Towards Ptolemaic metric properties of the z-normalized Euclidean distance for multivariate time series indexing

With the rapid proliferation of intelligent systems and modern sensors, time series data has become a prominent data type in various application areas, including finance, medicine, and industry. As modern sensors are able to capture different sources of information simultaneously, time series are becoming increasingly multivariate, in the sense that complex, multidimensional information is continuously captured at high frequency. This increase in complexity presents a challenge for state-of-the-art models and algorithms. This work-in-progress paper focuses on indexing multivariate time series, which is a foundational operation for efficient access and analysis of time series databases. To this end, we describe our latest theoretical and empirical findings regarding the z-normalized Euclidean distance justifying their metric and Ptolemaic properties. Additionally, we discuss the extension of that distance to multivariate time series and provide empirical evidence that this new distance induces a pseudometric space that also satisfies Ptolemy's inequality. We believe that our findings are useful for practitioners and scientists in this field, as well as for the development of efficient access methods for multivariate time series.