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Arnaud Poinas

Publications and source records attributed to Arnaud Poinas.

7 recordsLinked to original sources
arXiv2026

On two fundamental properties of the zeros of spectrograms of noisy signals

The spatial distribution of the zeros of the spectrogram is significantly altered when a signal is added to white Gaussian noise. The zeros tend to delineate the support of the signal, and deterministic structures form in the presence of interference, as if the zeros were trapped. While sophisticated methods have been proposed to detect signals as holes in the pattern of spectrogram zeros, few formal arguments have been made to support the delineation and trapping effects. Through detailed computations for simple toy signals, we show that two basic mathematical arguments, the intensity of zeros and Rouché's theorem, allow discussing delineation and trapping, and the influence of parameters like the signal-to-noise ratio. In particular, interfering chirps, even nearly superimposed, yield an easy-to-detect deterministic structure among zeros.

eess.SP↗
arXiv2024

A central limit theorem for a sequence of conditionally centered random fields

A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain conditional $α$-mixing conditions in space or time. Exploiting conditional centering and the space-time structure, the limiting normal distribution is obtained for increasing spatial domain, increasing length of the sequence, or both of these. The theorem is very well suited for establishing asymptotic normality in the context of unbiased estimating function inference for a wide range of space-time processes. This is pertinent given the abundance of space-time data. Two examples demonstrate the applicability of the theorem.

math.PR↗
arXiv2022

Asymptotic approximation of the likelihood of stationary determinantal point processes

Continuous determinantal point processes (DPPs) are a class of repulsive point processes on $\mathbb{R}^d$ with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behaviour when the observation window grows towards $\mathbb{R}^d$. This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on $\mathbb{R}^d$ and compare favourably to common alternative estimation methods for continuous DPPs.

math.ST↗
arXiv2021

On proportional volume sampling for experimental design in general spaces

Optimal design for linear regression is a fundamental task in statistics. For finite design spaces, recent progress has shown that random designs drawn using proportional volume sampling (PVS) lead to approximation guarantees for A-optimal design. PVS strikes the balance between design nodes that jointly fill the design space, while marginally staying in regions of high mass under the solution of a relaxed convex version of the original problem. In this paper, we examine some of the statistical implications of a new variant of PVS for (possibly Bayesian) optimal design. Using point process machinery, we treat the case of a generic Polish design space. We show that not only are the A-optimality approximation guarantees preserved, but we obtain similar guarantees for D-optimal design that tighten recent results. Moreover, we show that PVS can be sampled in polynomial time. Unfortunately, in spite of its elegance and tractability, we demonstrate on a simple example that the practical implications of general PVS are likely limited. In the second part of the paper, we focus on applications and investigate the use of PVS as a subroutine for stochastic search heuristics. We demonstrate that PVS is a robust addition to the practitioner's toolbox, especially when the regression functions are nonstandard and the design space, while low-dimensional, has a complicated shape (e.g., nonlinear boundaries, several connected components).

stat.CO↗
arXiv2019

Adaptive estimating function inference for non-stationary determinantal point processes

Estimating function inference is indispensable for many common point process models where the joint intensities are tractable while the likelihood function is not. In this paper we establish asymptotic normality of estimating function estimators in a very general setting of non-stationary point processes. We then adapt this result to the case of non-stationary determinantal point processes which are an important class of models for repulsive point patterns. In practice often first and second order estimating functions are used. For the latter it is common practice to omit contributions for pairs of points separated by a distance larger than some truncation distance which is usually specified in an ad hoc manner. We suggest instead a data-driven approach where the truncation distance is adapted automatically to the point process being fitted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies for non-stationary determinantal point processes and by an application to a real dataset.

math.ST↗
arXiv2018

Mixing properties and central limit theorem for associated point processes

Positively (resp. negatively) associated point processes are a class of point processes that induce attraction (resp. inhibition) between the points. As an important example, determinantal point processes (DPPs) are negatively associated. We prove $α$-mixing properties for associated spatial point processes by controlling their $α$-coefficients in terms of the first two intensity functions. A central limit theorem for functionals of associated point processes is deduced, using both the association and the $α$-mixing properties. We discuss in detail the case of DPPs, for which we obtain the limiting distribution of sums, over subsets of close enough points of the process, of any bounded function of the DPP. As an application, we get the asymptotic properties of the parametric two-step estimator of some inhomogeneous DPPs.

math.ST↗