The Internet-of-Things, complex sensor networks, multi-agent cyber-physical systems are all examples of spatially distributed systems that continuously evolve in time. Such systems generate huge amounts of spatio-temporal data, and system designers are often interested in analyzing and discovering structure within the data. There has been considerable interest in learning causal and logical properties of temporal data using logics such as Signal Temporal Logic (STL); however, there is limited work on discovering such relations on spatio-temporal data. We propose the first set of algorithms for unsupervised learning for spatio-temporal data. Our method does automatic feature extraction from the spatio-temporal data by projecting it onto the parameter space of a parametric spatio-temporal reach and escape logic (PSTREL). We propose an agglomerative hierarchical clustering technique that guarantees that each cluster satisfies a distinct STREL formula. We show that our method generates STREL formulas of bounded description complexity using a novel decision-tree approach which generalizes previous unsupervised learning techniques for Signal Temporal Logic. We demonstrate the effectiveness of our approach on case studies from diverse domains such as urban transportation, epidemiology, green infrastructure, and air quality monitoring.
