text: Learning Granger causality for general point processes is a very challenging task. In this paper, we propose an effective method, learning Granger causality, for a special but significant type of point processes --- Hawkes process. We reveal the relationship between Hawkes process's impact function and its Granger causality graph. Specifically, our model represents impact functions using a series of basis functions and recovers the Granger causality graph via group sparsity of the impact functions' coefficients. We propose an effective learning algorithm combining a maximum likelihood estimator (MLE) with a sparse-group-lasso (SGL) regularizer. Additionally, the flexibility of our model allows to incorporate the clustering structure event types into learning framework. We analyze our learning algorithm and propose an adaptive procedure to select basis functions. Experiments on both synthetic and real-world data show that our method can learn the Granger causality graph and the triggering patterns of the Hawkes processes simultaneously.
  @cite_33: We introduce the Piecewise-Constant Conditional Intensity Model, a model for learning temporal dependencies in event streams. We describe a closed-form Bayesian approach to learning these models, and describe an importance sampling algorithm for forecasting future events using these models, using a proposal distribution based on Poisson superposition. We then use synthetic data, supercomputer event logs, and web search query logs to illustrate that our learning algorithm can efficiently learn nonlinear temporal dependencies, and that our importance sampling algorithm can effectively forecast future events.
  @cite_7: The problem of estimating high-dimensional network models arises naturally in the analysis of many biological and socio-economic systems. In this work, we aim to learn a network structure from temporal panel data, employing the framework of Granger causal models under the assumptions of sparsity of its edges and inherent grouping structure among its nodes. To that end, we introduce a group lasso regression regularization framework, and also examine a thresholded variant to address the issue of group misspecification. Further, the norm consistency and variable selection consistency of the estimates are established, the latter under the novel concept of direction consistency. The performance of the proposed methodology is assessed through an extensive set of simulation studies and comparisons with existing techniques. The study is illustrated on two motivating examples coming from functional genomics and financial econometrics.
  @cite_36: 
  @cite_1: Information spreads across social and technological networks, but often the network structures are hidden from us and we only observe the traces left by the diffusion processes, called cascades. Can we recover the hidden network structures from these observed cascades? What kind of cascades and how many cascades do we need? Are there some network structures which are more difficult than others to recover? Can we design efficient inference algorithms with provable guarantees? Despite the increasing availability of cascade data and methods for inferring networks from these data, a thorough theoretical understanding of the above questions remains largely unexplored in the literature. In this paper, we investigate the network structure inference problem for a general family of continuous-time diffusion models using an @math -regularized likelihood maximization framework. We show that, as long as the cascade sampling process satisfies a natural incoherence condition, our framework can recover the correct network structure with high probability if we observe @math cascades, where @math is the maximum number of parents of a node and @math is the total number of nodes. Moreover, we develop a simple and efficient soft-thresholding inference algorithm, which we use to illustrate the consequences of our theoretical results, and show that our framework outperforms other alternatives in practice.
  @cite_0: A new non parametric approach to the problem of testing the independence of two random process is developed. The test statistic is the Hilbert Schmidt Independence Criterion (HSIC), which was used previously in testing independence for i.i.d pairs of variables. The asymptotic behaviour of HSIC is established when computed from samples drawn from random processes. It is shown that earlier bootstrap procedures which worked in the i.i.d. case will fail for random processes, and an alternative consistent estimate of the p-values is proposed. Tests on artificial data and real-world Forex data indicate that the new test procedure discovers dependence which is missed by linear approaches, while the earlier bootstrap procedure returns an elevated number of false positives. The code is available online: this https URL .
  @cite_40: 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.
  @cite_23: We describe an approach to learning causal models that leverages temporal information. We posit the existence of a graphical description of a causal process that generates observations through time. We explore assumptions connecting the graphical description with the statistical process and what one can infer about the causal structure of the process under these assumptions.
  @cite_34: A new class of graphical models capturing the dependence structure of events that occur in time is proposed. The graphs represent so-called local independences, meaning that the intensities of certain types of events are independent of some (but not necessarilly all) events in the past. This dynamic concept of independence is asymmetric, similar to Granger non-causality, so the corresponding local independence graphs differ considerably from classical graphical models. Hence a new notion of graph separation, which is called "d"-separation, is introduced and implications for the underlying model as well as for likelihood inference are explored. Benefits regarding facilitation of reasoning about and understanding of dynamic dependences as well as computational simplifications are discussed. Copyright 2008 Royal Statistical Society.
  @cite_20: A plausible representation of the relational information among entities in dynamic systems such as a living cell or a social community is a stochastic network that is topologically rewiring and semantically evolving over time. Although there is a rich literature in modeling static or temporally invariant networks, little has been done toward recovering the network structure when the networks are not observable in a dynamic context. In this article, we present a machine learning method called TESLA, which builds on a temporally smoothed l1-regularized logistic regression formalism that can be cast as a standard convex-optimization problem and solved efficiently by using generic solvers scalable to large networks. We report promising results on recovering simulated time-varying networks and on reverse engineering the latent sequence of temporally rewiring political and academic social networks from longitudinal data, and the evolving gene networks over >4,000 genes during the life cycle of Drosophila melanogaster from a microarray time course at a resolution limited only by sample frequency.
summary: Many efforts have been made to learn the Granger causality of point processes @cite_23 . For general random processes, a kernel independence test is developed in @cite_0 . Focusing on 1-D point process with simple piecewise constant conditional intensity, a model for capturing temporal dependencies between event types is proposed in @cite_33 . @cite_7 @cite_36 , the inherent grouping structure is considered when learning the Granger casuality on networks from discrete transition process. @cite_1 proposed a continuous-time diffusion network inference method based on parametric cascade generative process. In more general cases, a class of graphical models of marked point processes is proposed in @cite_34 to capture the local independence over various marks. Specializing the work for Hawkes processes, @cite_40 firstly connects Granger causality with impact functions. However, although applying lasso or its variants to capture the intra-structure of nodes @cite_20 is a common strategy, less work has been done on learning causality graph of Hawkes process with sparse-group-lasso as we do, which leads them to be sensitive to noisy and insufficient data.