We investigate spatio-temporal event analysis using point processes. Inferring the dynamics of event sequences spatiotemporally has many practical applications including crime prediction, social media analysis, and traffic forecasting. In particular, we focus on spatio-temporal Hawkes processes that are commonly used due to their capability to capture excitations between event occurrences. We introduce a novel inference framework based on randomized transformations and gradient descent to learn the process. We replace the spatial kernel calculations by randomized Fourier feature-based transformations. The introduced randomization by this representation provides flexibility while modeling the spatial excitation between events. Moreover, the system described by the process is expressed within closed-form in terms of scalable matrix operations. During the optimization, we use maximum likelihood estimation approach and gradient descent while properly handling positivity and orthonormality constraints. The experiment results show the improvements achieved by the introduced method in terms of fitting capability in synthetic and real-life datasets with respect to the conventional inference methods in the spatio-temporal Hawkes process literature. We also analyze the triggering interactions between event types and how their dynamics change in space and time through the interpretation of learned parameters.

Recurrent neural networks (RNNs) are widely used for online regression due to their ability to generalize nonlinear temporal dependencies. As an RNN model, long short-term memory networks (LSTMs) are commonly preferred in practice, as these networks are capable of learning long-term dependencies while avoiding the vanishing gradient problem. However, due to their large number of parameters, training LSTMs requires considerably longer training time compared to simple RNNs (SRNNs). In this article, we achieve the online regression performance of LSTMs with SRNNs efficiently. To this end, we introduce a first-order training algorithm with a linear time complexity in the number of parameters. We show that when SRNNs are trained with our algorithm, they provide very similar regression performance with the LSTMs in two to three times shorter training time. We provide strong theoretical analysis to support our experimental results by providing regret bounds on the convergence rate of our algorithm. Through an extensive set of experiments, we verify our theoretical work and demonstrate significant performance improvements of our algorithm with respect to LSTMs and the other state-of-the-art learning models.

We investigate nonlinear regression for nonstationary sequential data. In most real-life applications such as business domains including finance, retail, energy, and economy, time series data exhibit nonstationarity due to the temporally varying dynamics of the underlying system. We introduce a novel recurrent neural network (RNN) architecture, which adaptively switches between internal regimes in a Markovian way to model the nonstationary nature of the given data. Our model, Markovian RNN employs a hidden Markov model (HMM) for regime transitions, where each regime controls hidden state transitions of the recurrent cell independently. We jointly optimize the whole network in an end-to-end fashion. We demonstrate the significant performance gains compared to conventional methods such as Markov Switching ARIMA, RNN variants and recent statistical and deep learning-based methods through an extensive set of experiments with synthetic and real-life datasets. We also interpret the inferred parameters and regime belief values to analyze the underlying dynamics of the given sequences.

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