We propose an extension to Hawkes processes by treating the levels of self-excitation as a stochastic differential equation. Our new point process allows better approximation in application domains where events and intensities accelerate each other with correlated levels of contagion. We generalize a recent algorithm for simulating draws from Hawkes processes whose levels of excitation are stochastic processes, and propose a hybrid Markov chain Monte Carlo approach for model fitting. Our sampling procedure scales linearly with the number of required events and does not require stationarity of the point process. A modular inference procedure consisting of a combination between Gibbs and Metropolis Hastings steps is put forward. We recover expectation maximization as a special case. Our general approach is illustrated for contagion following geometric Brownian motion and exponential Langevin dynamics.
proposed an EM inference algorithm for Hawkes processes and applied to large social network datasets. Inspired by their latent variable set-up, we adapted some of their hidden variable formulation within the marked point process framework into our fully Bayesian inference setting. We have leveraged ideas from previous work on self-exciting processes to consequently treating the levels of excitation as random processes. introduced a multivariate point process combining self (Hawkes) and external (Cox) flavors to study latent networks in the data. These processes have also been proposed and applied in analyzing topic diffusion and user interactions @cite_11 @cite_12 . put forth a temporal point process model with one intensity being modulated by the other. Bounds of self exciting processes are also studied in @cite_4 . Differently from these, we breathe another dimension into Hawkes processes by modeling the contagion parameters as a stochastic differential equation equipped with general procedures for learning. This allows much more latitude in parameterizing the self-exciting processes as a basic building block before incorporating wider families of processes.
@cite_4 @cite_12 @cite_11
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Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive @math -penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Non-asymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activities inference, we finally lead a simulation study for multivariate Hawkes processes and compare our methodology with the adaptive Lasso procedure proposed by Zou in Zou . We observe an excellent behavior of our procedure with respect to the problem of supports recovery. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of Zou , our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience. Diffusion network inference and meme tracking have been two key challenges in viral diffusion. This paper shows that these two tasks can be addressed simultaneously with a probabilistic model involving a mixture of mutually exciting point processes. A fast learning algorithms is developed based on mean-field variational inference with budgeted diffusion bandwidth. The model is demonstrated with applications to the diffusion of viral texts in (1) online social networks (e.g., Twitter) and (2) the blogosphere on the Web. Time plays an essential role in the diffusion of information, influence and disease over networks. In many cases we only observe when a node copies information, makes a decision or becomes infected -- but the connectivity, transmission rates between nodes and transmission sources are unknown. Inferring the underlying dynamics is of outstanding interest since it enables forecasting, influencing and retarding infections, broadly construed. To this end, we model diffusion processes as discrete networks of continuous temporal processes occurring at different rates. Given cascade data -- observed infection times of nodes -- we infer the edges of the global diffusion network and estimate the transmission rates of each edge that best explain the observed data. The optimization problem is convex. The model naturally (without heuristics) imposes sparse solutions and requires no parameter tuning. The problem decouples into a collection of independent smaller problems, thus scaling easily to networks on the order of hundreds of thousands of nodes. Experiments on real and synthetic data show that our algorithm both recovers the edges of diffusion networks and accurately estimates their transmission rates from cascade data.