Nearest-Neighbour Markov Point Processes and Random Sets

Nearest-neighbour Markov point processes were introduced by Baddeley and Møller (1989) as generalizations of the Markov point processes of Ripley and Kelly. This note formulates and discusses a spatial Markov property for these point processes.

Download Full-text

Non-parametric confidence intervals for correlations in nearest-neighbour Markov point processes

Environmetrics ◽

10.1002/env.518 ◽

2002 ◽

Vol 13 (2) ◽

pp. 187-207

Author(s):

Andrea Pallini

Keyword(s):

Confidence Intervals ◽

Point Processes ◽

Nearest Neighbour ◽

Markov Point Processes ◽

Non Parametric

Download Full-text

Nearest-neighbour Markov point processes on graphs with Euclidean edges

Advances in Applied Probability ◽

10.1017/apr.2018.60 ◽

2018 ◽

Vol 50 (4) ◽

pp. 1275-1293 ◽

Cited By ~ 1

Author(s):

M. N. M. van Lieshout

Keyword(s):

Real Line ◽

Point Processes ◽

Local Geometry ◽

Renewal Processes ◽

Nearest Neighbour ◽

The Real ◽

Consistency Conditions ◽

Linear Networks ◽

Markov Point Processes ◽

Neighbour Point

Abstract We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles.

Download Full-text

A spatial Markov property for nearest-neighbour Markov point processes

Journal of Applied Probability ◽

10.2307/3214821 ◽

1990 ◽

Vol 27 (4) ◽

pp. 767-778 ◽

Cited By ~ 6

Author(s):

W. S. Kendall

Keyword(s):

Point Processes ◽

Markov Property ◽

Nearest Neighbour ◽

Markov Point Processes

Nearest-neighbour Markov point processes were introduced by Baddeley and Møller (1989) as generalizations of the Markov point processes of Ripley and Kelly. This note formulates and discusses a spatial Markov property for these point processes.

Download Full-text

Markov point processes

Advances in Applied Probability ◽

10.2307/1426871 ◽

1978 ◽

Vol 10 (2) ◽

pp. 262-263

Author(s):

Erhan Çinlar

Keyword(s):

Point Processes ◽

Markov Point Processes

Download Full-text

Densities for stationary random sets and point processes

Advances in Applied Probability ◽

10.1017/s0001867800022552 ◽

1984 ◽

Vol 16 (02) ◽

pp. 324-346 ◽

Cited By ~ 19

Author(s):

Wolfgang Weil ◽

John A. Wieacker

Keyword(s):

Poisson Process ◽

Point Processes ◽

Convex Set ◽

Compact Convex ◽

Random Sets ◽

Intensity Measure ◽

Additive Functionals ◽

Compact Convex Set ◽

Classical Integral

For certain stationary random setsX, densitiesDφ(X) of additive functionalsφare defined and formulas forare derived whenKis a compact convex set in. In particular, for the quermassintegrals and motioninvariantX, these formulas are in analogy with classical integral geometric formulas. The case whereXis the union set of a Poisson processYof convex particles is considered separately. Here, formulas involving the intensity measure ofYare obtained.

Download Full-text

Densities for stationary random sets and point processes

Advances in Applied Probability ◽

10.2307/1427072 ◽

1984 ◽

Vol 16 (2) ◽

pp. 324-346 ◽

Cited By ~ 37

Author(s):

Wolfgang Weil ◽

John A. Wieacker

Keyword(s):

Poisson Process ◽

Point Processes ◽

Convex Set ◽

Compact Convex ◽

Random Sets ◽

Intensity Measure ◽

Additive Functionals ◽

Compact Convex Set ◽

Classical Integral

For certain stationary random sets X, densities Dφ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals and motioninvariant X, these formulas are in analogy with classical integral geometric formulas. The case where X is the union set of a Poisson process Y of convex particles is considered separately. Here, formulas involving the intensity measure of Y are obtained.

Download Full-text