A variety of real-world processes (over networks) produce sequences of data whose complex temporal dynamics need to be studied. More especially, the event timestamps can carry important information about the underlying network dynamics, which otherwise are not available from the time-series evenly sampled from continuous signals. Moreover, in most complex processes, event sequences and evenly-sampled times series data can interact with each other, which renders joint modeling of those two sources of data necessary. To tackle the above problems, in this paper, we utilize the rich framework of (temporal) point processes to model event data and timely update its intensity function by the synergic twin Recurrent Neural Networks (RNNs). In the proposed architecture, the intensity function is synergistically modulated by one RNN with asynchronous events as input and another RNN with time series as input. Furthermore, to enhance the interpretability of the model, the attention mechanism for the neural point process is introduced. The whole model with event type and timestamp prediction output layers can be trained end-to-end and allows a black-box treatment for modeling the intensity. We substantiate the superiority of our model in synthetic data and three real-world benchmark datasets.
3) @cite_50 : Recently, Hawkes process has received a wide attention in network cascades modeling @cite_38 @cite_49 , community structure @cite_47 , viral diffusion and activity shaping @cite_25 , criminology @cite_18 , optimization and intervention in social networks @cite_31 , recommendation systems @cite_3 , and verification of crowd generated data @cite_37 . As an illustration example intensively used in this paper, we particularly write out its intensity function is: where @math is the infectivity matrix, indicating the directional influence strength from dimension @math to @math . It explicitly uses a triggering term to model the excitation effect from history events where the parameter @math denotes the decaying bandwidth. The model is originally motivated to analyze the earthquake and its aftershocks @cite_8 .
@cite_38 @cite_18 @cite_31 @cite_37 @cite_8 @cite_3 @cite_49 @cite_50 @cite_47 @cite_25
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IN contagious processes (e.g. measles, hijacking, etc.) the occurrence of events increases the probability of further events occurring in the near future. Also several series of events may interact with each other, for example one might consider notifications of some disease in a number of adjacent regions which would interact through infectives or carriers moving between the regions. In this paper we postulate a model for such processes and derive a general expression for the point spectral matrices. These theoretical spectra are useful for comparison with spectra estimated from data and thus provide a means of evaluating the fit of such a model in the manner of Bartlett (1963). The model studied was put forward in an earlier paper (Hawkes, 1971) but the solution was obtained only in special cases. In this paper an elegant solution is obtained for the. general case. Consider a stationary k-variate point process N(t), where Ni(t) represents the cumulative number of events in the ith process up to time t, with intensity vector X = of dN(t) dt and covariance density matrix Abstract The ability to make decisions based on quantities of interest that depend on variables inferred from measurement finds application in different fields of mechanics and physics. The evaluation of the inferred variables, and hence the quantities of interest, from the measurement typically requires the solution of an inverse problem. For example, in medical imaging the elastic heterogeneity of a tumor and its nonlinear elastic response can be used to distinguish benign tumors from their malignant counterparts. These images of linear and nonlinear elastic parameters of tissue are typically obtained by using a measured displacement field and solving a complex inverse elasticity problem. In this paper we consider circumventing the solution of the inverse problem by using measured displacements as input to a deep convolutional neural network (CNN) and training it to classify tumors on the basis of their elastic heterogeneity and nonlinearity. For a simple, 5-layer CNN trained with 8,000 samples for heterogeneity, and a 4-layer CNN trained with 4,000 samples for nonlinear elasticity we report classification accuracies in the range of 99 . 7 − 99 . 9 . The training and testing data are both obtained from the forward solution of finite element models of samples. We also analyze the weights of the trained model to understand the process through which the network extracts features of elastic moduli from the input displacement images. Finally, we apply the nonlinear elasticity classifier, which is trained entirely using simulated data, to displacement images obtained from ten patients with breast lesions and note that it correctly classifies eight out of ten cases. This application illustrates how data from physics-based models can be used in improving the performance of a data-driven algorithm in data-sparse scenarios. Learning the influence structure of multiple time series data is of great interest to many disciplines. This paper studies the problem of recovering the causal structure in network of multivariate linear Hawkes processes. In such processes, the occurrence of an event in one process affects the probability of occurrence of new events in some other processes. Thus, a natural notion of causality exists between such processes captured by the support of the excitation matrix. We show that the resulting causal influence network is equivalent to the Directed Information graph (DIG) of the processes, which encodes the causal factorization of the joint distribution of the processes. Furthermore, we present an algorithm for learning the support of excitation matrix (or equivalently the DIG). The performance of the algorithm is evaluated on synthesized multivariate Hawkes networks as well as a stock market and MemeTracker real-world dataset. Efficient and effective learning of social infectivity presents a critical challenge in modeling diffusion phenomena in social networks and other applications. Existing methods require substantial amount of event cascades to guarantee the learning accuracy and they only consider time-invariant infectivity. Our paper overcomes those two drawbacks by constructing a more compact model and parameterizing the infectivity using time-varying features, thus dramatically reduces the data requirement, and enables the learning of time-varying infectivity which also takes into account the underlying network topology. We replace the pairwise infectivity in the multidimensional Hawkes processes with linear combinations of those time-varying features, and optimize the associated coefficients with lasso-type of regularization. To efficiently solve the resulting optimization problem, we employ the technique of alternating direction method of multipliers which allows independent updating of the individual coefficients by optimizing a surrogate function upper-bounding the original objective function. On both synthetic and real world data, the proposed method performs better than alternatives in terms of both recovering the hidden diffusion network and predicting the occurrence time of social events.