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We propose a new Kalikow decomposition for continuous-time multivariate counting processes, on potentially infinite networks. We prove the existence of such a decomposition in various cases. This decomposition allows us to derive simulation algorithms that hold either for stationary processes with potentially infinite network but bounded intensities, or for processes with unbounded intensities in a finite network and with empty past before zero. The Kalikow decomposition is not unique, and we discuss the choice of the decomposition in terms of algorithmic efficiency in certain cases. We apply these methods to several examples: the linear Hawkes process, the age-dependent Hawkes process, the exponential Hawkes process, and the Galves–Löcherbach process.

When fitting the learning data of an individual to algorithm-like learning models, the observations are so dependent and non-stationary that one may wonder what the classical Maximum Likelihood Estimator (MLE) could do, even if it is the usual tool applied to experimental cognition. Our objective in this work is to show that the estimation of the learning rate cannot be efficient if the learning rate is constant in the classical Exp3 (Exponential weights for Exploration and Exploitation) algorithm. Secondly, we show that if the learning rate decreases polynomially with the sample size, then the prediction error and in some cases the estimation error of the MLE satisfy bounds in probability that decrease at a polynomial rate.

We develop several statistical tests of the determinant of the diffusion coefficient of a stochastic differential equation, based on discrete observations on a time interval $[0,T]$ sampled with a time step $\Delta$. Our main contribution is to control the test Type I and Type II errors in a non asymptotic setting, i.e. when the number of observations and the time step are fixed. The test statistics are calculated from the process increments. In dimension 1, the density of the test statistic is explicit. In dimension 2, the test statistic has no explicit density but upper and lower bounds are proved. We also propose a multiple testing procedure in dimension greater than 2. Every test is proved to be of a given non-asymptotic level and separability conditions to control their power are also provided. A numerical study illustrates the properties of the tests for stochastic processes with known or estimated drifts.

We propose a simple network of Hawkes processes as a cognitive model capable of learning to classify objects. Our learning algorithm, named EWAK for Exponentially Weighted Average and Kalikow decomposition, is based on a local synaptic learning rule based on firing rates at each output node. We were able to use local regret bounds to prove mathematically that the network is able to learn on average and even asymptotically under more restrictive assumptions.

We present here a new algorithm based on a random model for simulating efficiently large brain neuronal networks. Model parameters (mean firing rate, number of neurons, synaptic connection probability and postsynaptic duration) are easy to calibrate further on real data experiments. Based on time asynchrony assumption, both computational and memory complexities are proved to be theoretically linear with the number of neurons. These results are experimentally validated by sequential simulations of millions of neurons and billions of synapses in few minutes on a single processor desktop computer.

We derive new discrete event simulation algorithms for marked time point processes. The main idea is to couple a special structure, namely the associated local independence graph, as defined by Didelez arXiv:0710.5874, with the activity tracking algorithm [muzy, 2019] for achieving high performance asynchronous simulations. With respect to classical algorithm, this allows reducing drastically the computational complexity, especially when the graph is sparse. [muzy, 2019] A. Muzy. 2019. Exploiting activity for the modeling and simulation of dynamics and learning processes in hierarchical (neurocognitive) systems. (Submitted to) Magazine of Computing in Science & Engineering (2019)

We derive new discrete event simulation algorithms for marked time point processes. The main idea is to couple a special structure, namely the associated local independence graph, as defined by Didelez arXiv:0710.5874, with the activity tracking algorithm [muzy, 2019] for achieving high performance asynchronous simulations. With respect to classical algorithm, this allows reducing drastically the computational complexity, especially when the graph is sparse. [muzy, 2019] A. Muzy. 2019. Exploiting activity for the modeling and simulation of dynamics and learning processes in hierarchical (neurocognitive) systems. (Submitted to) Magazine of Computing in Science & Engineering (2019)

