THE SPECIFIC AREA OF INHOMOGENEOUS BOOLEAN MODELS. EXISTENCE RESULTS AND APPLICATIONS

2011 ◽  
Vol 29 (2) ◽  
pp. 111 ◽  
Author(s):  
Elena Villa
Keyword(s):  
Measure Theory ◽  
Geometric Measure Theory ◽  
Specific Area ◽  
Boolean Models ◽  
Closed Set ◽  
Geometric Measure ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set ◽  
The Mean

The problem of the evaluation of the so-called specific area of a random closed set, in connection with its mean boundary measure, is mentioned in the classical book by Matheron on random closed sets (Matheron, 1975, p. 50); it is still an open problem, in general. We offer here an overview of some recent results concerning the existence of the specific area of inhomogeneous Boolean models, unifying results from geometric measure theory and from stochastic geometry. A discussion of possible applications to image analysis concerning the estimation of the mean surface density of random closed sets, and, in particular, to material science concerning birth-and-growth processes, is also provided.

scholarly journals Spectral theory for random closed sets and estimating the covariance via frequency space

2003 ◽  
Vol 35 (3) ◽  
pp. 603-613 ◽  
Author(s):  
Karsten Koch ◽  
Joachim Ohser ◽  
Katja Schladitz
Keyword(s):  
Power Spectrum ◽  
Spectral Theory ◽  
Real World ◽  
Second Order ◽  
Mathematical Basis ◽  
Closed Set ◽  
Frequency Space ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set

A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. The definition and a proof of the existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khinchin theorem for the power spectrum are used to two ends. First, well-known second-order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second-order characteristics in frequency space. Examples show that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real-world counterparts.

Palm representation and approximation of the covariance of random closed sets

2003 ◽  
Vol 35 (02) ◽  
pp. 295-302 ◽  
Author(s):  
Stephan Böhm ◽  
Volker Schmidt
Keyword(s):  
Approximation Formula ◽  
Second Derivative ◽  
Exponential Approximation ◽  
Closed Set ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Palm Calculus ◽  
Random Closed Set ◽  
Very High ◽  
Isotropic Random

The covariance C(r), r ≥ 0, of a stationary isotropic random closed set Ξ is typically complicated to evaluate. This is the reason that an exponential approximation formula for C(r) has been widely used in the literature, which matches C(0) and C (1)(0), and in many cases also lim r→∞ C(r). However, for 0 < r < ∞, the accuracy of this approximation is not very high in general. In the present paper, we derive representation formulae for the covariance C(r) and its derivative C (1)(r) using Palm calculus, where r ≥ 0 is arbitrary. As a consequence, an explicit expression is obtained for the second derivative C (2)(0). These results are then used to get a refined exponential approximation for C(r), which additionally matches the second derivative C (2)(0).

Palm representation and approximation of the covariance of random closed sets

2003 ◽  
Vol 35 (2) ◽  
pp. 295-302 ◽  
Author(s):  
Stephan Böhm ◽  
Volker Schmidt
Keyword(s):  
Approximation Formula ◽  
Second Derivative ◽  
Exponential Approximation ◽  
Closed Set ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Palm Calculus ◽  
Random Closed Set ◽  
Very High ◽  
Isotropic Random

The covariance C(r), r ≥ 0, of a stationary isotropic random closed set Ξ is typically complicated to evaluate. This is the reason that an exponential approximation formula for C(r) has been widely used in the literature, which matches C(0) and C(1)(0), and in many cases also limr→∞C(r). However, for 0 < r < ∞, the accuracy of this approximation is not very high in general. In the present paper, we derive representation formulae for the covariance C(r) and its derivative C(1)(r) using Palm calculus, where r ≥ 0 is arbitrary. As a consequence, an explicit expression is obtained for the second derivative C(2)(0). These results are then used to get a refined exponential approximation for C(r), which additionally matches the second derivative C(2)(0).

scholarly journals NUMERICAL EXPERIMENTS FOR THE ESTIMATION OF MEAN DENSITIES OF RANDOM SETS

2014 ◽  
Vol 33 (2) ◽  
pp. 83 ◽  
Author(s):  
Federico Camerlenghi ◽  
Vincenzo Capasso ◽  
Elena Villa
Keyword(s):  
Numerical Experiments ◽  
Random Sets ◽  
Density Estimator ◽  
Absolutely Continuous ◽  
Minkowski Content ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Spatially Inhomogeneous ◽  
The Subject ◽  
Theoretical Results

Many real phenomena may be modelled as random closed sets in ℝd, of different Hausdorff dimensions. The problem of the estimation of pointwise mean densities of absolutely continuous, and spatially inhomogeneous, random sets with Hausdorff dimension n < d, has been the subject of extended mathematical analysis by the authors. In particular, two different kinds of estimators have been recently proposed, the first one is based on the notion of Minkowski content, the second one is a kernel-type estimator generalizing the well-known kernel density estimator for random variables. The specific aim of the present paper is to validate the theoretical results on statistical properties of those estimators by numerical experiments. We provide a set of simulations which illustrates their valuable properties via typical examples of lower dimensional random sets.

scholarly journals STATISTICS FOR PATCH OBSERVATIONS

2016 ◽  
Vol XLI-B6 ◽  
pp. 235-242
Author(s):  
K. L. Hingee
Keyword(s):  
Remote Sensing ◽  
High Resolution ◽  
Land Cover ◽  
Tree Canopy ◽  
Closed Set ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Contact Distribution ◽  
Land Cover Maps ◽  
Underlying Processes

In the application of remote sensing it is common to investigate processes that generate patches of material. This is especially true when using categorical land cover or land use maps. Here we view some existing tools, landscape pattern indices (LPI), as non-parametric estimators of random closed sets (RACS). This RACS framework enables LPIs to be studied rigorously. A RACS is any random process that generates a closed set, which encompasses any processes that result in binary (two-class) land cover maps. RACS theory, and methods in the underlying field of stochastic geometry, are particularly well suited to high-resolution remote sensing where objects extend across tens of pixels, and the shapes and orientations of patches are symptomatic of underlying processes. For some LPI this field already contains variance information and border correction techniques. After introducing RACS theory we discuss the core area LPI in detail. It is closely related to the spherical contact distribution leading to conditional variants, a new version of contagion, variance information and multiple border-corrected estimators. We demonstrate some of these findings on high resolution tree canopy data.

scholarly journals Nonlinear expectations of random sets

2020 ◽  
Vol 25 (1) ◽  
pp. 5-41
Author(s):  
Ilya Molchanov ◽  
Anja Mühlemann
Keyword(s):  
Random Sets ◽  
Linear Spaces ◽  
Infinite Dimensional ◽  
Closed Sets ◽  
Construction Methods ◽  
Dual Representations ◽  
Random Closed Sets ◽  
Sublinear Functionals ◽  
Primal And Dual ◽  
Nonlinear Expectations

AbstractSublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space) or, equivalently, on random closed sets. This calls for a separate study of sublinear and superlinear expectations, since a change of sign does not alter the direction of the inclusion in the set-valued setting.We identify the extremal expectations as those arising from the primal and dual representations of nonlinear expectations. Several general construction methods for nonlinear expectations are presented and the corresponding duality representation results are obtained. On the application side, sublinear expectations are naturally related to depth trimming of multivariate samples, while superlinear ones can be used to assess utilities of multiasset portfolios.

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