Nonparametric estimation of the pair correlation function of replicated inhomogeneous point processes

Abstract We consider a Markov chain of point processes such that each state is a superposition of an independent cluster process with the previous state as its centre process together with some independent noise process and a thinned version of the previous state. The model extends earlier work by Felsenstein (1975) and Shimatani (2010) describing a reproducing population. We discuss when closed-form expressions of the first- and second-order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

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Second-order variational equations for spatial point processes with a view to pair correlation function estimation

Spatial Statistics ◽

10.1016/j.spasta.2019.03.001 ◽

2019 ◽

Vol 30 ◽

pp. 103-115

Author(s):

Jean-François Coeurjolly ◽

Francisco Cuevas-Pacheco ◽

Rasmus Waagepetersen

Keyword(s):

Correlation Function ◽

Pair Correlation Function ◽

Point Processes ◽

Pair Correlation ◽

Second Order ◽

Function Estimation ◽

Spatial Point Processes ◽

Variational Equations ◽

Spatial Point

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The distance decay of similarity in tropical rainforests. A spatial point processes analytical formulation

10.1101/216929 ◽

2017 ◽

Author(s):

Anna Tovo ◽

Marco Favretti

Keyword(s):

Correlation Function ◽

Pair Correlation Function ◽

Point Processes ◽

Species Turnover ◽

Pair Correlation ◽

Analytical Formula ◽

Barro Colorado Island ◽

Spatial Density ◽

Spatial Point Processes ◽

Spatial Point

AbstractIn this paper we are concerned with the analytical description of the change in floristic composition (species turnover) with the distance between two plots of a tropical rainforest due to the clustering of the individuals of the different species. We describe the plant arrangement by a superposition of spatial point processes and in this framework we introduce an analytical function which represents the average spatial density of the Sørensen similarity between two infinitesimal plots at distancer. We see that the decay in similarity with the distance is essentially described by the pair correlation function of the superposed process and that it is governed by the most abundant species. We test our analytical model with empirical data obtained for the Barro Colorado Island and Pasoh rainforests. To this end we adopt the statistical estimator for the pair correlation function in [1] and we design a novel one for the Sørensen similarity. Furthermore, we test our analytical formula by modelling the forest study area with Neyman-Scott point processes. We conclude comparing the advantages of our approach with other ones existing in literature.

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