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Register a new userSurfacing Estimation Uncertainty in the Decay Parameters of Hawkes Processes with Exponential Kernels
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Added by
Tiago Santos
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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As a tool for capturing irregular temporal dependencies (rather than resorting to binning temporal observations to construct time series), Hawkes processes with exponential decay have seen widespread adoption across many application domains, such as predicting the occurrence time of the next earthquake or stock market spike. However, practical applications of Hawkes processes face a noteworthy challenge: There is substantial and often unquantified variance in decay parameter estimations, especially in the case of a small number of observations or when the dynamics behind the observed data suddenly change. We empirically study the cause of these practical challenges and we develop an approach to surface and thereby mitigate them. In particular, our inspections of the Hawkes process likelihood function uncover the properties of the uncertainty when fitting the decay parameter. We thus propose to explicitly capture this uncertainty within a Bayesian framework. With a series of experiments with synthetic and real-world data from domains such as classical earthquake modeling or the manifestation of collective emotions on Twitter, we demonstrate that our proposed approach helps to quantify uncertainty and thereby to understand and fit Hawkes processes in practice.
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There has recently been a concerted effort to derive mechanisms in vision and machine learning systems to offer uncertainty estimates of the predictions they make. Clearly, there are enormous benefits to a system that is not only accurate but also has a sense for when it is not sure. Existing proposals center around Bayesian interpretations of modern deep architectures -- these are effective but can often be computationally demanding. We show how classical ideas in the literature on exponential families on probabilistic networks provide an excellent starting point to derive uncertainty estimates in Gated Recurrent Units (GRU). Our proposal directly quantifies uncertainty deterministically, without the need for costly sampling-based estimation. We demonstrate how our model can be used to quantitatively and qualitatively measure uncertainty in unsupervised image sequence prediction. To our knowledge, this is the first result describing sampling-free uncertainty estimation for powerful sequential models such as GRUs.
Asynchronous events on the continuous time domain, e.g., social media actions and stock transactions, occur frequently in the world. The ability to recognize occurrence patterns of event sequences is crucial to predict which typeof events will happen next and when. A de facto standard mathematical framework to do this is the Hawkes process. In order to enhance expressivity of multivariate Hawkes processes, conventional statistical methods and deep recurrent networks have been employed to modify its intensity function. The former is highly interpretable and requires small size of training data but relies on correct model design while the latter has less dependency on prior knowledge and is more powerful in capturing complicated patterns. We leverage pros and cons of these models and propose a self-attentive Hawkes process(SAHP). The proposed method adapts self-attention to fit the intensity function of Hawkes processes. This design has two benefits:(1) compared with conventional statistical methods, the SAHP is more powerful to identify complicated dependency relationships between temporal events; (2)compared with deep recurrent networks, the self-attention mechanism is able to capture longer historical information, and is more interpretable because the learnt attention weight tensor shows contributions of each historical event. Experiments on four real-world datasets demonstrate the effectiveness of the proposed method.
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We propose a novel framework for modeling multiple multivariate point processes, each with heterogeneous event types that share an underlying space and obey the same generative mechanism. Focusing on Hawkes processes and their variants that are associated with Granger causality graphs, our model leverages an uncountable event type space and samples the graphs with different sizes from a nonparametric model called {it graphon}. Given those graphs, we can generate the corresponding Hawkes processes and simulate event sequences. Learning this graphon-based Hawkes process model helps to 1) infer the underlying relations shared by different Hawkes processes; and 2) simulate event sequences with different event types but similar dynamics. We learn the proposed model by minimizing the hierarchical optimal transport distance between the generated event sequences and the observed ones, leading to a novel reward-augmented maximum likelihood estimation method. We analyze the properties of our model in-depth and demonstrate its rationality and effectiveness in both theory and experiments.
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