Given a times series ${\bf Y}$ in $\mathbb{R}^n$, with a piece-wise contant mean and independent components, the twin problems of change-point detection and change-point localization respectively amount to detecting the existence of times where the mean varies and estimating the positions of those change-points. In this work, we tightly characterize optimal rates for both problems and uncover the phase transition phenomenon from a global testing problem to a local estimation problem. Introducing a suitable definition of the energy of a change-point, we first establish in the single change-point setting that the optimal detection threshold is $\sqrt{2\log\log(n)}$. When the energy is just above the detection threshold, then the problem of localizing the change-point becomes purely parametric: it only depends on the difference in means and not on the position of the change-point anymore. Interestingly, for most change-point positions, it is possible to detect and localize them at a much smaller energy level. In the multiple change-point setting, we establish the energy detection threshold and show similarly that the optimal localization error of a specific change-point becomes purely parametric. Along the way, tight optimal rates for Hausdorff and $l_1$ estimation losses of the vector of all change-points positions are also established. Two procedures achieving these optimal rates are introduced. The first one is a least-squares estimator with a new multiscale penalty that favours well spread change-points. The second one is a two-step multiscale post-processing procedure whose computational complexity can be as low as $O(n\log(n))$. Notably, these two procedures accommodate with the presence of possibly many low-energy and therefore undetectable change-points and are still able to detect and localize high-energy change-points even with the presence of those nuisance parameters., Comment: 73 pages

Inspired by Kalikow-type decompositions, we introduce a new stochastic model of infinite neuronal networks, for which we establish sharp oracle inequalities for Lasso methods and restricted eigenvalue properties for the associated Gram matrix with high probability. These results hold even if the network is only partially observed. The main argument rely on the fact that concentration inequalities can easily be derived whenever the transition probabilities of the underlying process admit a sparse space-time representation., Comment: 40 pages

International audience; In this note, we prove some non-asymptotic concentration inequalities for functionals, called innovations, of inhomogeneous Neymann-Scott point processes, a particular class of spatial point process models. Innovation is a functional built from the counting measure minus its integral compensator. The result is then applied to obtain almost sure rate of convergence for such functionals.

International audience

Background: Statistical models that predict neuron spike occurrence from the earlier spiking activity of the whole recorded network are promising tools to reconstruct functional connectivity graphs. Some of the previously used methods were in the general statistical framework of the multivariate Hawkes processes but they often required huge amount of data, prior knowledge about the recorded network, and may generate non stationary models that could not be directly used in simulation. New Method: Here, we present a method, based on least-square estimators and LASSO penalty criteria, optimizing Hawkes models that can be used for simulation. Results: Challenging our method to multiple Integrate and Fire models of neuron networks demonstrated that it eciently detects both excitatory and inhibitory connections. The few errors that occasionally occurred with complex networks including common inputs, weak and chained connections, could easily be discarded based on objective criteria. Conclusions: The present method is robust, stable, applicable with an experimentally realistic amount of data, and does not require any prior knowledge of the studied network. Therefore, it can be used on a personal computer as a turn-key procedure to infer connectivity graphs and generate simulation models from simultaneous spike train recordings

We investigate several distribution free dependence detection procedures, mainly based on bootstrap principles and their approximation properties. Thanks to this study, we introduce a new distribution free Unitary Events (UE) method, named Permutation UE, which consists in a multiple testing procedure based on permutation and delayed coincidence count. Each involved single test of this procedure achieves the prescribed level, so that the corresponding multiple testing procedure controls the False Discovery Rate (FDR), and this with as few assumptions as possible on the underneath distribution. Some simulations show that this method outperforms the trial-shuffling and the MTGAUE method in terms of single levels and FDR, for a comparable amount of false negatives. Application on real data is also provided., Comment: 45 pages, 8 figures

