This paper introduces the factorial marked temporal point process model and presents efficient learning methods. In conventional (multi-dimensional) marked temporal point process models, event is often encoded by a single discrete variable i.e. a marker. In this paper, we describe the factorial marked point processes whereby time-stamped event is factored into multiple markers. Accordingly the size of the infectivity matrix modeling the effect between pairwise markers is in power order w.r.t. the number of the discrete marker space. We propose a decoupled learning method with two learning procedures: i) directly solving the model based on two techniques: Alternating Direction Method of Multipliers and Fast Iterative Shrinkage-Thresholding Algorithm; ii) involving a reformulation that transforms the original problem into a Logistic Regression model for more efficient learning. Moreover, a sparse group regularizer is added to identify the key profile features and event labels. Empirical results on real world datasets demonstrate the efficiency of our decoupled and reformulated method. The source code is available online.
Point process is a powerful tool for modeling event sequence with timestamp in continuous time space. Early work dates back to the Hawkes processes @cite_24 which shows appropriateness for self-exciting and mutual-exciting process like earthquake and its aftershock @cite_5 @cite_25 . The learning is fulfilled by maximum likelihood estimation by directly computing the gradient and Hessian matrix w.r.t. the log-likelihood function @cite_9 . Recently more modernized machine learning approaches devise new efficient algorithms for learning the parameters of the specified point process. Nonparametric Expectation-Maximization (EM) algorithm is proposed in @cite_15 for multiscale Hawkes Processes using the majorization-minimization framework, which shows superior efficiency and robustness compared with sampling based estimation methods. @cite_18 extends the technique to handle the multi-dimensional Hawkes process by adding a low-rank and sparsity regularization term in the maximum likelihood estimation (MLE) based loss function.
@cite_18 @cite_9 @cite_24 @cite_5 @cite_15 @cite_25
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A maximum likelihood estimation procedure of Hawkes' self-exciting point process model is proposed with explicit presentations of the log-likelihood of the model and its gradient and Hessian. A simulation method of the process is also presented. Some numerical results are given. This paper presents a factorial marked temporal point process model and presents efficient learning methods. In conventional (multi-dimensional) marked temporal point process models, an event is often encoded by a single discrete variable (marker). We describe the factorial marked point processes whereby time-stamped event is factored into multiple markers. Accordingly the size of the infectivity matrix modeling the effect between pairwise markers is in exponential order regarding the number of discrete markers. We propose a decoupled learning method with two learning procedures: i) directly solving the model based on two techniques: Alternating Direction Method of Multipliers and Fast Iterative Shrinkage-Thresholding Algorithm; ii) involving a reformulation that transforms the original problem into a Logistic Regression model for more efficient learning. Moreover, a sparse group regularizer is added to identify the key profile features and event labels. Empirical results on real world datasets demonstrate the efficiency of our decoupled and reformulated method. We propose a Multi-task Multi-dimensional Hawkes Process (MMHP) for modeling event sequences where there exist multiple triggering patterns within sequences and structures across sequences. MMHP is able to model the dynamics of multiple sequences jointly by imposing structural constraints and thus systematically uncover clustering structure among sequences. We propose an effective and robust optimization algorithm to learn MMHP models, which takes advantage of alternating direction method of multipliers (ADMM), majorization minimization and Euler-Lagrange equations. Our experimental results demonstrate that MMHP performs well on both synthetic and real data. Hawkes (1971) introduced a powerful multivariate point process model of mutually exciting processes to explain causal structure in data. In this paper it is shown that the Granger causality structure of such processes is fully encoded in the corresponding link functions of the model. A new nonparametric estimator of the link functions based on a time-discretized version of the point process is introduced by using an infinite order autoregression. Consistency of the new estimator is derived. The estimator is applied to simulated data and to neural spike train data from the spinal dorsal horn of a rat.