Causality graphs are routinely estimated in social sciences, natural sciences, and engineering due to their capacity to efficiently represent the spatiotemporal structure of multivariate data sets in a format amenable for human interpretation, forecasting, and anomaly detection. A popular approach to mathematically formalize causality is based on vector autoregressive (VAR) models and constitutes an alternative to the well-known, yet usually intractable, Granger causality. Relying on such a VAR causality notion, this paper develops two algorithms with complementary benefits to track time-varying causality graphs in an online fashion. Their constant complexity per update also renders these algorithms appealing for big-data scenarios. Despite using data sequentially, both algorithms are shown to asymptotically attain the same average performance as a batch estimator which uses the entire data set at once. To this end, sublinear (static) regret bounds are established. Performance is also characterized in time-varying setups by means of dynamic regret analysis. Numerical results with real and synthetic data further support the merits of the proposed algorithms in static and dynamic scenarios.

The availability of automatic identification system (AIS) data for tracking vessels has paved the way for improvements in maritime safety and efficiency. However, one of the main challenges in using AIS data is often the low quality of the data. Practically, AIS-based trajectory data of vessels are available at irregular time intervals; consequently, large temporal gaps often exist in the historical AIS data. Meanwhile, certain tasks such as waypoint detection using historical data, which involves finding locations along the trajectory where the vessel changes its course (and possibly speed, acceleration, etc.), require AIS messages with a high temporal resolution. High-resolution AIS data are especially required for waypoint detection in critical areas where vessels maneuver carefully because of, e.g., narrow pathways or the presence of islands. One possible solution to address the problem of insufficient AIS data in vessel trajectories is interpolation. In this paper, we address the problem of detecting waypoints in a single representative trajectory with insufficient data using various interpolation-based methods. To this end, a two-step approach is proposed, in which the trajectories are first interpolated, and then the waypoint detection method is applied to the merged trajectory containing both interpolated and observed AIS messages. The numerical results demonstrate the effectiveness of exploiting various interpolation methods for waypoint detection. Moreover, the results of the numerical experiments show that the proposed methodology is effective for waypoint detection in envisaged settings with insufficient data, and outperforms the competing algorithm.

Identifying the topology underlying a set of time series is useful for tasks such as prediction, denoising, and data completion. Vector autoregressive (VAR) model-based topologies capture dependencies among time series and are often inferred from observed spatio-temporal data. When data are affected by noise and/or missing samples, topology identification and signal recovery (reconstruction) tasks must be performed jointly. Additional challenges arise when i) the underlying topology is time-varying, ii) data become available sequentially, and iii) no delay is tolerated. This study proposes an online algorithm to overcome these challenges in estimating VAR model-based topologies, having constant complexity per iteration, which makes it interesting for big-data scenarios. The inexact proximal online gradient descent framework is used to derive a performance guarantee for the proposed algorithm, in the form of a dynamic regret bound. Numerical tests are also presented, showing the ability of the proposed algorithm to track time-varying topologies with missing data in an online fashion.

Mobility data of a moving object, called trajectory data, are continuously generated by vessel navigation systems, wearable devices, and drones, to name a few. Trajectory data consist of samples that include temporal, spatial, and other descriptive features of object movements. One of the main challenges in trajectory data analysis is to divide trajectory data into meaningful segments based on certain criteria. Most of the available segmentation algorithms are limited to processing data offline, i.e., they cannot segment a stream of trajectory samples. In this work, we propose an approach called Reactive Buffering Window - Trajectory Segmentation (RBW-TS), which partitions trajectory data into segments while receiving a stream of trajectory samples. Another novelty compared to existing work is that the proposed algorithm is based on multidimensional features of trajectories, and it can incorporate as many relevant features of the underlying trajectory as needed. This makes RBW-TS general and applicable to numerous domains by simply selecting trajectory features relevant for segmentation purposes. The proposed online algorithm incurs lower computational and memory requirements. Furthermore, it is robust to noisy samples and outliers. We validate RBW-TS on three use cases: (a) segmenting human-movement trajectories in different modes of transportation, (b) segmenting trajectories generated by vessels in the maritime domain, and (c) segmenting human-movement trajectories in a commercial shopping center. The numerical results detailed in the paper demonstrate that (i) RBW-TS is capable of detecting the true breakpoints of segments in all three usecases while processing a stream of trajectory points; (ii) despite low memory and computational requirements, the performance in terms of the harmonic mean of purity and coverage is comparable to that of state-of-the-art batch and online algorithms; (iii) RBW-TS achieves different levels of accuracy depending on the various internal parameter estimation methods used; and (iv) RBW-TS can tackle real-world trajectory data for segmentation purposes.