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Maximum error-bounded Piecewise Linear Representation for online stream approximation

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Abstract

Given a time series data stream, the generation of error-bounded Piecewise Linear Representation (error-bounded PLR) is to construct a number of consecutive line segments to approximate the stream, such that the approximation error does not exceed a prescribed error bound. In this work, we consider the error bound in \(L_\infty \) norm as approximation criterion, which constrains the approximation error on each corresponding data point, and aim on designing algorithms to generate the minimal number of segments. In the literature, the optimal approximation algorithms are effectively designed based on transformed space other than time-value space, while desirable optimal solutions based on original time domain (i.e., time-value space) are still lacked. In this article, we proposed two linear-time algorithms to construct error-bounded PLR for data stream based on time domain, which are named OptimalPLR and GreedyPLR, respectively. The OptimalPLR is an optimal algorithm that generates minimal number of line segments for the stream approximation, and the GreedyPLR is an alternative solution for the requirements of high efficiency and resource-constrained environment. In order to evaluate the superiority of OptimalPLR, we theoretically analyzed and compared OptimalPLR with the state-of-art optimal solution in transformed space, which also achieves linear complexity. We successfully proved the theoretical equivalence between time-value space and such transformed space, and also discovered the superiority of OptimalPLR on processing efficiency in practice. The extensive results of empirical evaluation support and demonstrate the effectiveness and efficiency of our proposed algorithms.

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Notes

  1. It should be noted that \(S[i-1,j]\) can be \(\delta \)-representable even if \(S[i,j]\) is maximally \(\delta \)-representable.

  2. Without loss of generality, we assume that \(x_i<x_j\).

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Acknowledgments

This research is partially supported by Natural Science Foundation of China (Grant No.61232006) and the Australian Research Council (Grant No. DP140103171 and DP130103051).

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Authors and Affiliations

  1. Division of CEMSE, KAUST, Thuwal, Saudi Arabia

    Qing Xie & Xiangliang Zhang

  2. AEHRC, CSIRO, Brisbane, Australia

    Chaoyi Pang

  3. Zhejiang University (NIT), Ningbo, China

    Chaoyi Pang

  4. Hebei Academy of Sciences, Hebei, China

    Chaoyi Pang

  5. School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, Australia

    Xiaofang Zhou

  6. School of Computer Science and Technology, Soochow University, Suzhou, China

    Xiaofang Zhou

  7. Huawei Noah’s Ark Research Lab, Hong Kong, China

    Ke Deng

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  1. Qing Xie
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  2. Chaoyi Pang
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  3. Xiaofang Zhou
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  4. Xiangliang Zhang
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  5. Ke Deng
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Correspondence to Xiangliang Zhang.

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Xie, Q., Pang, C., Zhou, X. et al. Maximum error-bounded Piecewise Linear Representation for online stream approximation. The VLDB Journal 23, 915–937 (2014). https://doi.org/10.1007/s00778-014-0355-0

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