In this thesis, we want to adapt model selection technics tothe problem of estimating the intensity of point processes. More precisely, wewant to prove that penalized projection estimators of the intensity areadaptive either in a family of projection estimators or for the minimax risk.We restrict ourselves to the inhomogeneous Poisson processes and to thecounting processes with Aalen's multiplicative intensity.In both cases, we want to prove an oracle type inequality whichshows that penalized projection estimators do quite as well as the bestprojection estimator for a given family of models. The key to derive oracletype inequalities is the concentration phenomena and more precisely theknowledge of some exponential inequalities which allow us to control inprobability the deviations of chi-square type statistics upon their mean. Weprove two different concentration inequalities. The first one is only validfor Poisson processes. It has the same orders of magnitude as the one due toM. Talagrand for suprema of empirical processes. The second one does notachieve these orders of magnitude but is valid for more general countingprocesses.This last result uses martingales technics which can be applied alsoto derive concentration inequalities for degenerate U-statistics of order 2andalso for double integrals with respect to a centred Poisson measure. We also compute some lower bounds for minimax risks and we showthat penalized projection estimators reach these lower bounds up to someconstants., L'objet de cette thèse est d'adapter des techniquesde sélection de modèle au cadre particulier de l'estimation d'intensité deprocessus ponctuels. Plus précisément, nous voulons montrer que lesestimateurs par projection pénalisés de l'intensité sont adaptatifs soit dansune famille d'estimateurs par projection, soit pour le risque minimax. Nousnous sommes restreints à deux cas particuliers : les processus de Poissoninhomogènes et les processus de comptage à intensitémultiplicative d'Aalen.Dans les deux cas, nous voulons trouver une inégalité de typeoracle, qui garantit que les estimateurs par projection pénalisés ont un risquedu même ordre de grandeur que le meilleur estimateur par projection pour unefamille de modèles donnés. La clé qui permet de prouver des inégalités detype oracle est le phénomène de concentration de la mesure ou plus précisémentla connaissance d'inégalités exponentielles, qui permettent de contrôler enprobabilité les déviations de statistiques de type khi-deux au dessus de leurmoyenne. Nous avons prouvé deux types d'inégalités de concentration. Lapremière n'est valable que pour les processus de Poisson. Elle est comparableen terme d'ordre de grandeur à l'inégalité de M. Talagrand pour les suprema deprocessus empiriques. La deuxième est plus grossière mais elle est valablepour des processus de comptage beaucoup plus généraux.Cette dernière inégalité met en oeuvre des techniques demartingales dont nous nous sommes inspirés pour prouver des inégalités deconcentration pour des U-statistiques dégénérées d'ordre 2 ainsi que pour desintégrales doubles par rapport à une mesure de Poisson recentrée.Nous calculons aussi certaines bornes inférieures pour lesrisques minimax et montrons que les estimateurs par projection pénalisésatteignent ces vitesses.

En neurosciences, le principal objet d'étude est le train de spike car il est considéré comme le vecteur principal de transmission de l'information de l'activité cérébrale. Au fil des différentes études, plusieurs modélisations pour les trains de spikes ont été proposées, plus pour des raisons biologiques que mathématiques. Nous proposons ici des procédures statistiques permettant de tester les diverses modélisations.

International audience; Considering either two independent i.i.d. samples, or two independent samples generated from a heteroscedastic regression model, or two independent Poisson processes, we address the question of testing equality of their respective distributions. We first propose single testing procedures based on a general symmetric kernel. The corresponding critical values are chosen from a wild or permutation bootstrap approach, and the obtained tests are exactly (and not just asymptotically) of level . We then introduce an aggregation method, which enables to overcome the difficulty of choosing a kernel and/or the parameters of the kernel. We derive non-asymptotic properties for the aggregated tests, proving that they may be optimal in a classical statistical sense.

In this paper, we deal with the problem of calibrating thresholding rules in the setting of Poisson intensity estimation. By using sharp concentration inequalities, oracle inequalities are derived and we establish the optimality of our estimate up to a logarithmic term. This result is proved under mild assumptions and we do not impose any condition on the support of the signal to be estimated. Our procedure is based on data-driven thresholds. As usual, they depend on a threshold parameter $\gamma$ whose optimal value is hard to estimate from the data. Our main concern is to provide some theoretical and numerical results to handle this issue. In particular, we establish the existence of a minimal threshold parameter from the theoretical point of view: taking $\gamma, Comment: 10 figures

The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\tilde{f}_{n,\gamma}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a logarithmic term. Oracle inequalities allow to derive the maxiset of $\tilde{f}_{n,\gamma}$. Then, minimax properties of $\tilde{f}_{n,\gamma}$ are established. We first prove that the rate of this estimator on Besov spaces ${\cal B}^\al_{p,q}$ when $p\leq 2$ is $(\ln(n)/n)^{\al/(1+2\al)}$. This result has two consequences. First, it establishes that the minimax rate of Besov spaces ${\cal B}^\al_{p,q}$ with $p\leq 2$ when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. This result is new. Furthermore, $\tilde{f}_{n,\gamma}$ is adaptive minimax up to a logarithmic term. When $p>2$, the situation changes dramatically and the rate of $\tilde{f}_{n,\gamma}$ on Besov spaces ${\cal B}^\al_{p,q}$ is worse than $(\ln(n)/n)^{\al/(1+2\al)}$. Finally, the random threshold depends on a parameter $\gamma$ that has to be suitably chosen in practice. Some theoretical results provide upper and lower bounds of $\gamma$ to obtain satisfying oracle inequalities. Simulations reinforce these results.

In contemporary applied and computational mathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent random matrices. This quantity is controlled by the norm of the expected square of the random matrix and the expectation of the maximum squared norm achieved by one of the summands; there is also a weak dependence on the dimension of the random matrix. The purpose of this paper is to give a complete, elementary proof of this important inequality.