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.

An analysis of the Pólya point process

1983 ◽  
Vol 15 (01) ◽  
pp. 39-53 ◽  
Author(s):  
Ed Waymire ◽  
Vijay K. Gupta
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Critical Value ◽  
General Representation ◽  
Infinitely Divisible ◽  
Generating Functional ◽  
Representation Problem ◽  
Polya Process ◽  
Pólya Process ◽  
Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible 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.

On a class of parabolic differential equations driven with stochastic point processes

1975 ◽  
Vol 12 (01) ◽  
pp. 98-106
Author(s):  
K. Gopalsamy ◽  
A. T. Bharucha-Reid
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Point Process ◽  
Point Processes ◽  
Boundary Value ◽  
Parabolic Differential Equation ◽  
Parabolic Differential Equations ◽  
Stochastic Point Process ◽  
Stochastic Point Processes

This paper is concerned with the solution of an initial and boundary value problem for a parabolic differential equation driven by a stochastic point process.

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.

An analysis of the Pólya point process

1983 ◽  
Vol 15 (1) ◽  
pp. 39-53 ◽  
Author(s):  
Ed Waymire ◽  
Vijay K. Gupta
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Critical Value ◽  
General Representation ◽  
Infinitely Divisible ◽  
Generating Functional ◽  
Representation Problem ◽  
Polya Process ◽  
Pólya Process ◽  
Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

scholarly journals Analysis, Synthesis, and Estimation of Fractal-Rate Stochastic Point Processes

Fractals ◽  
1997 ◽  
Vol 05 (04) ◽  
pp. 565-595 ◽  
Author(s):  
Stefan Thurner ◽  
Steven B. Lowen ◽  
Markus C. Feurstein ◽  
Conor Heneghan ◽  
Hans G. Feichtinger ◽  
...  
Keyword(s):  
Point Process ◽  
Action Potentials ◽  
Point Processes ◽  
Computer Network ◽  
Amorphous Semiconductors ◽  
Data Sets ◽  
Different Types ◽  
Design Values ◽  
Low Jitter ◽  
Stochastic Point Processes

Fractal and fractal-rate stochastic point processes (FSPPs and FRSPPs) provide useful models for describing a broad range of diverse phenomena, including electron transport in amorphous semiconductors, computer-network traffic, and sequences of neuronal action potentials. A particularly useful statistic of these processes is the fractal exponent α, which may be estimated for any FSPP or FRSPP by using a variety of statistical methods. Simulated FSPPs and FRSPPs consistently exhibit bias in this fractal exponent, however, rendering the study and analysis of these processes non-trivial. In this paper, we examine the synthesis and estimation of FRSPPs by carrying out a systematic series of simulations for several different types of FRSPP over a range of design values for α. The discrepancy between the desired and achieved values of α is shown to arise from finite data size and from the character of the point-process generation mechanism. In the context of point-process simulation, reduction of this discrepancy requires generating data sets with either a large number of points, or with low jitter in the generation of the points. In the context of fractal data analysis, the results presented here suggest caution when interpreting fractal exponents estimated from experimental data sets.

On a class of parabolic differential equations driven with stochastic point processes

1975 ◽  
Vol 12 (1) ◽  
pp. 98-106 ◽  
Author(s):  
K. Gopalsamy ◽  
A. T. Bharucha-Reid
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Point Process ◽  
Point Processes ◽  
Boundary Value ◽  
Parabolic Differential Equation ◽  
Parabolic Differential Equations ◽  
Stochastic Point Process ◽  
Stochastic Point Processes

This paper is concerned with the solution of an initial and boundary value problem for a parabolic differential equation driven by a stochastic point process.

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.

SEGMENTED POINT PROCESS MODELS FOR MAINTAINED SYSTEMS

2007 ◽  
Vol 14 (05) ◽  
pp. 431-458 ◽  
Author(s):  
A. SYAMSUNDAR ◽  
V. N. A. NAIKAN
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Single Point ◽  
Process Models ◽  
Parameter Estimates ◽  
Minimal Repair ◽  
Homogeneous Poisson Process ◽  
Stable Conditions ◽  
Point Process Models

A maintained system is generally modeled using point processes. The most common processes used are the renewal process and the non homogeneous Poisson process corresponding to maximal and minimal repair situations with homogeneous Poisson process being a special case of both. A general repair formulation with a factor indicating the degree of repair is introduced into the minimal repair model to form an Arithmetic Reduction of Intensity model. These processes are generally able to model maintained systems with a fair degree of accuracy when the system is operating under stable conditions. However whenever there is a change in the environment these models which are monotonic in nature are not able to accommodate this change. Such systems operating under different environments need to be modelled by segmented models with the system domain divided into segments at the points of changes in the environment. The individual segments can then be modeled by any of the above point process models and these can be combined to form a composite model. This paper proposes a statistical model of such an operating/maintenance environment. Its purpose is to quantify the impacts of changes in the environment on the failure intensities. Field data from an industrial-setting demonstrate that appropriate parameter estimates for such phenomena can be obtained and such models are shown to more accurately describe the maintained system in a changing environment than the single point process models usually used.

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