Random Marked Sets

Abstract

We aim to link random fields and marked point processes, and, therefore, introduce a new class of stochastic processes which are defined on a random set in R d . Unlike for random fields, the mark covariance function of a random marked set is in general not positive definite. This implies that in many situations the use of simple geostatistical methods appears to be questionable. Surprisingly, for a special class of processes based on Gaussian random fields, we do have positive definiteness for the corresponding mark covariance function and mark correlation function.

The Correct Asymptotic Variance for the Sample Mean of a Homogeneous Poisson Marked Point Process

“…The setup of data from a marked point process is still at the forefront of active research; see, for example, Ballani et al (2012) and the references therein. Nevertheless, the subject has been under investigation for several decades; see Masry (1983) or Kutoyants (1984aKutoyants ( ), (1984b.…”

Nonparametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets

“…Let X =( X 1 , X 2 ) be a bivariate random closed set and Γ=(Γ 1 ,Γ 2 ) a bivariate random field taking almost surely nonnegative values. Suppose that X and Γ are independent, and set Ψ=(Ψ 1 ,Ψ 2 ), where The univariate case was dubbed a random field model by Ballani, Kabluchko, and Schlather () for which, under the assumption that both X and Γ are stationary, Koubek, Pawlas, Brereton, Kriesche, and Schmidt () employed the R 12 ‐function for testing purposes.…”

Sliced Inverse Regression and Independence in Random Marked Sets with Covariates

“…The notation λ = λ 1 will be used. Ballani et al (2012) defined a random marked closed set as a pair (Y, ), which is a random element in a measurable space of hypographs of random upper-semicontinuous functions (a mark) defined on a random closed set Y ∈ R d . Thus, for a random marked H k -set (Y, ), we have the notion of a marked point, fibre, or surface process.…”