This paper proposes a new method for determining similarity and anomalies between time series, most practically effective in large collections of (likely related) time series, by measuring distances between structural breaks within such a collection. We introduce a class of semi-metric distance measures, which we term MJ distances. These semi-metrics provide an advantage over existing options such as the Hausdorff and Wasserstein metrics. We prove they have desirable properties, including better sensitivity to outliers, while experiments on simulated data demonstrate that they uncover similarity within collections of time series more effectively. Semi-metrics carry a potential disadvantage: without the triangle inequality, they may not satisfy a “transitivity property of closeness.” We analyse this failure with proof and introduce an computational method to investigate, in which we demonstrate that our semi-metrics violate transitivity infrequently and mildly. Finally, we apply our methods to cryptocurrency and measles data, introducing a judicious application of eigenvalue analysis.

本文提出了一种确定时间序列之间的相似性和异常的新方法，该方法通过测量时间序列之间的结构性断裂之间的距离，在（可能相关的）时间序列的大集合中最有效。我们介绍一类半度量距离度量，我们称其为MJ距离。与现有选项（例如Hausdorff和Wasserstein指标）相比，这些半指标具有优势。我们证明了它们具有理想的属性，包括对异常值的更好敏感性，而对模拟数据的实验表明，它们可以更有效地发现时间序列集合内的相似性。半度量有一个潜在的缺点：没有三角不等式，它们可能无法满足“紧密度的传递性”。我们用证据分析了这种失败，并引入了一种计算方法进行调查，在该方法中，我们证明了我们的半度量很少偶尔地违反传递性。最后，我们将我们的方法应用于加密货币和麻疹数据，引入了特征值分析的明智应用。