A Generalization of Poisson's Binomial Limit For use in Ecology

We introduce a new class of Cox cluster processes called generalised shotnoise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process which drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and next on how to make simulation for GSNCPs.Particularly, results for first and second order moment measures, reduced Palm distributions, the J-function, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given.Our results are exemplified for special important cases of GSNCPs, and we discuss the relation to corresponding results for SNCPs.

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“…For the Gaussian kernel (3) we have a (modified) Thomas process [44], and for the uniform kernel (4) we have a Matérn cluster process [24,25].…”

Section: Example 1 a Particular Tractable Model Class Is Obtained Whmentioning

confidence: 99%

“…Next we establish V -uniform ergodicity by checking the geometric drift condition (44) with N > max{β, (1 − q 1 )/q 1 }M µ(D). Assume that the current state φ of the Metropolis-Hastings chain is such that φ / ∈ S N , and set k = card(φ).…”

Section: Conditional Simulationmentioning

confidence: 99%

Shot noise Cox processes

Møller

2003

Adv. Appl. Probab.

150

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“…This is realised using the Thomas cluster process, a generalisation of the Neymann-Scott cluster process, where parent points are replaced by clusters of offspring points whose locations being isotropic Gaussian displacements from the cluster parent location [Thomas, 1949]. The artificial point pattern is created applying the rThomas-function from the package spatstat [Baddeley and Turner, 2005] with the same intensity as in the previous case and a homogeneous offspring with intensity 2 that is comparable to the intensity of megalithic grave clusters (Fig.…”

Section: Data Influenced By First-and/or Second-order Effectsmentioning

confidence: 99%

Point Pattern Analysis as Tool for Digital Geoarchaeology: A Case Study of Megalithic Graves in Schleswig-Holstein, Germany

Knitter

Nakoinz

2017

Natural Science in Archaeology

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In this contribution we apply different methods of spatial and geomorphometric analysis in order to present a general approach of data exploration in areas where detailed local information is absent. Our data are based on locations of megalithic graves from Funnel Beaker societies (3700-2800 BCE) in the area of Schleswig-Holstein, Germany. Using these locations, we apply methods of point pattern analysis in order to reconstruct the spatial processes that created the sample: We use density based measures to show the influence of first order effects on the dataset. While first order effects are related to the underlying areal characteristics of the point locations and hence are determinant of their intensity, second-order effects are the result of interactions between points. We conduct distance related approaches, e.g. focusing on nearest neighbor characteristics, in order to investigate the interaction between the points. The point pattern analyses is complemented by integrating geomorphometric measures that are indirectly indicative for some general environmental conditions, even in prehistoric times. This helps (a) to relate first-order effects to societal or environmental features and (b) to understand the specific pattern of interactions between the points. The necessary raw data in form of digital elevation models are freely available for large parts of the globe. All analyses are conducted using free and open source software in order to provide their limitless application.

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“…The simplest example of a shot noise Cox process is the so-called modified Thomas process (Thomas, 1949) (in the sequel we will call it shortly a Thomas process). In R 2 it is given by the choice…”

Section: Cox Processesmentioning

confidence: 99%

Inhomogeneity in Spatial Cox Point Processes – Location Dependent Thinning Is Not the Only Option

Prokešová

2010

Image Anal Stereol

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In the literature on point processes the by far most popular option for introducing inhomogeneity into a point process model is the location dependent thinning (resulting in a second-order intensity-reweighted stationary point process). This produces a very tractable model and there are several fast estimation procedures available. Nevertheless, this model dilutes the interaction (or the geometrical structure) of the original homogeneous model in a special way. When concerning the Markov point processes several alternative inhomogeneous models were suggested and investigated in the literature. But it is not so for the Cox point processes, the canonical models for clustered point patterns. In the contribution we discuss several other options how to define inhomogeneous Cox point process models that result in point patterns with different types of geometric structure. We further investigate the possible parameter estimation procedures for such models.

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A Generalization of Poisson's Binomial Limit For use in Ecology

Cited by 101 publications

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Shot noise Cox processes

Shot noise Cox processes

Point Pattern Analysis as Tool for Digital Geoarchaeology: A Case Study of Megalithic Graves in Schleswig-Holstein, Germany

Inhomogeneity in Spatial Cox Point Processes – Location Dependent Thinning Is Not the Only Option

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