We develop a method to construct distribution-free prediction intervals for dynamic time-series, called \Verb|EnbPI| that wraps around any bootstrap ensemble estimator to construct sequential prediction intervals. \Verb|EnbPI| is closely related to the conformal prediction (CP) framework but does not require data exchangeability. Theoretically, these intervals attain finite-sample, \textit{approximately valid} marginal coverage for broad classes of regression functions and time-series with strongly mixing stochastic errors. Computationally, \Verb|EnbPI| avoids overfitting and requires neither data-splitting nor training multiple ensemble estimators; it efficiently aggregates bootstrap estimators that have been trained. In general, \Verb|EnbPI| is easy to implement, scalable to producing arbitrarily many prediction intervals sequentially, and well-suited to a wide range of regression functions. We perform extensive real-data analyses to demonstrate its effectiveness.

This paper presents a novel approach for hierarchical time series forecasting that produces coherent, probabilistic forecasts without requiring any explicit post-processing reconciliation. Unlike the state-of-the-art, the proposed method simultaneously learns from all time series in the hierarchy and incorporates the reconciliation step into a single trainable model. This is achieved by applying the reparameterization trick and casting reconciliation as an optimization problem with a closed-form solution. These model features make end-to-end learning of hierarchical forecasts possible, while accomplishing the challenging task of generating forecasts that are both probabilistic and coherent. Importantly, our approach also accommodates general aggregation constraints including grouped and temporal hierarchies. An extensive empirical evaluation on real-world hierarchical datasets demonstrates the advantages of the proposed approach over the state-of-the-art.

Smooth dynamics interrupted by discontinuities are known as hybrid systems and arise commonly in nature. Latent ODEs allow for powerful representation of irregularly sampled time series but are not designed to capture trajectories arising from hybrid systems. Here, we propose the Latent Segmented ODE (LatSegODE), which uses Latent ODEs to perform reconstruction and changepoint detection within hybrid trajectories featuring jump discontinuities and switching dynamical modes. Where it is possible to train a Latent ODE on the smooth dynamical flows between discontinuities, we apply the pruned exact linear time (PELT) algorithm to detect changepoints where latent dynamics restart, thereby maximizing the joint probability of a piece-wise continuous latent dynamical representation. We propose usage of the marginal likelihood as a score function for PELT, circumventing the need for model-complexity-based penalization. The LatSegODE outperforms baselines in reconstructive and segmentation tasks including synthetic data sets of sine waves, Lotka Volterra dynamics, and UCI Character Trajectories.

There recently has been a surge of interest in developing a new class of deep learning (DL) architectures that integrate an explicit time dimension as a fundamental building block of learning and representation mechanisms. In turn, many recent results show that topological descriptors of the observed data, encoding information on the shape of the dataset in a topological space at different scales, that is, persistent homology of the data, may contain important complementary information, improving both performance and robustness of DL. As convergence of these two emerging ideas, we propose to enhance DL architectures with the most salient time-conditioned topological information of the data and introduce the concept of zigzag persistence into time-aware graph convolutional networks (GCNs). Zigzag persistence provides a systematic and mathematically rigorous framework to track the most important topological features of the observed data that tend to manifest themselves over time. To integrate the extracted time-conditioned topological descriptors into DL, we develop a new topological summary, zigzag persistence image, and derive its theoretical stability guarantees. We validate the new GCNs with a time-aware zigzag topological layer (Z-GCNETs), in application to traffic forecasting and Ethereum blockchain price prediction. Our results indicate that Z-GCNET outperforms 13 state-of-the-art methods on 4 time series datasets.

Continuous-time event sequences represent discrete events occurring in continuous time. Such sequences arise frequently in real-life. Usually we expect the sequences to follow some regular pattern over time. However, sometimes these patterns may be interrupted by unexpected absence or occurrences of events. Identification of these unexpected cases can be very important as they may point to abnormal situations that need human attention. In this work, we study and develop methods for detecting outliers in continuous-time event sequences, including unexpected absence and unexpected occurrences of events. Since the patterns that event sequences tend to follow may change in different contexts, we develop outlier detection methods based on point processes that can take context information into account. Our methods are based on Bayesian decision theory and hypothesis testing with theoretical guarantees. To test the performance of the methods, we conduct experiments on both synthetic data and real-world clinical data and show the effectiveness of the proposed methods.

In this work, we propose TimeGrad, an autoregressive model for multivariate probabilistic time series forecasting which samples from the data distribution at each time step by estimating its gradient. To this end, we use diffusion probabilistic models, a class of latent variable models closely connected to score matching and energy-based methods. Our model learns gradients by optimizing a variational bound on the data likelihood and at inference time converts white noise into a sample of the distribution of interest through a Markov chain using Langevin sampling. We demonstrate experimentally that the proposed autoregressive denoising diffusion model is the new state-of-the-art multivariate probabilistic forecasting method on real-world data sets with thousands of correlated dimensions. We hope that this method is a useful tool for practitioners and lays the foundation for future research in this area.

Multivariate Hawkes processes (MHPs) are widely used in a variety of fields to model the occurrence of causally related discrete events in continuous time. Most state-of-the-art approaches address the problem of learning MHPs from perfect traces without noise. In practice, the process through which events are collected might introduce noise in the timestamps. In this work, we address the problem of learning the causal structure of MHPs when the observed timestamps of events are subject to random and unknown shifts, also known as random translations. We prove that the cumulants of MHPs are invariant to random translations, and therefore can be used to learn their underlying causal structure. Furthermore, we empirically characterize the effect of random translations on state-of-the-art learning methods. We show that maximum likelihood-based estimators are brittle, while cumulant-based estimators remain stable even in the presence of significant time shifts.