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.
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References
Byrd RH, Liu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16:1190–1208
Casarin R, Craiu R, Leisen R (2013) Interacting multiple try algorithms with different proposal distributions. Stat Comput 23:185–200
Chen D, Morin P, Wagner U (2013) Absolute approximation of Tukey depth: theory and experiments. Comput Geom 46:566–573
Cuesta-Albertos JA, Nieto-Reyes A (2008) The random Tukey depth. Comput Stat Data Anal 52:4979–4988
Dutta S, Ghosh AK (2012) On robust classification using projection depth. Ann Inst Stat Math 64:657–676
Dyckerhoff R, Mozharovskyi P (2016) Exact computation of the halfspace depth. Comput Stat Data Anal 98:19–30
Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell 6:721–741
Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109
Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49:409–436
Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680
Lange T, Mosler K, Mozharovskyi P (2014) Fast nonparametric classification based on data depth. Stat Pap 55:49–69
Liang F, Liu C, Carroll RJ (2011) Advanced markov chain Monte Carlo methods: learning from past samples. Wiley, West Sussex
Liang F, Cheng Y, Lin G (2014) Simulated stochastic approximation annealing for global optimization with a square-root cooling schedule. J Am Stat Assoc 109:847–863
Liu JS (2001) Monte Carlo strategies in scientific computing. Springer, New York
Liu X (2017) Fast implementation of the Tukey depth. Comput Stat 32:1395–1410
Liu JS, Liang F, Wong WH (2000) The multiple-try and local optimization in Metropolis sampling. J Am Stat Assoc 95:121–134
Liu X, Zuo Y (2014) Computing halfspace depth and regression depth. Commun Stat Simul Comput 43:969–985
Liu X, Zuo Y, Wang Q (2017) Finite sample breakdown point of Tukey’s halfspace median. Sci China Math 60(5):861–874
Martino L, Olmo VPD, Read J (2012) A multi-point Metropolis scheme with generic weight functions. Stat Probab Lett 82:1445–1453
Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equations of state calculations by fast computing machines. J Chem Phys 21:1087–1092
Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313
Pandolfi S, Bartolucci F, Friel N (2010) A generalization of the multiple-try Metropolis algorithm for Bayesian estimation and model selection. J Mach Learn Res 9:581–588
R Core Team (2016) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. https://www.R-project.org/
Rousseeuw PJ, Hubert M (1999) Regression depth. J Am Stat Assoc 94:388–402
Rousseeuw PJ, Struyf A (1998) Computing location depth and regression depth in higher dimensions. Stat Comput 8:193–203
Ruszczynski A (2006) Nonlinear optimization. Princeton University Press, Princeton
Shao W, Guo G (2018) Multiple-try simulated annealing algorithm for global optimization. Math Probl Eng 2018:1–11
Shao W, Zuo Y (2012) Simulated annealing for higher dimensional projection depth. Comput Stat Data Anal 56:4026–4036
Shao W, Guo G, Zhao G, Meng F (2014) Simulated annealing for the bounds of Kendall’s tau and Spearman’s rho. J Stat Comput Simul 84:2688–2699
Tukey JW (1975) Mathematrics and the picturing of data. Proc Int Cong Math 2:523–531
Zuo Y, Serfling R (2000) General notions of statistical depth function. Ann Stat 28:461–482
Acknowledgements
The authors would like to thank the Editor, Associate Editor and reviewers for their insightful comments and constructive suggestions, which led to distinctive improvements in this paper. The first author’s research was partially supported by the National Natural Science Foundation of China (11501320, 71471101, 11426143), the Natural Science Foundation of Shandong Province (ZR2014AP008) and the Natural Science Foundation of Qufu Normal University (bsqd20130114).
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Shao, W., Zuo, Y. Computing the halfspace depth with multiple try algorithm and simulated annealing algorithm. Comput Stat 35, 203–226 (2020). https://doi.org/10.1007/s00180-019-00906-x
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DOI: https://doi.org/10.1007/s00180-019-00906-x