scholarly journals Transforming Spatial Point Processes into Poisson Processes Using Random Superposition

2012 ◽  
Vol 44 (1) ◽  
pp. 42-62 ◽  
Author(s):  
Jesper Møller ◽  
Kasper K. Berthelsen
Keyword(s):  
Model Checking ◽  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Intensity Function ◽  
Poisson Processes ◽  
Process Models ◽  
Death Process ◽  
Spatial Point Process ◽  
Spatial Point

Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable function β defined on the space for the points of the process. It is possible to superpose a locally stable spatial point process X with a complementary spatial point process Y to obtain a Poisson process X ⋃ Y with intensity function β. Underlying this is a bivariate spatial birth-death process (Xt, Yt) which converges towards the distribution of (X, Y). We study the joint distribution of X and Y, and their marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure for Y conditional on X. This may be used for model checking: given a model for the Papangelou intensity of the original spatial point process, this model is used to generate the complementary process, and the resulting superposition is a Poisson process with intensity function β if and only if the true Papangelou intensity is used. Whether the superposition is actually such a Poisson process can easily be examined using well-known results and fast simulation procedures for Poisson processes. We illustrate this approach to model checking in the case of a Strauss process.

scholarly journals Transforming Spatial Point Processes into Poisson Processes Using Random Superposition

2012 ◽  
Vol 44 (01) ◽  
pp. 42-62 ◽  
Author(s):  
Jesper Møller ◽  
Kasper K. Berthelsen
Keyword(s):  
Model Checking ◽  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Intensity Function ◽  
Poisson Processes ◽  
Process Models ◽  
Death Process ◽  
Spatial Point Process ◽  
Spatial Point

Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable function β defined on the space for the points of the process. It is possible to superpose a locally stable spatial point process X with a complementary spatial point process Y to obtain a Poisson process X ⋃ Y with intensity function β. Underlying this is a bivariate spatial birth-death process (X t , Y t ) which converges towards the distribution of (X, Y). We study the joint distribution of X and Y, and their marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure for Y conditional on X. This may be used for model checking: given a model for the Papangelou intensity of the original spatial point process, this model is used to generate the complementary process, and the resulting superposition is a Poisson process with intensity function β if and only if the true Papangelou intensity is used. Whether the superposition is actually such a Poisson process can easily be examined using well-known results and fast simulation procedures for Poisson processes. We illustrate this approach to model checking in the case of a Strauss process.

scholarly journals Can the Spatial Point Patterns of Animal Distributions Be Detected Using Sparse Samples? A Case Study of Four Soricomorpha (Mammalia) Species in Poland / Czy Przestrzenny Wzorzec Rozkładu Punktów W Dystrybucji Zwierząt Może Zostać Określony Na Podstawie Rzadkiego Próbkowania? Studium Przypadku Na Czterech Gatunkach Soricomorpha (Mammalia) Występujących W Polsce.

2014 ◽  
Vol 59 (1-4) ◽  
pp. 11-24
Author(s):  
Youhua Chen
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Process Models ◽  
Spatial Point Process ◽  
Data Set ◽  
Point Patterns ◽  
Spatial Point ◽  
Spatial Point Patterns ◽  
Point Process Models ◽  
K Function

Abstract In the present study, Riley's K function and alternative spatial point process models are calculated and compared for the hybrid distributional records of four Soricomorpha species (Talpa europaea, Sorex araneus, Sorex minutus, and Neomys fodiens) in Poland over different sampling sizes. The following spatial point process models are fitted and compared: homogeneous Poisson process (HPP) and inhomogeneous Poisson process (IPP) models. For IPP models, the covariates explaining the trend are latitude and longitude. Spatial process models and true distributional aggregation status (using K function) of the four species are also calculated based on the full observed data set for the purpose to check how many grids are required to sample so as to reflect the true spatial distributional point patterns. When performind tha sampling, the sanpling size 5, 10, 30, 60 and 100 are considered. For each sampling size, 500 replicates are performed to keep consistence and reduce uncertainty. The results showed that, for the full observed data set over the whole territory of Poland, IPP models were much better than the null HPP model for explaining the distribution of Soricomorpha species. For every sample size, the true aggregation status and the associated spatial point process models of each species over the studied area can be perfectly identified when using the information derived from limiting samples only. Based on the results, it is found that around 20% of grid cells should be used as the minimum threshold for accurately detecting the true spatial point patterns

