A CONTINUOUS WAVELET-BASED APPROACH TO DETECT ANISOTROPIC PROPERTIES IN SPATIAL POINT PROCESSES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350017 ◽  
Author(s):  
ROBERTO D'ERCOLE ◽  
JORGE MATEU
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Unitary Operator ◽  
Random Measure ◽  
Continuous Wavelet ◽  
Two Dimensional ◽  
Mother Wavelet ◽  
Spatial Point Processes ◽  
Stochastic Point Process ◽  
Dirac Measures

A two-dimensional stochastic point process can be regarded as a random measure and thus represented as a (countable) sum of Delta Dirac measures concentrated at some points. Integration with respect to the point process itself leads to the concept of the continuous wavelet transform of a point process. Applying then suitable translation, rotation and dilation operations through a non unitary operator, we obtain a transformed point process which highlights main properties of the original point process. The choice of the mother wavelet is relevant and we thus conduct a detailed analysis proposing three two-dimensional mother wavelets. We use this approach to detect main directions present in the point process, and to test for anisotropy.

A wavelet-based approach to quantify the anisotropy degree of spatial random point configurations

2014 ◽  
Vol 12 (06) ◽  
pp. 1450037 ◽  
Author(s):  
Roberto D'Ercole ◽  
Jorge Mateu
Keyword(s):  
Point Process ◽  
Random Measure ◽  
Statistical Hypothesis ◽  
Continuous Wavelet ◽  
Two Dimensional ◽  
Anisotropic Structure ◽  
Point Configurations ◽  
Dirac Measures ◽  
Anisotropy Degree ◽  
Anisotropy Direction

A two-dimensional point process, if considered as a random measure, can be expressed as a countable sum of Delta Dirac measures concentrated at some random points. Then a continuous wavelet transform can be applied to obtain information on some structural properties. We introduce the notions of wavelet-based isotropy, main anisotropy direction and anisotropy degree to characterize the implicit anisotropic structure of the point process. We propose several statistical hypothesis tests that are proved to be useful to test for the presence of anisotropy. An application to a real case is also included.

Detecting the filamentary structure of 3D point clouds: Proposal of a wavelet-based method

2017 ◽  
Vol 08 (03) ◽  
pp. 1750041 ◽  
Author(s):  
Roberto D’Ercole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Three Dimensional ◽  
Random Measure ◽  
Point Clouds ◽  
Filamentary Structure ◽  
Point Patterns ◽  
3D Point Clouds ◽  
Mexican Hat ◽  
Dirac Measures

The analysis of the filamentary structure of the cosmo as well as that of the internal structure of the polar ice suggests the development of models based on three-dimensional (3D) point processes. A point process, regarded as a random measure, can be expressed as a sum of Delta Dirac measures concentrated at some random points. The integration with respect to the point process leads the continuous wavelet transform of the process itself. As possible mother wavelets, we propose the application of the Mexican hat and the Morlet wavelet in order to implement the scale-angle energy density of the process, depending on the dilation parameter and on the three angles which define the direction in the Euclidean space. Such indicator proves to be a sensitive detector of any variation in the direction and it can be successfully implemented to study the isotropy or the filamentary structure in 3D point patterns.

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).

Structural and hidden properties of 1D point processes: A wavelet-based study

2017 ◽  
Vol 15 (05) ◽  
pp. 1750041 ◽  
Author(s):  
Roberto D’Ercole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Measure ◽  
Mathematical Tool ◽  
One Dimensional ◽  
Homogeneous Cluster ◽  
Mexican Hat ◽  
Depth Analysis ◽  
Definition Of ◽  
Dirac Measures

A one-dimensional (1D) point process, if considered as a random measure, can be represented by a sum, at most countable, of Delta Dirac measures concentrated at some random points. The integration with respect to the point process leads to the definition of the continuous wavelet transform of the process itself. As a possible choice of the mother wavelet, we propose the Mexican hat and the Morlet wavelet in order to implement the energy density of the process as a function of two wavelet parameters. Such mathematical tool works as a microscope to process an in-depth analysis of some classes of processes, in particular homogeneous, cluster, and locally scaled processes.

Rarefactions of compound point processes

1984 ◽  
Vol 21 (04) ◽  
pp. 710-719
Author(s):  
Richard F. Serfozo
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Classical Result ◽  
Point Processes ◽  
Rare Events ◽  
Random Measure ◽  
Bernoulli Trials ◽  
Random Measures ◽  
Infinitely Divisible ◽  
Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

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.

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (03) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k } are the points of a renewal process and {Ak } are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (3) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k} are the points of a renewal process and {Ak} are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

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.

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