TY - JOUR A1 - Brill, Markus A1 - Fluschnik, Till A1 - Froese, Vincent A1 - Jain, Brijnesh A1 - Niedermeier, Rolf A1 - Schultz, David T1 - Exact mean computation in dynamic time warping spaces JF - Data Mining and Knowledge Discovery N2 - Averaging time series under dynamic time warping is an important tool for improving nearest-neighbor classifiers and formulating centroid-based clustering. The most promising approach poses time series averaging as the problem of minimizing a Fréchet function. Minimizing the Fréchet function is NP-hard and so far solved by several heuristics and inexact strategies. Our contributions are as follows: we first discuss some inaccuracies in the literature on exact mean computation in dynamic time warping spaces. Then we propose an exponential-time dynamic program for computing a global minimum of the Fréchet function. The proposed algorithm is useful for benchmarking and evaluating known heuristics. In addition, we present an exact polynomial-time algorithm for the special case of binary time series. Based on the proposed exponential-time dynamic program, we empirically study properties like uniqueness and length of a mean, which are of interest for devising better heuristics. Experimental evaluations indicate substantial deficits of state-of-the-art heuristics in terms of their output quality. Y1 - 2019 U6 - https://doi.org/10.1007/s10618-018-0604-8 VL - 33 SP - 252 EP - 291 PB - Springer Nature ER - TY - JOUR A1 - Jain, Brijnesh A1 - Froese, Vincent A1 - Schultz, David T1 - An average-compress algorithm for the sample mean problem under dynamic time warping JF - Journal of Global Optimization N2 - Computing a sample mean of time series under dynamic time warping is NP-hard. Consequently, there is an ongoing research effort to devise efficient heuristics. The majority of heuristics have been developed for the constrained sample mean problem that assumes a solution of predefined length. In contrast, research on the unconstrained sample mean problem is underdeveloped. In this article, we propose a generic average-compress (AC) algorithm to address the unconstrained problem. The algorithm alternates between averaging (A-step) and compression (C-step). The A-step takes an initial guess as input and returns an approximation of a sample mean. Then the C-step reduces the length of the approximate solution. The compressed approximation serves as initial guess of the A-step in the next iteration. The purpose of the C-step is to direct the algorithm to more promising solutions of shorter length. The proposed algorithm is generic in the sense that any averaging and any compression method can be used. Experimental results show that the AC algorithm substantially outperforms current state-of-the-art algorithms for time series averaging. KW - Time series averaging KW - Fréchet function KW - Heuristic KW - Nonconvex optimization Y1 - 2023 U6 - https://doi.org/10.1007/s10898-023-01294-9 PB - Springer ER -