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Jiong Yang, Wei Wang, Philip S. Yu, "Mining Asynchronous Periodic Patterns in Time Series Data," IEEE Transactions on Knowledge and Data Engineering, vol. 15, no. 3, pp. 613628, May/June, 2003.  
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@article{ 10.1109/TKDE.2003.1198394, author = {Jiong Yang and Wei Wang and Philip S. Yu}, title = {Mining Asynchronous Periodic Patterns in Time Series Data}, journal ={IEEE Transactions on Knowledge and Data Engineering}, volume = {15}, number = {3}, issn = {10414347}, year = {2003}, pages = {613628}, doi = {http://doi.ieeecomputersociety.org/10.1109/TKDE.2003.1198394}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
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TY  JOUR JO  IEEE Transactions on Knowledge and Data Engineering TI  Mining Asynchronous Periodic Patterns in Time Series Data IS  3 SN  10414347 SP613 EP628 EPD  613628 A1  Jiong Yang, A1  Wei Wang, A1  Philip S. Yu, PY  2003 KW  Asynchronous periodic pattern KW  segmentbased approach KW  partial periodicity. VL  15 JA  IEEE Transactions on Knowledge and Data Engineering ER   
Abstract—Periodicy detection in time series data is a challenging problem of great importance in many applications. Most previous work focused on mining synchronous periodic patterns and did not recognize the misaligned presence of a pattern due to the intervention of random noise. In this paper, we propose a more flexible model of asynchronous periodic pattern that may be present only within a subsequence and whose occurrences may be shifted due to disturbance. Two parameters min_rep and max_dis are employed to specify the minimum number of repetitions that is required within each segment of nondisrupted pattern occurrences and the maximum allowed disturbance between any two successive valid segments. Upon satisfying these two requirements, the longest valid subsequence of a pattern is returned. A twophase algorithm is devised to first generate potential periods by distancebased pruning followed by an iterative procedure to derive and validate candidate patterns and locate the longest valid subsequence. We also show that this algorithm cannot only provide linear time complexity with respect to the length of the sequence but also achieve space efficiency.
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