Power spectra of random spike fields and related processes

This article reviews known results and contains 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 spectrum), and signals or fields sampled at random times or points (where again the sampling point process is 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 spectrum of general spatial linear birth and death processes. Finally we adress the issue of random sampling, and of the linear reconstruction of a signal from its random samples, reviewing and extending former results.

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Pierre Bremaud, Andrea Ridolfi
2005
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https://infoscience.epfl.ch/record/51032?ln=fr

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Power spectra of random spike fields and related processes

Pierre Bremaud, Andrea Ridolfi

This paper presents general methods for obtaining power spectra of a large class of signals and random fields driven by an underlying point processes, in particular spatial shot noises with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectrum, and signals or fields sampled at random times or points, where again the sampling point process is quite general. The formulas obtained clearly show the interaction between the underlying counting process, the sampled process or the impulse response. 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. Finally we obtain the Cramèr spectrum of general spatial birth and death processes.

2002

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Ultrawide Bandwidth Signals as Shot-Noise: a Unifying Approach

Andrea Ridolfi

This paper presents general methods for obtaining power spectra of a large class of signals and random fields driven by an underlying point processes, in particular spatial shot noises with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectrum, and signals or fields sampled at random times or points, where again the sampling point process is quite general. The formulas obtained clearly show the interaction between the underlying counting process, the sampled process or the impulse response. 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. Finally we obtain the Cram'er spectrum of general spatial birth and death processes.

2005

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Efficient inference for spatial extreme value processes associated to log-Gaussian random functions

Jonathan A. Tawn, Jennifer Lynne Wadsworth

Max-stable processes arise as the only possible nontrivial limits for maxima of affinely normalized identically distributed stochastic processes, and thus form an important class of models for the extreme values of spatial processes. Until recently, inference for max-stable processes has been restricted to the use of pairwise composite likelihoods, due to intractability of higher-dimensional distributions. In this work we consider random fields that are in the domain of attraction of a widely used class of max-stable processes, namely those constructed via manipulation of log-Gaussian random functions. For this class, we exploit limiting d-dimensional multivariate Poisson process intensities of the underlying process for inference on all d-vectors exceeding a high marginal threshold in at least one component, employing a censoring scheme to incorporate information below the marginal threshold. We also consider the d-dimensional distributions for the equivalent max-stable process, and perform full likelihood inference by exploiting the methods of Stephenson & Tawn (2005), where information on the occurrence times of extreme events is shown to dramatically simplify the likelihood. The Stephenson-Tawn likelihood is in fact simply a special case of the censored Poisson process likelihood. We assess the improvements in inference from both methods over pairwise likelihood methodology by simulation.

Oxford University Press2014

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