Abstract
For the offline segmentation of long hydrometeological time series, a new algorithm which combines the dynamic programming with the recently introduced remaining cost concept of branch-and-bound approach is developed. The algorithm is called modified dynamic programming (mDP) and segments the time series based on the first-order statistical moment. Experiments are performed to test the algorithm on both real world and artificial time series comprising of hundreds or even thousands of terms. The experiments show that the mDP algorithm produces accurate segmentations in much shorter time than previously proposed segmentation algorithms.
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Notes
The recursive solution of an entire sequence of minimization problems makes the DP algorithm very attractive for online operation; the same feature, however, is the main difficulty in converting the algorithm to online operation. Indeed, the two inner loops in the Minimization section of the algorithm show that, if a new datum x T+1 is added to the time series, the costs e s,t must be recomputed for every pair (s,t) with s < t ≤ T + 1. Hence, for every new datum T + 1 additional computation must be performed; as T (i.e., the length of the time series) increases the amount of computation required becomes prohibitive, especially for online operation.
Note that here too the assumption of offline segmentation is crucial. Namely, to implement the reduction of upper bound u is possible only if the costs d s,t are known for every value of s,t and, in particular, for t = T, which implies that the final time T is known; this would not be the case for online segmentation.
All experiments were performed by running a Microsoft Visual Studio 2005 C# implementation of DP and mDP. The executable was run on a Windows PC with HT processor running at 3.00 GHz (CPU) and 2 GB memory (RAM).
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Acknowledgments
The authors thank the IAHS Secretary General Dr. Pierre Hubert of Université P. & M. Curie, Paris, France, for sharing his software of automatic segmentation algorithm online. The user-friendly version of the algorithms can be supplied to those who show interest and make a request to the authors. This manuscript has been submitted when the second author (H. Aksoy) was working at Leuphana Universität Lüneburg, Campus Suderburg in Germany as an experienced researcher invited by the Alexander von Humboldt Foundation of Germany.
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Appendices
Appendix A (Dynamic Programming)
Appendix B (Modified Dynamic Programming)
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Gedikli, A., Aksoy, H., Erdem Unal, N. et al. Modified dynamic programming approach for offline segmentation of long hydrometeorological time series. Stoch Environ Res Risk Assess 24, 547–557 (2010). https://doi.org/10.1007/s00477-009-0335-x
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DOI: https://doi.org/10.1007/s00477-009-0335-x