Space-Time Models in Stochastic Geometry

Beneš, Viktor; Prokešová, Michaela; Helisová, Kateřina; Zikmundová, Markéta

doi:10.1007/978-3-319-10064-7_7

Cited by 3 publications

(4 citation statements)

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“…In general, the strongest form of separability may be defined by the requirement that the distribution of a STPP is equal to the product of the distributions of the marginal processes in space and time. This form of separability is equivalent to the independence of the spatial and temporal components of the point process, and under this separability, the spatial and temporal components can be modeled completely separately (Beneš et al;. Diggle (2013, page 220) uses the term no spatio-temporal interaction for a point process with independent spatial and temporal components.…”

Section: Introductionmentioning

confidence: 99%

Testing the first-order separability hypothesis for spatio-temporal point patterns

Ghorbani,

Vafaei,

Dvořák

et al. 2020

Preprint

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First-order separability of a spatio-temporal point process plays a fundamental role in the analysis of spatio-temporal point pattern data. While it is often a convenient assumption that simplifies the analysis greatly, existing non-separable structures should be accounted for in the model construction. We propose three different tests to investigate this hypothesis as a step of preliminary data analysis. The first two tests are exact or asymptotically exact for Poisson processes. The first test based on permutations and global envelopes allows us to detect at which spatial and temporal locations or lags the data deviate from the null hypothesis. The second test is a simple and computationally cheap χ 2 -test. The third test is based on statistical reconstruction method and can be generally applied for non-Poisson processes. The performance of the first two tests is studied in a simulation study for Poisson and non-Poisson models. The third test is applied to the real data of the UK 2001 epidemic foot and mouth disease.

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Section: Introductionmentioning

confidence: 99%

Testing the first-order separability hypothesis for spatio-temporal point patterns

Ghorbani,

Vafaei,

Dvořák

et al. 2020

Preprint

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“…This was further developed by Møller & Ghorbani (). With a well‐defined space‐time K ‐function, parameter estimation of an inhomogeneous space‐time SNCP may be performed by a two‐step estimation procedure with minimum contrast estimation in the second step as discussed by Beneš et al (). Such a procedure is a generalization from the purely spatial case, which was discussed by Waagepetersen & Guan ().…”

Section: Introductionmentioning

confidence: 99%

“…Such a procedure is a generalization from the purely spatial case, which was discussed by Waagepetersen & Guan (). However, the main problem, identified by Beneš et al (), is the higher dimensionality of the space‐time K ‐function and the consequent low stability of its non‐parametric estimate. This makes the minimum contrast estimation numerically unstable even for point patterns including several hundreds of points.…”

Section: Introductionmentioning

confidence: 99%

Parameter Estimation for Inhomogeneous Space‐Time Shot‐Noise Cox Point Processes

Dvořák

Prokešová

2016

Scandinavian J Statistics

Self Cite

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We consider the problem of parameter estimation for inhomogeneous space-time shot-noise Cox point processes. We explore the possibility of using a stepwise estimation method and dimensionality-reducing techniques to estimate different parts of the model separately. We discuss the estimation method using projection processes and propose a refined method that avoids projection to the temporal domain. This remedies the main flaw of the method using projection processes -possible overlapping in the projection process of clusters, which are clearly separated in the original space-time process. This issue is more prominent in the temporal projection process where the amount of information lost by projection is higher than in the spatial projection process. For the refined method, we derive consistency and asymptotic normality results under the increasing domain asymptotics and appropriate moment and mixing assumptions. We also present a simulation study that suggests that cluster overlapping is successfully overcome by the refined method.

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“…where κ is a kernel function (weight) and L is a non-negative Lévy basis (see Beneš et al, 2015;Hellmund et al, 2008), we have a spatio-temporal Lévy-driven Cox process. Under some regularity conditions, Λ has an equivalent shot-noise representation (Møller, 2003) with additional random noise.…”

Section: Spatio-temporal Stationary Poisson Cluster and Shotnoise Coxmentioning

confidence: 99%

Statistical tests for comparisons of spatial and spatio-temporal point patterns

González¹

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The director and the main author hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other university. This dissertation contains nothing which is the outcome of work done in collaboration with others, except as specified in the text and Acknowledgements. Castellón, January 2018.

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Space-Time Models in Stochastic Geometry

Cited by 3 publications

References 0 publications

Testing the first-order separability hypothesis for spatio-temporal point patterns

Testing the first-order separability hypothesis for spatio-temporal point patterns

Parameter Estimation for Inhomogeneous Space‐Time Shot‐Noise Cox Point Processes

Statistical tests for comparisons of spatial and spatio-temporal point patterns

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