Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes

2000 ◽  
Vol 32 (03) ◽  
pp. 844-865 ◽  
Author(s):  
Wilfrid S. Kendall ◽  
Jesper Møller
Keyword(s):  
Point Process ◽  
Discrete Time ◽  
Point Processes ◽  
Random Point ◽  
Perfect Simulation ◽  
Stable Point ◽  
Birth And Death Processes ◽  
The Past ◽  
Coupling From The Past ◽  
Equilibrium Distributions

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.

Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes

2000 ◽  
Vol 32 (3) ◽  
pp. 844-865 ◽  
Author(s):  
Wilfrid S. Kendall ◽  
Jesper Møller
Keyword(s):  
Point Process ◽  
Discrete Time ◽  
Point Processes ◽  
Random Point ◽  
Perfect Simulation ◽  
Stable Point ◽  
Birth And Death Processes ◽  
The Past ◽  
Coupling From The Past ◽  
Equilibrium Distributions

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.

scholarly journals Power spectra of random spike fields and related processes

2005 ◽  
Vol 37 (4) ◽  
pp. 1116-1146 ◽  
Author(s):  
Pierre Brémaud ◽  
Laurent Massoulié ◽  
Andrea Ridolfi
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Power Spectra ◽  
Sampling Point ◽  
Birth And Death Processes ◽  
Large Classes ◽  
Random Samples ◽  
Branching Point ◽  
Random Times ◽  
Random Impulse

In this article, we review known results and present new ones concerning the power spectra of large classes of signals and random fields driven by an underlying point process, such as spatial shot noises (with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectra) and signals or fields sampled at random times or points (where the sampling point process is again quite general). We also obtain the Bartlett spectrum for the general linear Hawkes spatial branching point process (with random fertility rate and general immigrant process described by its Bartlett spectrum). We then obtain the Bochner spectra of general spatial linear birth and death processes. Finally, we address the issues of random sampling and linear reconstruction of a signal from its random samples, reviewing and extending former results.

scholarly journals Random point processes and tilings arising from Gaussian analytic functions

2014 ◽  
Vol 70 (a1) ◽  
pp. C523-C523
Author(s):  
Michael Baake ◽  
Holger Koesters ◽  
Robert Moody
Keyword(s):  
Point Process ◽  
Analytic Functions ◽  
Point Processes ◽  
Logarithmic Potential ◽  
Basins Of Attraction ◽  
Random Point ◽  
Specific Class ◽  
Determinantal Point Processes ◽  
Point Sets ◽  
Gaussian Analytic Functions

Getting a grasp of what aperiodic order really entails is going to require collecting and understanding many diverse examples. Aperiodic crystals are at the top of the largely unknown iceberg beneath. Here we present a recently studied form of random point process in the (complex) plane which arises as the sets of zeros of a specific class of analytic functions given by power series with randomly chosen coefficients: Gaussian analytic functions (GAF). These point sets differ from Poisson processes by having a sort of built in repulsion between points, though the resulting sets almost surely fail both conditions of the Delone property. Remarkably the point sets that arise as the zeros of GAFs determine a random point process which is, in distribution, invariant under rotation and translation. In addition, there is a logarithmic potential function for which the zeros are the attractors, and the resulting basins of attraction produce tilings of the plane by tiles which are, almost surely, all of the same area. We discuss GAFs along with their tilings and diffraction, and as well note briefly their relationship to determinantal point processes, which are also of physical interest.

scholarly journals Exact Monte Carlo simulation for fork-join networks

2011 ◽  
Vol 43 (02) ◽  
pp. 484-503 ◽  
Author(s):  
Hongsheng Dai
Keyword(s):  
Monte Carlo ◽  
Response Times ◽  
Simulation Method ◽  
Perfect Simulation ◽  
Analytic Approximations ◽  
The Past ◽  
Coupling From The Past ◽  
Service Stations ◽  
Explicit Formulae ◽  
Queue Lengths

In a fork-join network each incoming job is split into K tasks and the K tasks are simultaneously assigned to K parallel service stations for processing. For the distributions of response times and queue lengths of fork-join networks, no explicit formulae are available. Existing methods provide only analytic approximations for the response time and the queue length distributions. The accuracy of such approximations may be difficult to justify for some complicated fork-join networks. In this paper we propose a perfect simulation method based on coupling from the past to generate exact realisations from the equilibrium of fork-join networks. Using the simulated realisations, Monte Carlo estimates for the distributions of response times and queue lengths of fork-join networks are obtained. Comparisons of Monte Carlo estimates and theoretical approximations are also provided. The efficiency of the sampling algorithm is shown theoretically and via simulation.

