Toward accurate dynamic time warping in linear time and space

ion speeds up the DTW algorithm by operating on a reduced representation of the data. These algorithms include IDDTW [3], PDTW [13], and COW [2] . The left side of Figure 5 shows a full-resolution cost matrix for which a minimum-distance warp path must be found. Rather than running DTW on the full resolution (1/1) cost matrix, the time series are reduced in size to make the number of cells in the cost matrix more manageable. A warp path is found for the lowerresolution time series and is mapped back to full resolution. The resulting speedup depends on how much abstraction is used. Obviously, the calculated warp path becomes increasingly inaccurate as the level of abstraction increases. Projecting the low resolution warp path to the full resolution usually creates a warp path that is far from optimal. This is because even IF an optimal warp path passes through the low-resolution cell, projecting the warp path to the higher resolution ignores local variations in the warp path that can be very significant. Indexing [9][14] uses lower-bounding functions to prune the number of times DTW is run for similarity search [17]. Indexing speeds up applications in which DTW is used, but it does not make up the actual DTW calculation any more efficient. Our FastDTW algorithm uses ideas from both the constraints and abstraction approaches. Using a combination of both overcomes many limitations of using either method individually, and yields an accurate algorithm that is O(N) in both time and space complexity. Our multi-level approach is superficially similar to IDDTW [3] because they both evaluate several different resolutions. However, IDDTW simply executes PDTW [13] at increasingly higher resolutions until a desired “accuracy” is achieved. IDDTW does not project low resolution solutions to higher resolutions. In Section 4, we will demonstrate that these methods are more inaccurate than our method given the same amount of execution time. Projecting warp paths to higher resolutions is also done in the construction of “Match Webs” [15]. However, their approach is still O(N) due to the simultaneous search for many warp paths (they call them “chains”). A multi-resolution approach in their application also could not continue down to the low resolutions without severely reducing the number of “chains” that could be found. Some recent research [18] asserts that there is no need to speed up the original DTW algorithm. However, this is only true under the following (common) conditions: 1) Tight Constraints A relatively strict near-linear warp path is allowable. 2) Short Time Series All time series are short enough for the DTW algorithm to execute quickly. (~3,000 points if a warp path is needed, or ~100,000 if no warp path is needed and the user has a lot of patience).