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Computing the halfspace depth with multiple try algorithm and simulated annealing algorithm

Author

Listed:
  • Wei Shao

    (Qufu Normal University
    Qufu Normal University)

  • Yijun Zuo

    (Michigan State University)

Abstract

The halfspace depth is a powerful tool for the nonparametric multivariate analysis. However, its computation is very challenging for it involves the infimum over infinitely many directional vectors. The exact computation of the halfspace depth is a NP-hard problem if both sample size n and dimension d are parts of input. The approximate algorithms often can not get accurate (exact) results in high dimensional cases within limited time. In this paper, we propose a new general stochastic optimization algorithm, which is the combination of simulated annealing and the multiple try Metropolis algorithm. As a by product of the new algorithm, it is then successfully applied to the computation of the halfspace depth of data sets which are not necessarily in general position. The simulation and real data examples indicate that the new algorithm is highly competitive to, especially in the high dimension and large sample cases, other (exact and approximate) algorithms, including the simulated annealing and the quasi-Newton method and so on, both in accuracy and efficiency.

Suggested Citation

  • Wei Shao & Yijun Zuo, 2020. "Computing the halfspace depth with multiple try algorithm and simulated annealing algorithm," Computational Statistics, Springer, vol. 35(1), pages 203-226, March.
    • Handle: RePEc:spr:compst:v:35:y:2020:i:1:d:10.1007_s00180-019-00906-x
    DOI: 10.1007/s00180-019-00906-x
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    References listed on IDEAS

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    1. Faming Liang & Yichen Cheng & Guang Lin, 2014. "Simulated Stochastic Approximation Annealing for Global Optimization With a Square-Root Cooling Schedule," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(506), pages 847-863, June.
    2. Tatjana Lange & Karl Mosler & Pavlo Mozharovskyi, 2014. "Fast nonparametric classification based on data depth," Statistical Papers, Springer, vol. 55(1), pages 49-69, February.
    3. Xiaohui Liu, 2017. "Fast implementation of the Tukey depth," Computational Statistics, Springer, vol. 32(4), pages 1395-1410, December.
    4. Dyckerhoff, Rainer & Mozharovskyi, Pavlo, 2016. "Exact computation of the halfspace depth," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 19-30.
    5. Shao, Wei & Zuo, Yijun, 2012. "Simulated annealing for higher dimensional projection depth," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 4026-4036.
    6. Subhajit Dutta & Anil Ghosh, 2012. "On robust classification using projection depth," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(3), pages 657-676, June.
    7. Cuesta-Albertos, J.A. & Nieto-Reyes, A., 2008. "The random Tukey depth," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4979-4988, July.
    8. Martino, Luca & Del Olmo, Victor Pascual & Read, Jesse, 2012. "A multi-point Metropolis scheme with generic weight functions," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1445-1453.
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