Temporal point process is widely used for sequential data modeling. In this paper, we focus on the problem of modeling sequential event propagation in graph, such as retweeting by social network users, news transmitting between websites, etc. Given a collection of event propagation sequences, conventional point process model consider only the event history, i.e. embed event history into a vector, not the latent graph structure. We propose a Graph Biased Temporal Point Process (GBTPP) leveraging the structural information from graph representation learning, where the direct influence between nodes and indirect influence from event history is modeled respectively. Moreover, the learned node embedding vector is also integrated into the embedded event history as side information. Experiments on a synthetic dataset and two real-world datasets show the efficacy of our model compared to conventional methods and state-of-the-art.
Second, Temporal point processes with conditional intensity functions is a more general framework for sequential event data modeling. Temporal Point Process (TPP) is powerful for modeling event sequence with time-stamp in continuous time space. Early work dates back to the Hawkes processes @cite_40 which shows appropriateness for self-exciting and mutual-exciting process like earthquake and its aftershock @cite_15 @cite_26 . As an effective model for event sequence modeling, TPP has widely used in various applications, including data mining tasks e.g. social infectivity learning @cite_37 , conflict analysis @cite_7 , crime modeling @cite_31 , email network analytics @cite_39 and extremal behavior of stock price @cite_23 , and event prediction tasks e.g. failure prediction @cite_21 , sales outcome forecasting @cite_2 , literature citation prediction @cite_11 .
@cite_37 @cite_26 @cite_31 @cite_7 @cite_21 @cite_39 @cite_40 @cite_23 @cite_2 @cite_15 @cite_11
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Large volumes of event data are becoming increasingly available in a wide variety of applications, such as healthcare analytics, smart cities and social network analysis. The precise time interval or the exact distance between two events carries a great deal of information about the dynamics of the underlying systems. These characteristics make such data fundamentally different from independently and identically distributed data and time-series data where time and space are treated as indexes rather than random variables. Marked temporal point processes are the mathematical framework for modeling event data with covariates. However, typical point process models often make strong assumptions about the generative processes of the event data, which may or may not reflect the reality, and the specifically fixed parametric assumptions also have restricted the expressive power of the respective processes. Can we obtain a more expressive model of marked temporal point processes? How can we learn such a model from massive data? In this paper, we propose the Recurrent Marked Temporal Point Process (RMTPP) to simultaneously model the event timings and the markers. The key idea of our approach is to view the intensity function of a temporal point process as a nonlinear function of the history, and use a recurrent neural network to automatically learn a representation of influences from the event history. We develop an efficient stochastic gradient algorithm for learning the model parameters which can readily scale up to millions of events. Using both synthetic and real world datasets, we show that, in the case where the true models have parametric specifications, RMTPP can learn the dynamics of such models without the need to know the actual parametric forms; and in the case where the true models are unknown, RMTPP can also learn the dynamics and achieve better predictive performance than other parametric alternatives based on particular prior assumptions. Determinantal point processes (DPP) serve as a practicable modeling for many applications of repulsive point processes. A known approach for simulation was proposed in Hough(2006) , which generate the desired distribution point wise through rejection sampling. Unfortunately, the size of rejection could be very large. In this paper, we investigate the application of perfect simulation via coupling from the past (CFTP) on DPP. We give a general framework for perfect simulation on DPP model. It is shown that the limiting sequence of the time-to-coalescence of the coupling is bounded by @math An application is given to the stationary models in DPP. Event sequence, asynchronously generated with random timestamp, is ubiquitous among applications. The precise and arbitrary timestamp can carry important clues about the underlying dynamics, and has lent the event data fundamentally different from the time-series whereby series is indexed with fixed and equal time interval. One expressive mathematical tool for modeling event is point process. The intensity functions of many point processes involve two components: the background and the effect by the history. Due to its inherent spontaneousness, the background can be treated as a time series while the other need to handle the history events. In this paper, we model the background by a Recurrent Neural Network (RNN) with its units aligned with time series indexes while the history effect is modeled by another RNN whose units are aligned with asynchronous events to capture the long-range dynamics. The whole model with event type and timestamp prediction output layers can be trained end-to-end. Our approach takes an RNN perspective to point process, and models its background and history effect. For utility, our method allows a black-box treatment for modeling the intensity which is often a pre-defined parametric form in point processes. Meanwhile end-to-end training opens the venue for reusing existing rich techniques in deep network for point process modeling. We apply our model to the predictive maintenance problem using a log dataset by more than 1000 ATMs from a global bank headquartered in North America. This paper presents an inference algorithm that can discover temporal logic properties of a system from data. Our algorithm operates on finite time system trajectories that are labeled according to whether or not they demonstrate some desirable system properties (e.g. "the car successfully stops before hitting an obstruction"). A temporal logic formula that can discriminate between the desirable behaviors and the undesirable ones is constructed. The formulae also indicate possible causes for each set of behaviors (e.g. "If the speed of the car is greater than 15 m s within 0.5s of brake application, the obstruction will be struck") which can be used to tune designs or to perform on-line monitoring to ensure the desired behavior. We introduce reactive parameter signal temporal logic (rPSTL), a fragment of parameter signal temporal logic (PSTL) that is expressive enough to capture causal, spatial, and temporal relationships in data. We define a partial order over the set of rPSTL formulae that is based on language inclusion. This order enables a directed search over this set, i.e. given a candidate rPSTL formula that does not adequately match the observed data, we can automatically construct a formula that will fit the data at least as well. Two case studies, one involving a cattle herding scenario and one involving a stochastic hybrid gene circuit model, are presented to illustrate our approach.