Statistical inference, modeling, and learning of point processes

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Li, Shuang
Xie, Yao
Song, Le
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Complex systems, such as healthcare systems, cities, and information networks, often produce a large volume of time series data, along with ordered event data, which are discrete in time and space, and rich in other features (e.g., markers or texts). We model the asynchronous event data as point processes. It is essential to understand and model the complex dynamics of these time series and event data so that accurate prediction, reliable detection, or smart intervention can be carried out for social goods. Specifically, my thesis focuses on the following aspects: (1) new statistical models and effective learning algorithms for complex dynamics exhibited in event data; (2) new inference algorithms for change-point detection, and temporal logic reasoning involving time series and event data. In Chapter 1, we propose a kernel-based nonparametric change-point detection method for high-dimensional streaming data. Change-point detection is an essential topic in modern complex systems. For example, wearable sensors are nowadays common in healthcare systems, which make it possible to monitor patients' health status in real time. Early event detection of deterioration is helpful and can even save patients' lives. However, it is challenging to aggregate measurements from different sensors to form one indicator, and it is not clear how to define pre- and post- change-point distributions. To tackle this problem, in Chapter 1, we propose a distribution-free and computationally efficient kernel-based nonparametric change-point detection method, which enjoys fewer assumptions on the distributions and can handle high-dimensional streaming data. Theoretical tail probability approximation of the nonparametric statistic is also proposed, which provides a statistically principled way to determine the detection thresholds. The proposed nonparametric method shows excellent performance on real human-activity detection dataset and speech dataset. In Chapter 2, we model networked asynchronous event data as point processes and propose a continuous-time change-point detection framework to detect dynamic changes in networks. We cast the problem into a sequential hypothesis test, and derive the generalized likelihood-ratio (GLR) statistic for networked point processes by considering the network topology. The constructed statistic can achieve weak signal detection by aggregating local statistics over time and networks. We further propose to evaluate the proposed GLR statistic via an efficient EM-like algorithm which can be implemented in a distributed fashion across dimensions. Similarly, we obtain a highly accurate theoretical threshold characterization for the proposed GLR statistic and demonstrate the excellent performance of our method on real social media datasets, such as Twitter and Memetracker. In Chapter 3, we propose an expressive model for the event data and further propose an adversarial learning framework to uncover the temporal dynamics. When modeling event data as point processes, instead of hand-crafting the occurrence intensity function by a parametric form, we leverage recent advances in deep learning and parameterize the intensity function as a recurrent neural network (RNN). RNN is a composition of a series of highly flexible nonlinear functions, which allows the model to capture complex dynamics in event data and make the generative process mimic the real data much better than the prior art. Fitting neural network models for even data is challenging. We develop a novel adversarial learning framework to address this challenge and further avoid model-misspecification. Our method provides a novel connection of such event data fitting method to inverse reinforcement learning, where a stochastic policy and the associated reward function are learned simultaneously. The proposed framework has been evaluated on real crime, social network, and healthcare datasets, and outperforms the state-of-the-art methods in data description. In Chapter 4, we propose a unified framework to integrate first-order temporal logic rules into point process models for event data. The proposed modeling framework excels in small data regime and has the ability to incorporate domain knowledge. The proposed temporal logic point processes model the intensity function of the event starts and ends via a set of first-order temporal logic rules. Using softened representation of temporal relations, and a weighted combination of logic rules, our framework can also deal with uncertainty in event data. Furthermore, many existing point process models can be interpreted as special cases of our framework given simple temporal logic rules. We derive a maximum likelihood estimation procedure for the proposed temporal logic point processes, and show that it can lead to accurate predictions when data are sparse and domain knowledge is critical. The proposed framework has been evaluated on real healthcare datasets, and outperforms the neural network models in event predication on small data and is easy to interpret.
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