Lévy-based Cox point processes

Hellmund, Gunnar; Prokešová, Michaela; Jensen, Eva B. Vedel

doi:10.1017/s0001867800002718

Cited by 24 publications

(42 citation statements)

References 37 publications

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“…A space‐time SNCP is a Cox process with the driving field Λ of the form

Λ (u, t) = \sum_{(r, v, s) \in normalΦ} r k ((u, t), (v, s)), (u, t) \in {double-struckR}^{2} \times double-struckR,

where Φ is a Poisson process on

(0, \infty) \times {double-struckR}^{2} \times double-struckR

with intensity measure U and k is a smoothing kernel, that is, a non‐negative function integrable in both coordinates. Under some basic integrability assumptions, is an almost surely locally integrable random field, and the corresponding point process X is a well‐defined Cox process (Møller, , or Hellmund et al, ).…”

Section: Space‐time Shot‐noise Cox Processesmentioning

confidence: 99%

“…Conditionally on Φ, the cluster processes X ( v , s ) are independent Poisson processes with intensity function r k (·,( v , s )); that is, r corresponds to the weight of the cluster. When

\int_{0}^{1} V (normald r) = \infty

, then even for a compact set

A \in scriptℬ ({double-struckR}^{2} \times double-struckR)

, the number of cluster centres in A is infinite almost surely; see Møller () and Hellmund et al () for details. However, under the basic integrability assumptions mentioned earlier, almost surely only finitely many of the cluster centres produce at least one point in the cluster.…”

Section: Space‐time Shot‐noise Cox Processesmentioning

confidence: 99%

“…However, because the weights of the majority of the clusters are very small, X is still a well‐defined Cox process. The name Gamma SNCP refers to the fact that V is the Lévy measure of a Gamma‐distributed random variable (Hellmund et al, , sec. 4).…”

Section: Space‐time Shot‐noise Cox Processesmentioning

confidence: 99%

See 2 more Smart Citations

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

Dvořák

Prokešová

2016

Scandinavian J Statistics

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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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“…A space‐time SNCP is a Cox process with the driving field Λ of the form

Λ (u, t) = \sum_{(r, v, s) \in normalΦ} r k ((u, t), (v, s)), (u, t) \in {double-struckR}^{2} \times double-struckR,

where Φ is a Poisson process on

(0, \infty) \times {double-struckR}^{2} \times double-struckR

Section: Space‐time Shot‐noise Cox Processesmentioning

confidence: 99%

“…Conditionally on Φ, the cluster processes X ( v , s ) are independent Poisson processes with intensity function r k (·,( v , s )); that is, r corresponds to the weight of the cluster. When

\int_{0}^{1} V (normald r) = \infty

, then even for a compact set

A \in scriptℬ ({double-struckR}^{2} \times double-struckR)

Section: Space‐time Shot‐noise Cox Processesmentioning

confidence: 99%

See 1 more Smart Citation

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

Dvořák

Prokešová

2016

Scandinavian J Statistics

Self Cite

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“…is the von Mises-Fisher kernel proposed by Hansen et al (2011). It has a parameter˛> 0, and d.u; v/ denotes the great circle distance between u and v. The short terminology of a Lévy basis has been introduced in Barndorff-Nielsen & Schmiegel (2004); see also Hellmund et al (2008) and Jónsdóttir et al (2008).…”

Section: Description Of the Modelmentioning

confidence: 99%

Estimating Particle Shape and Orientation Using Volume Tensors

Ziegel

Nyengaard

Jensen

2015

Scandinavian J Statistics

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We develop statistical procedures for estimating shape and orientation of arbitrary three-dimensional particles. We focus on the case where particles cannot be observed directly, but only via sections. Volume tensors are used for describing particle shape and orientation, and we derive stereological estimators of the tensors. These estimators are combined to provide consistent estimators of the moments of the so-called particle cover density. The covariance structure associated with the particle cover density depends on the orientation and shape of the particles. For instance, if the distribution of the typical particle is invariant under rotations, then the covariance matrix is proportional to the identity matrix. We develop a non-parametric test for such isotropy. A flexible Lévy-based particle model is proposed, which may be analysed using a generalized method of moments in which the volume tensors enter. The developed methods are used to study the cell organization in the human brain cortex.

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“…Note in this relation that, so far, only independently scattered stable measures have received much attention in the literature; see [28] and, more recently, [7,16] on the subject. It should be noted that Cox processes driven by various random measures are often used in spatial statistics (see [17,21,22]). A particularly novel feature of discrete stable processes is that the point counts have discrete α-stable distributions, which have infinite expectations unless for the degenerate case of α = 1.…”

Section: Introductionmentioning

confidence: 99%

Stability for random measures, point processes and discrete semigroups

Davydov¹,

Molchanov²,

Zuyev³

2011

Bernoulli

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Discrete stability extends the classical notion of stability to random elements in discrete spaces by defining a scaling operation in a randomised way: an integer is transformed into the corresponding binomial distribution. Similarly defining the scaling operation as thinning of counting measures we characterise the corresponding discrete stability property of point processes. It is shown that these processes are exactly Cox (doubly stochastic Poisson) processes with strictly stable random intensity measures. We give spectral and LePage representations for general strictly stable random measures without assuming their independent scattering. As a consequence, spectral representations are obtained for the probability generating functional and void probabilities of discrete stable processes. An alternative cluster representation for such processes is also derived using the so-called Sibuya point processes, which constitute a new family of purely random point processes. The obtained results are then applied to explore stable random elements in discrete semigroups, where the scaling is defined by means of thinning of a point process on the basis of the semigroup. Particular examples include discrete stable vectors that generalise discrete stable random variables and the family of natural numbers with the multiplication operation, where the primes form the basis.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ301 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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Lévy-based Cox point processes

Cited by 24 publications

References 37 publications

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

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

Estimating Particle Shape and Orientation Using Volume Tensors

Stability for random measures, point processes and discrete semigroups

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