Scalable Methods for Multiple Time Series Comparison in Second Order Dynamics

Statistical comparison of multiple time series in their underlying frequency patterns has many real applications. However, existing methods are only applicable to a small number of mutually independent time series, and empirical results for dependent time series are only limited to comparing two time series. We propose scalable methods based on a new algorithm that enables us to compare the spectral density of a large number of time series. The new algorithm helps us efficiently obtain all pairwise feature differences in frequency patterns between M time series, which plays an essential role in our methods. When all M time series are independent of each other, we derive the joint asymptotic distribution of their pairwise feature differences. The asymptotic dependence structure between the feature differences motivates our proposed test for multiple mutually independent time series. We then adapt this test to the case of multiple dependent time series by partially accounting for the underlying dependence structure. Additionally, we introduce a global test to further enhance the approach. To examine the finite sample performance of our proposed methods, we conduct simulation studies. The new approaches demonstrate the ability to compare a large number of time series, whether independent or dependent, while exhibiting competitive power. Finally, we apply our methods to compare multiple mechanical vibrational time series.