A limit theorem for spatial point processes

1986 ◽  
Vol 18 (3) ◽  
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.

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.

Algorithmically deconstructing shot locations as a method for shot quality in hockey

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Devan G. Becker ◽  
Douglas G. Woolford ◽  
Charmaine B. Dean
Keyword(s):  
Point Process ◽  
Process Model ◽  
Point Processes ◽  
Learning Algorithm ◽  
Basis Functions ◽  
Cox Process ◽  
Spatial Point Processes ◽  
Professional Basketball ◽  
Spatial Point ◽  
Log Gaussian Cox Process

AbstractSpatial point processes have been successfully used to model the relative efficiency of shot locations for each player in professional basketball games. Those analyses were possible because each player makes enough baskets to reliably fit a point process model. Goals in hockey are rare enough that a point process cannot be fit to each player’s goal locations, so novel techniques are needed to obtain measures of shot efficiency for each player. A Log-Gaussian Cox Process (LGCP) is used to model all shot locations, including goals, of each NHL player who took at least 500 shots during the 2011–2018 seasons. Each player’s LGCP surface is treated as an image and these images are then used in an unsupervised statistical learning algorithm that decomposes the pictures into a linear combination of spatial basis functions. The coefficients of these basis functions are shown to be a very useful tool to compare players. To incorporate goals, the locations of all shots that resulted in a goal are treated as a “perfect player” and used in the same algorithm (goals are further split into perfect forwards, perfect centres and perfect defence). These perfect players are compared to other players as a measure of shot efficiency. This analysis provides a map of common shooting locations, identifies regions with the most goals relative to the number of shots and demonstrates how each player’s shot location differs from scoring locations.

scholarly journals Thinning spatial point processes into Poisson processes

2010 ◽  
Vol 42 (2) ◽  
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.

A new graph related to the directions of nearest neighbours in a point process

2003 ◽  
Vol 35 (1) ◽  
pp. 47-55 ◽  
Author(s):  
S. N. Chiu ◽  
I. S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Point Process ◽  
Process Simulation ◽  
Point Processes ◽  
Nearest Neighbour ◽  
Simulation Studies ◽  
Nearest Neighbours ◽  
Typical Point

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

scholarly journals Modeling Forest Tree Data Using Sequential Spatial Point Processes

2021 ◽  
Author(s):  
Adil Yazigi ◽  
Antti Penttinen ◽  
Anna-Kaisa Ylitalo ◽  
Matti Maltamo ◽  
Petteri Packalen ◽  
...  
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Process Models ◽  
Aerial Image ◽  
Tree Age ◽  
Data Set ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
Forest Inventories

AbstractThe spatial structure of a forest stand is typically modeled by spatial point process models. Motivated by aerial forest inventories and forest dynamics in general, we propose a sequential spatial approach for modeling forest data. Such an approach is better justified than a static point process model in describing the long-term dependence among the spatial location of trees in a forest and the locations of detected trees in aerial forest inventories. Tree size can be used as a surrogate for the unknown tree age when determining the order in which trees have emerged or are observed on an aerial image. Sequential spatial point processes differ from spatial point processes in that the realizations are ordered sequences of spatial locations, thus allowing us to approximate the spatial dynamics of the phenomena under study. This feature is useful in interpreting the long-term dependence and spatial history of the locations of trees. For the application, we use a forest data set collected from the Kiihtelysvaara forest region in Eastern Finland.

scholarly journals On the conditional distributions of spatial point processes

2011 ◽  
Vol 43 (2) ◽  
pp. 301-307 ◽  
Author(s):  
François Caron ◽  
Pierre Del Moral ◽  
Arnaud Doucet ◽  
Michele Pace
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Measure ◽  
Theoretic Approach ◽  
Conditional Distributions ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
Observation Process ◽  
Original Analysis ◽  
Latent Process

We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

scholarly journals On the conditional distributions of spatial point processes

2011 ◽  
Vol 43 (02) ◽  
pp. 301-307 ◽  
Author(s):  
François Caron ◽  
Pierre Del Moral ◽  
Arnaud Doucet ◽  
Michele Pace
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Measure ◽  
Theoretic Approach ◽  
Conditional Distributions ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
Observation Process ◽  
Original Analysis ◽  
Latent Process

We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

Sign in / Sign up

Export Citation Format

Share Document