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Register a new userGaussian approximation of nonlinear Hawkes processes
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Authors
Giovanni Luca Torrisi
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We give a general Gaussian bound for the first chaos (or innovation) of point processes with stochastic intensity constructed by embedding in a bivariate Poisson process. We apply the general result to nonlinear Hawkes processes, providing quantitative central limit theorems.
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In this paper we consider some non linear Hawkes processes with signed reproduction function (or memory kernel) thus exhibiting both self-excitation and inhibition. We provide a Law of Large Numbers, a Central Limit Theorem and large deviation results, as time growths to infinity. The proofs lie on a renewal structure for these processes introduced in Costa et al. (2020) which leads to a comparison with cumulative processes. Explicit computations are made on some examples. Similar results have been obtained in the literature for self-exciting Hawkes processes only.
Hawkes processes are a class of point processes that have the ability to model the self- and mutual-exciting phenomena. Although the classic Hawkes processes cover a wide range of applications, their expressive ability is limited due to three key hypotheses: parametric, linear and homogeneous. Recent work has attempted to address these limitations separately. This work aims to overcome all three assumptions simultaneously by proposing the flexible state-switching Hawkes processes: a flexible, nonlinear and nonhomogeneous variant where a state process is incorporated to interact with the point processes. The proposed model empowers Hawkes processes to be applied to time-varying systems. For inference, we utilize the latent variable augmentation technique to design two efficient Bayesian inference algorithms: Gibbs sampler and mean-field variational inference, with analytical iterative updates to estimate the posterior. In experiments, our model achieves superior performance compared to the state-of-the-art competitors.
Traditionally, Hawkes processes are used to model time--continuous point processes with history dependence. Here we propose an extended model where the self--effects are of both excitatory and inhibitory type and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and our approach dispenses with the necessity of estimating a branching structure for the posterior, as we perform inference on an aggregated sum of Gaussian Processes. Efficient approximate Bayesian inference is achieved via data augmentation, and we describe a mean--field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from three different domains and compare it to previously reported results.
The origin(s) of the ubiquity of Zipfs law (an inverse power law form for the probability density function (PDF) with exponent $1+1$) is still a matter of fascination and investigation in many scientific fields from linguistic, social, economic, computer sciences to essentially all natural sciences. In parallel, self-excited dynamics is a prevalent characteristic of many systems, from seismicity, financial volatility and financial defaults, to sociology, consumer behaviors, computer sciences, The Internet, neuronal discharges and spike trains in biological neuron networks, gene expression and even criminology. Motivated by financial and seismic modelling, we bring the two threads together by introducing a general class of nonlinear self-excited point processes with fast-accelerating intensities as a function of tension. Solving the corresponding master equations, we find that a wide class of such nonlinear Hawkes processes have the PDF of their intensities described by Zipfs law on the condition that (i) the intensity is a fast-accelerating function of tension and (ii) the distribution of the point fertilities is symmetric. This unearths a novel mechanism for Zipfs law, providing a new understanding of its ubiquity.
This paper investigates Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of this point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows to apply results known for Galton-Watson trees. In the present paper, we establish limit theorems for Hawkes process with signed reproduction functions by using renewal techniques. We notably prove exponential concentration inequalities, and thus extend results of Reynaud-Bouret and Roy (2007) which were proved for nonnegative reproduction functions using this cluster representation which is no longer valid in our case. An important step for this is to establish the existence of exponential moments for renewal times of M/G/infinity queues that appear naturally in our problem. These results have their own interest, independently of the original problem for the Hawkes processes.
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