Random Closed Sets Research Articles

Martin–Löf random generalized Poisson processes

Martin–Löf randomness was originally defined and studied in the context of the Cantor space 2ω. In [2] probability theoretic random closed sets (RACS) are used as the foundation for the study of Martin–Löf randomness in spaces of closed sets. We use that framework to explore Martin–Löf randomness for the space of closed subsets of R and a particular family of measures on this space, the generalized Poisson processes. This gives a novel class of Martin–Löf random closed subsets of R. We describe some of the properties of these Martin–Löf random closed sets; one result establishes that a real number is Martin–Löf random if and only if it is contained in some Martin–Löf random closed set.

Description of the symmetric convex random closed sets as zonotopes from their Feret diameters

In this paper, the 2-D random closed sets (RACS) are studied by means of the Feret diameter, also known as the caliper diameter. More specifically, it is shown that a 2-D symmetric convex RACS can be approximated as precisely as we want by some random zonotopes (polytopes formed by the Minkowski sum of line segments) in terms of the Hausdorff distance. Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS. Particularly, the moments of the random vector representing the face lengths of the zonotope approximation are related to the moments of the Feret diameter random process of the RACS.

STATISTICS FOR PATCH OBSERVATIONS

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.

STATISTICS FOR PATCH OBSERVATIONS

Abstract. 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.

Poisson hail on a hot ground

We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACSs) of the Euclidean space. These RACSs arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACSs can be served simultaneously and service is in the first-in–first-out order, i.e. only the hailstones in contact with the ground melt at speed 1, whereas the others are queued. A tagged RACS waits until all RACSs that arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided that the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.

Poisson hail on a hot ground

We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACSs) of the Euclidean space. These RACSs arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACSs can be served simultaneously and service is in the first-in–first-out order, i.e. only the hailstones in contact with the ground melt at speed 1, whereas the others are queued. A tagged RACS waits until all RACSs that arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided that the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.

Statistical environment representation for navigation in natural environments

In this paper we present a novel approach to mobile robot navigation based on environment representation by statistical models. Natural unstructured scenes are interpreted as realizations of Random Closed Sets (RCSs), whose macroscopic characteristics are mapped. Contrary to feature based approaches, this environment representation does not require the existence of remarkable objects in the workspace, and is robust with respect to small dynamic changes. We address the relocalization problem assuming that a statistical model, serving as a map of the environment, is available a priori. Simulation results demonstrate the feasibility of our approach.

Linear Statistical Inference for Random Fuzzy Data

The paper presents first steps towards linear statistical inference with random fuzzy data. In establishing best linear unbiased estimators (BLUE) or best linear predictors (BLP) it is necessary to consider a suitable notion of expectation and variance for random fuzzy variables. Using x-cuts of the random fuzzy variables, the mathematical problem can be formulated within the theory of random closed sets (RCS). As a methodological guide we use the Fréchet-approach which leads for a given metric to an associated expectation and variance. In particular, we prove that the well known Aumann expectation is a Fréchet expectation and that the associated variance appears as a reasonable definition for a variance of RCS's. This associated variance has the advantage that at least special fuzzy number data can formally be handled like Euclidean vectors. Due to the special addition and scalar multiplication these “vectors”, however, fail to build a linear space which leads to somewhat more complicated optimization procedures for BLUE and BLP. This is demonstrated for linear regression and for prediction of stationary processes. In special cases, however, the fuzzified versions of the classical BLUE and BLB keep their optimality

On statistical inference for random closed sets in compact subsets of ℝd

Quite often in the statistical analysis of medical and biological problems, data are images corresponding to entire objects that include smaller objects within them. In these cases, we need models of random closed sets (RACS) confined to compact subsets of the plane. There is no room for stationarity hypotheses and the increase of statistical information comes from independent replicates of the same phenomena rather than increasing our sample window. We investigate practical methods of modelling RACS by means of circumscribed balls, leading to a natural definition of location, size and shape. We discuss the possibilities of using these random variables in order to define statistical spaces of RACS that will allow us to use maximum likelihood methods.

On statistical inference for random closed sets in compact subsets of ℝd

Quite often in the statistical analysis of medical and biological problems, data are images corresponding to entire objects that include smaller objects within them. In these cases, we need models of random closed sets (RACS) confined to compact subsets of the plane. There is no room for stationarity hypotheses and the increase of statistical information comes from independent replicates of the same phenomena rather than increasing our sample window. We investigate practical methods of modelling RACS by means of circumscribed balls, leading to a natural definition of location, size and shape. We discuss the possibilities of using these random variables in order to define statistical spaces of RACS that will allow us to use maximum likelihood methods.

Application of stochastic geometry to problems in plankton ecology

The most fundamental linkages in ecosystem dynamics are trophodynamic. A trophodynamic theory requires a framework based upon inter-organism or interparticle distance, a metric important in its own right, and an essential component relating trophodynamics and the kinetic environment. It is typically assumed that interparticle distances are drawn from a random distribution, even though particles are known to be distributed in patches. Both random and patch-structure interparticle distance are analysed using the theory of stochastic geometry. Aspects of stochastic geometry - point processes and random closed sets (RCS) - useful for studying plankton ecology are presented. For point-process theory, the interparticle distances, random -distribution order statistics, transitions from random to patch structures, and second-order-moment functions are described. For RCS-theory, the volume fractions, contact distributions, and covariance functions are given. Applications of stochastic-geometry theory relate to nutrient flux among organisms, grazing, and coupling between turbulent flow and biological processes. The theory shows that particles are statistically closer than implied by the literature, substantially resolving the troublesome issues of autotroph-heterotroph nutrient exchange; that the microzone notion can be extended by RCS; that patch structure can substantially modify predator-prey encounter rates, even though the number of prey is fixed; and that interparticle distances and the RCS covariance function provide a fundamental coupling with physical processes. In addition to contributing to the understanding of plankton ecology, stochastic geometry is a potentially useful for improving the design of acoustic and optical sensors

Morphological modeling of images by sequential random functions

After a brief reminder of the Mathematical Morphology (MM) tools used to characterize Random Closed Sets (RACS) and random functions (RF), the construction of sequential RF models with support in R ″ is developed. For each point x in R ″, the models combine families of independent RF, indexed by a parameter t, in different ways: masking, taking the supremum or the infinum of observed values, adding. By the masking process, the sequential RF with Markovian jumps and the Dead Leaves Functions models simulate random images with objects in the foreground partially masking objects in the background, as seen in perspective views. The other constructions of RF give, among other, the ∨ and the ∧ Boolean RF, and the Dilution RF. The main properties and the main distribution functions of the models (e.g. statistical behavior of the RF under the non-linear operations provided by dilation or erosion) are presented. These models illustrate the MM approach for random structure modeling.