Time series is traditionally treated with two main approaches, i.e., the time domain approach and the frequency domain approach. These approaches must rely on a sliding window so that time-shift versions of a sequence can be measured to be similar. Coupled with the use of a root point-to-point measure, existing methods often have quadratic time complexity. We offer the third R\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}$$\end{document} domain approach. It begins with an insight that sequences in a stationary time series can be treated as sets of independent and identically distributed (iid) points generated from an unknown distribution in R\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}$$\end{document}. This R\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}$$\end{document} domain treatment enables two new possibilities: (a) The similarity between two sequences can be computed using a distributional measure such as Wasserstein distance (WD), kernel mean embedding or isolation distributional kernel (KI\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {K}_I$$\end{document}), and (b) these distributional measures become non-sliding-window-based. Together, they offer an alternative that has more effective similarity measurements and runs significantly faster than the point-to-point and sliding-window-based measures. Our empirical evaluation shows that KI\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {K}_I$$\end{document} is an effective and efficient distributional measure for time series; and KI\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {K}_I$$\end{document}-based detectors have better detection accuracy than existing detectors in two tasks: (i) anomalous sequence detection in a stationary time series and (ii) anomalous time series detection in a dataset of non-stationary time series. The insight makes underutilized “old things new again” which gives existing distributional measures and anomaly detectors a new life in time series anomaly detection that would otherwise be impossible.
