# Online Topology Identification from Vector Autoregressive Time Series

## TL;DR

This paper introduces two online algorithms for real-time identification of causality graphs from multivariate time series using VAR models, suitable for big data and dynamic environments, with proven asymptotic optimality.

## Contribution

It develops two novel online algorithms for tracking time-varying causality graphs from VAR models, with theoretical performance guarantees and applicability to large-scale data.

## Key findings

- Algorithms achieve asymptotic performance matching batch estimators.
- Algorithms have constant complexity per update, suitable for big data.
- Numerical results validate effectiveness in static and dynamic scenarios.

## Abstract

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.