Matrix Profile XXI: A Geometric Approach to Time Series Chains Improves Robustness

Time series motifs have become a fundamental tool to characterize repeated and conserved structure in systems, such as manufacturing telemetry, economic activities, and both human physiological and cultural behaviors. Recently time series chains were introduced as a generalization of time series motifs to represent evolving patterns in time series, in order to characterize the evolution of systems. Time series chains are a very promising primitive; however, we have observed that the original definition can be brittle in the sense that a small fluctuation in time series may "cut" a chain. Furthermore, the original definition does not provide a measure of the "significance" of a chain, and therefore cannot support top-k search for chains or provide a mechanism to discard spurious chains that might be discovered when searching large datasets. Inspired by observations from dynamical systems theory, this paper introduces two novel quality metrics for time series chains, directionality and graduality, to improve robustness and to enable top-K search. With extensive empirical work we show that our proposed definition is much more robust to the vagaries of real-word datasets and allows us to find unexpected regularities in time series datasets.