Markov property of point processes

1987 ◽  
Vol 76 (1) ◽  
pp. 71-80 ◽  
Author(s):  
Hans G. Kellerer
Keyword(s):  
Point Processes ◽  
Markov Property

A spatial Markov property for nearest-neighbour Markov point processes

1990 ◽  
Vol 27 (04) ◽  
pp. 767-778 ◽  
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.

A spatial Markov property for nearest-neighbour Markov point processes

1990 ◽  
Vol 27 (4) ◽  
pp. 767-778 ◽  
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.

Decomposition of gamma-distributed domains constructed from Poisson point processes

2003 ◽  
Vol 35 (01) ◽  
pp. 56-69 ◽  
Author(s):  
Richard Cowan ◽  
Malcolm Quine ◽  
Sergei Zuyev
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Markov Property ◽  
Simple Algorithm ◽  
Theoretical Development ◽  
Classical Theorem ◽  
Type Result ◽  
Delaunay Tessellations ◽  
Natural Decomposition ◽  
Strong Markov

A known gamma-type result for the Poisson process states that certain domains defined through configuration of the points (or ‘particles’) of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples—based on the Voronoi and Delaunay tessellations—where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.

Decomposition of gamma-distributed domains constructed from Poisson point processes

2003 ◽  
Vol 35 (1) ◽  
pp. 56-69 ◽  
Author(s):  
Richard Cowan ◽  
Malcolm Quine ◽  
Sergei Zuyev
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Markov Property ◽  
Simple Algorithm ◽  
Theoretical Development ◽  
Classical Theorem ◽  
Type Result ◽  
Delaunay Tessellations ◽  
Natural Decomposition ◽  
Strong Markov

A known gamma-type result for the Poisson process states that certain domains defined through configuration of the points (or ‘particles’) of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples—based on the Voronoi and Delaunay tessellations—where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.

scholarly journals Existence of Gibbs point processes with stable infinite range interaction

2020 ◽  
Vol 57 (3) ◽  
pp. 775-791
Author(s):  
David Dereudre ◽  
Thibaut Vasseur
Keyword(s):  
Point Processes ◽  
Existence Results ◽  
Pairwise Interaction ◽  
Range Interaction ◽  
Pairwise Interactions ◽  
Gibbs Point Processes ◽  
Infinite Range Interactions ◽  
Finite Configuration ◽  
Multi Body ◽  
Infinite Range

AbstractWe provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

A review of multistate modelling approaches in monitoring disease progression: Bayesian estimation using the Kolmogorov-Chapman forward equations

2021 ◽  
pp. 096228022199750
Author(s):  
Zvifadzo Matsena Zingoni ◽  
Tobias F Chirwa ◽  
Jim Todd ◽  
Eustasius Musenge
Keyword(s):  
Bayesian Estimation ◽  
Transition Probabilities ◽  
Markov Property ◽  
Likelihood Estimation ◽  
Single Individual ◽  
Transition Rates ◽  
Multistate Models ◽  
Time To Event ◽  
Modeling Framework ◽  
Forward Equations

There are numerous fields of science in which multistate models are used, including biomedical research and health economics. In biomedical studies, these stochastic continuous-time models are used to describe the time-to-event life history of an individual through a flexible framework for longitudinal data. The multistate framework can describe more than one possible time-to-event outcome for a single individual. The standard estimation quantities in multistate models are transition probabilities and transition rates which can be mapped through the Kolmogorov-Chapman forward equations from the Bayesian estimation perspective. Most multistate models assume the Markov property and time homogeneity; however, if these assumptions are violated, an extension to non-Markovian and time-varying transition rates is possible. This manuscript extends reviews in various types of multistate models, assumptions, methods of estimation and data features compatible with fitting multistate models. We highlight the contrast between the frequentist (maximum likelihood estimation) and the Bayesian estimation approaches in the multistate modeling framework and point out where the latter is advantageous. A partially observed and aggregated dataset from the Zimbabwe national ART program was used to illustrate the use of Kolmogorov-Chapman forward equations. The transition rates from a three-stage reversible multistate model based on viral load measurements in WinBUGS were reported.

Sign in / Sign up

Export Citation Format

Share Document