Fast Warping Distance for Sparse Time Series

Dynamic Time Warping (DTW) distance has been effectively used in mining time series data in a multitude of domains. However, DTW, in its original formulation, is extremely inefficient in comparing long sparse time series, which mostly contain zeros and unevenly spaced non-zero observations. Original DTW distance does not take advantage of the sparsity, and thus, incur a prohibitively large computational cost for long time series. We derive a new time warping similarity measure (AWarp) for sparse time series that works on run-length encoded representation of sparse time series. The complexity of AWarp is quadratic on the number of observations as opposed to the range of time of the time series. Therefore, AWarp can be several orders of magnitude faster than DTW on sparse time series. AWarp is exact for binary-valued time series and close approximation of the original DTW distance for any-valued series. We discuss useful variants of AWarp: bounded (both upper and lower), constrained, and multidimensional. We show applications of AWarp to three data mining tasks including clustering, classification, and outlier detection, which are otherwise not feasible using classic DTW, while producing equivalent results. Potential areas of applications include bot detection, human activity classification, and unusual review pattern mining.