A Structured Study of Multivariate Time-Series Distance Measures

Distance measures are fundamental to time series analysis and have been extensively studied for decades. Until now, research efforts mainly focused on univariate time series, leaving multivariate cases largely under-explored. Furthermore, the existing experimental studies on multivariate distances have critical limitations: (a) focusing only on lock-step and elastic measures while ignoring categories such as sliding and kernel measures; (b) considering only one normalization technique; and (c) placing limited focus on statistical analysis of findings. Motivated by these shortcomings, we present the most complete evaluation of multivariate distance measures to date. Our study examines 30 standalone measures across 8 categories, 2 channel-dependency models, and considers 13 normalizations. We perform a comprehensive evaluation across 30 datasets and 3 downstream tasks, accompanied by rigorous statistical analysis. To ensure fairness, we conduct a thorough investigation of parameters for methods in both a supervised and an unsupervised manner. Our work verifies and extends earlier findings, showing that insights from univariate distance measures also apply to the multivariate case: (a) alternative normalization methods outperform Z-score, and for the first time, we demonstrate statistical differences in certain categories for the multivariate case; (b) multiple lock-step measures are better suited than Euclidean distance, when it comes to multivariate time series; and (c) newer elastic measures outperform the widely adopted Dynamic Time Warping distance, especially with proper parameter tuning in the supervised setting. Moreover, our results reveal that (a) sliding measures offer the best trade-off between accuracy and runtime; (b) current normalization techniques fail to significantly enhance accuracy on multivariate time series and, surprisingly, do not outperform the no normalization case, indicating a lack of appropriate solutions for normalizing multivariate time series; and (c) independent consideration of time series channels is beneficial only for elastic measures. In summary, we offer guidelines to aid in designing and selecting preprocessing strategies and multivariate distance measures for our community.