A limit theorem for spatial point processes

1986 ◽  
Vol 18 (03) ◽  
pp. 646-659 ◽  
Author(s):  
Steven P. Ellis
Keyword(s):  
Limit Theorem ◽  
Poisson Process ◽  
Asymptotic Distribution ◽  
Point Process ◽  
Point Processes ◽  
Affine Transformations ◽  
Density Estimates ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
As If

Spatial point processes are considered whose points are subjected to certain classes of affine transformations indexed by some variable, T. Under some hypotheses, for large T integrals with respect to such a point process behave approximately as if the process were Poisson. Under stronger hypotheses, the transformed process converges as a process to a Poisson process. The result gives the asymptotic distribution of certain density estimates.

scholarly journals Thinning spatial point processes into Poisson processes

2010 ◽  
Vol 42 (02) ◽  
pp. 347-358 ◽  
Author(s):  
Jesper Møller ◽  
Frederic Paik Schoenberg
Keyword(s):  
Point Process ◽  
Process Model ◽  
Goodness Of Fit ◽  
Point Processes ◽  
Death Process ◽  
Cox Process ◽  
Birth And Death Process ◽  
Conditional Intensity ◽  
Spatial Point Processes ◽  
Spatial Point

In this paper we describe methods for randomly thinning certain classes of spatial point processes. In the case of a Markov point process, the proposed method involves a dependent thinning of a spatial birth-and-death process, where clans of ancestors associated with the original points are identified, and where we simulate backwards and forwards in order to obtain the thinned process. In the case of a Cox process, a simple independent thinning technique is proposed. In both cases, the thinning results in a Poisson process if and only if the true Papangelou conditional intensity is used, and, thus, can be used as a graphical exploratory tool for inspecting the goodness-of-fit of a spatial point process model. Several examples, including clustered and inhibitive point processes, are considered.

scholarly journals DETECTING LINEAR FEATURES BY SPATIAL POINT PROCESSES

2016 ◽  
Vol XLI-B3 ◽  
pp. 841-848
Author(s):  
Dengfeng Chai ◽  
Alena Schmidt ◽  
Christian Heipke
Keyword(s):  
Point Process ◽  
Feature Detection ◽  
Point Processes ◽  
Spatial Point Process ◽  
Linear Features ◽  
Spatial Point Processes ◽  
Linear Feature ◽  
Spatial Point ◽  
Novel Approach ◽  
Optimal Linear

This paper proposes a novel approach for linear feature detection. The contribution is twofold: a novel model for spatial point processes and a new method for linear feature detection. It describes a linear feature as a string of points, represents all features in an image as a configuration of a spatial point process, and formulates feature detection as finding the optimal configuration of a spatial point process. Further, a prior term is proposed to favor straight linear configurations, and a data term is constructed to superpose the points on linear features. The proposed approach extracts straight linear features in a global framework. The paper reports ongoing work. As demonstrated in preliminary experiments, globally optimal linear features can be detected.

scholarly journals DETECTING LINEAR FEATURES BY SPATIAL POINT PROCESSES

2016 ◽  
Vol XLI-B3 ◽  
pp. 841-848
Author(s):  
Dengfeng Chai ◽  
Alena Schmidt ◽  
Christian Heipke
Keyword(s):  
Point Process ◽  
Feature Detection ◽  
Point Processes ◽  
Spatial Point Process ◽  
Linear Features ◽  
Spatial Point Processes ◽  
Linear Feature ◽  
Spatial Point ◽  
Novel Approach ◽  
Optimal Linear

This paper proposes a novel approach for linear feature detection. The contribution is twofold: a novel model for spatial point processes and a new method for linear feature detection. It describes a linear feature as a string of points, represents all features in an image as a configuration of a spatial point process, and formulates feature detection as finding the optimal configuration of a spatial point process. Further, a prior term is proposed to favor straight linear configurations, and a data term is constructed to superpose the points on linear features. The proposed approach extracts straight linear features in a global framework. The paper reports ongoing work. As demonstrated in preliminary experiments, globally optimal linear features can be detected.

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