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.

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.

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.

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.

On arrivals that see time averages: a martingale approach

1990 ◽  
Vol 27 (2) ◽  
pp. 376-384 ◽  
Author(s):  
Benjamin Melamed ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Point Process ◽  
Point Processes ◽  
Steady State Distribution ◽  
State Distribution ◽  
Martingale Theory ◽  
Stochastic Intensity ◽  
Sample Paths ◽  
Time Averages ◽  
Definition Of

This paper is a sequel to our previous paper investigating when arrivals see time averages (ASTA) in a stochastic model; i.e., when the steady-state distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the steady-state distribution of the observed process. The relation between the two distributions was also characterized when ASTA does not hold. These results were obtained using the conditional intensity of the point process given the present state of the observed process (assumed to be well defined) and basic properties of Riemann–Stieltjes integrals. Here similar results are obtained using the stochastic intensity associated with the martingale theory of point processes, as in Brémaud (1981). In the martingale framework, the ASTA result is almost an immediate consequence of the definition of a stochastic intensity. In a stationary framework, the results characterize the Palm distribution, but stationarity is not assumed here. Watanabe's (1964) martingale characterization of a Poisson process is also applied to establish a general version of anti–PASTA: if the points of the point process are appropriately generated by the observed process and the observed process is Markov with left-continuous sample paths, then ASTA implies that the point process must be Poisson.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (4) ◽  
pp. 732-739 ◽  
Author(s):  
Bruce M. Brown ◽  
Sidney I. Resnick
Keyword(s):  
Stochastic Processes ◽  
Point Process ◽  
Time Scales ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extreme Values ◽  
Weak Limit ◽  
One Dimensional ◽  
Limit Process ◽  
Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (04) ◽  
pp. 732-739 ◽  
Author(s):  
Bruce M. Brown ◽  
Sidney I. Resnick
Keyword(s):  
Stochastic Processes ◽  
Point Process ◽  
Time Scales ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extreme Values ◽  
Weak Limit ◽  
One Dimensional ◽  
Limit Process ◽  
Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

On arrivals that see time averages: a martingale approach

1990 ◽  
Vol 27 (02) ◽  
pp. 376-384 ◽  
Author(s):  
Benjamin Melamed ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Point Process ◽  
Point Processes ◽  
Steady State Distribution ◽  
State Distribution ◽  
Martingale Theory ◽  
Stochastic Intensity ◽  
Sample Paths ◽  
Time Averages ◽  
Definition Of

This paper is a sequel to our previous paper investigating whenarrivals see time averages(ASTA) in a stochastic model; i.e., when the steady-state distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the steady-state distribution of the observed process. The relation between the two distributions was also characterized when ASTA does not hold. These results were obtained using the conditional intensity of the point process given the present state of the observed process (assumed to be well defined) and basic properties of Riemann–Stieltjes integrals. Here similar results are obtained using the stochastic intensity associated with the martingale theory of point processes, as in Brémaud (1981). In the martingale framework, the ASTA result is almost an immediate consequence of the definition of a stochastic intensity. In a stationary framework, the results characterize the Palm distribution, but stationarity is not assumed here. Watanabe's (1964) martingale characterization of a Poisson process is also applied to establish a general version of anti–PASTA: if the points of the point process are appropriately generated by the observed process and the observed process is Markov with left-continuous sample paths, then ASTA implies that the point process must be Poisson.

scholarly journals Limits of compound and thinned point processes

1975 ◽  
Vol 12 (2) ◽  
pp. 269-278 ◽  
Author(s):  
Olav Kallenberg
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Random Measure ◽  
Poisson Processes ◽  
Limiting Behaviour ◽  
Mutually Independent

Let η =Σjδτj be a point process on some space S and let β,β1,β2, … be identically distributed non-negative random variables which are mutually independent and independent of η. We can then form the compound point process ξ = Σjβjδτj which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ξ as . In the particular case when β takes the values 1 and 0 with probabilities p and 1 –p respectively, ξ becomes a p-thinning of η and our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.

NONPARAMETRIC DETECTION OF DEPENDENCES IN STOCHASTIC POINT PROCESSES

2004 ◽  
Vol 14 (06) ◽  
pp. 1987-1993 ◽  
Author(s):  
ANDREAS KAISER ◽  
THOMAS SCHREIBER
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Single Point ◽  
Crucial Point ◽  
Process Dynamics ◽  
Time Index ◽  
Common Time ◽  
Nonparametric Detection ◽  
Definition Of ◽  
Stochastic Point Processes

A new, parameter-free approach based on information theoretical tools is presented which allows the detection of dependences in the dynamics between two point processes. The crucial point is the definition of sequences of inter-event intervals between the events of two stochastic point processes where these sequences are ordered to only one common time index. This is an enhancement of the concept of event intervals of a single point process and makes the analysis of the process dynamics of more than one point processes possible. An application of this method is also illustrated using a model consisting of two synaptically coupled Hindmarsh–Rose neurons.

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).

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