scholarly journals Exact Monte Carlo simulation for fork-join networks

2011 ◽  
Vol 43 (2) ◽  
pp. 484-503 ◽  
Author(s):  
Hongsheng Dai
Keyword(s):  
Monte Carlo ◽  
Response Times ◽  
Simulation Method ◽  
Perfect Simulation ◽  
Analytic Approximations ◽  
The Past ◽  
Coupling From The Past ◽  
Service Stations ◽  
Explicit Formulae ◽  
Queue Lengths

In a fork-join network each incoming job is split into K tasks and the K tasks are simultaneously assigned to K parallel service stations for processing. For the distributions of response times and queue lengths of fork-join networks, no explicit formulae are available. Existing methods provide only analytic approximations for the response time and the queue length distributions. The accuracy of such approximations may be difficult to justify for some complicated fork-join networks. In this paper we propose a perfect simulation method based on coupling from the past to generate exact realisations from the equilibrium of fork-join networks. Using the simulated realisations, Monte Carlo estimates for the distributions of response times and queue lengths of fork-join networks are obtained. Comparisons of Monte Carlo estimates and theoretical approximations are also provided. The efficiency of the sampling algorithm is shown theoretically and via simulation.

scholarly journals Attractive regular stochastic chains: perfect simulation and phase transition

2013 ◽  
Vol 34 (5) ◽  
pp. 1567-1586 ◽  
Author(s):  
SANDRO GALLO ◽  
DANIEL Y. TAKAHASHI
Keyword(s):  
Phase Transition ◽  
Infinite Order ◽  
Concentration Of Measure ◽  
Finite Alphabet ◽  
Perfect Simulation ◽  
Exponential Rate ◽  
The Past ◽  
Coupling From The Past ◽  
Probability Kernel ◽  
Independent And Identically Distributed

AbstractWe prove that uniqueness of the stationary chain, or equivalently, of the$g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an independent and identically distributed (i.i.d.) process with countable alphabet; (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson–Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order.

scholarly journals Likelihood-based inference for Matérn type-III repulsive point processes

2009 ◽  
Vol 41 (4) ◽  
pp. 958-977 ◽  
Author(s):  
Mark L. Huber ◽  
Robert L. Wolpert
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Simulation Method ◽  
Perfect Simulation ◽  
Accurate Approximation ◽  
Intensity Parameter ◽  
Type Iii ◽  
Competition For Resources ◽  
As If

In a repulsive point process, points act as if they are repelling one another, leading to underdispersed configurations when compared to a standard Poisson point process. Such models are useful when competition for resources exists, as in the locations of towns and trees. Bertil Matérn introduced three models for repulsive point processes, referred to as types I, II, and III. Matérn used types I and II, and regarded type III as intractable. In this paper an algorithm is developed that allows for arbitrarily accurate approximation of the likelihood for data modeled by the Matérn type-III process. This method relies on a perfect simulation method that is shown to be fast in practice, generating samples in time that grows nearly linearly in the intensity parameter of the model, while the running times for more naive methods grow exponentially.

scholarly journals Likelihood-based inference for Matérn type-III repulsive point processes

2009 ◽  
Vol 41 (04) ◽  
pp. 958-977 ◽  
Author(s):  
Mark L. Huber ◽  
Robert L. Wolpert
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Simulation Method ◽  
Perfect Simulation ◽  
Accurate Approximation ◽  
Intensity Parameter ◽  
Type Iii ◽  
Competition For Resources ◽  
As If

In a repulsive point process, points act as if they are repelling one another, leading to underdispersed configurations when compared to a standard Poisson point process. Such models are useful when competition for resources exists, as in the locations of towns and trees. Bertil Matérn introduced three models for repulsive point processes, referred to as types I, II, and III. Matérn used types I and II, and regarded type III as intractable. In this paper an algorithm is developed that allows for arbitrarily accurate approximation of the likelihood for data modeled by the Matérn type-III process. This method relies on a perfect simulation method that is shown to be fast in practice, generating samples in time that grows nearly linearly in the intensity parameter of the model, while the running times for more naive methods grow exponentially.

scholarly journals Power spectra of random spike fields and related processes

2005 ◽  
Vol 37 (04) ◽  
pp. 1116-1146 ◽  
Author(s):  
Pierre Brémaud ◽  
Laurent Massoulié ◽  
Andrea Ridolfi
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Power Spectra ◽  
Sampling Point ◽  
Birth And Death Processes ◽  
Large Classes ◽  
Random Samples ◽  
Branching Point ◽  
Random Times ◽  
Random Impulse

In this article, we review known results and present new ones concerning the power spectra of large classes of signals and random fields driven by an underlying point process, such as spatial shot noises (with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectra) and signals or fields sampled at random times or points (where the sampling point process is again quite general). We also obtain the Bartlett spectrum for the general linear Hawkes spatial branching point process (with random fertility rate and general immigrant process described by its Bartlett spectrum). We then obtain the Bochner spectra of general spatial linear birth and death processes. Finally, we address the issues of random sampling and linear reconstruction of a signal from its random samples, reviewing and extending former results.

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