Elastic statistical analysis of interval-valued time series.

We investigate the problem of statistical analysis of interval-valued time series data - two nonintersecting real-valued functions, representing lower and upper limits, over a period of time. Specifically, we pay attention to the two concepts of phase (or horizontal) variability and amplitude (or vertical) variability, and propose a phase-amplitude separation method. We view interval-valued time series as elements of a function (Hilbert) space and impose a Riemannian structure on it. We separate phase and amplitude variability in observed interval functions using a metric-based alignment solution. The key idea is to map an interval to a point in , view interval-valued time series as parameterized curves in , and borrow ideas from elastic shape analysis of planar curves, including PCA, to perform registration, summarization, analysis, and modeling of multiple series. The proposed phase-amplitude separation provides a new way of PCA and modeling for interval-valued time series, and enables shape clustering of interval-valued time series. We apply this framework to three different applications, including finance, meteorology and physiology, proves the effectiveness of proposed methods, and discovers some underlying patterns in the data. Experimental results on simulated data show that our method applies to the point-valued time series.