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Multivariate mode hunting: Data analytic tools with measures of significance

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  • Burman, Prabir
  • Polonik, Wolfgang

Abstract

Multivariate mode hunting is of increasing practical importance. Only a few such methods exist, however, and there usually is a trade-off between practical feasibility and theoretical justification. In this paper we attempt to do both. We propose a method for locating isolated modes (or better, modal regions) in a multivariate data set without pre-specifying their total number. Information on significance of the findings is provided by means of formal testing for the presence of antimodes. Critical values of the tests are derived from large sample considerations. The method is designed to be computationally feasible in moderate dimensions, and it is complemented by diagnostic plots. Since the null hypothesis under consideration is highly composite the proposed tests involve calibration in order to ensure a correct (asymptotic) level. Our methods are illustrated by application to real data sets.

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  • Burman, Prabir & Polonik, Wolfgang, 2009. "Multivariate mode hunting: Data analytic tools with measures of significance," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1198-1218, July.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:6:p:1198-1218
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    1. Ghislaine Gayraud & Judith Rousseau, 2005. "Rates of Convergence for a Bayesian Level Set Estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 639-660, December.
    2. Gregory Rozál & J. Hartigan, 1994. "The MAP test for multimodality," Journal of Classification, Springer;The Classification Society, vol. 11(1), pages 5-36, March.
    3. Burman, P. & Nolan, D., 1992. "Location-adaptive density estimation and nearest-neighbor distance," Journal of Multivariate Analysis, Elsevier, vol. 40(1), pages 132-157, January.
    4. Bai, Zhidong & Chen, Zehua & Wu, Yaohua, 2003. "Convergence rate of the best-r-point-average estimator for the maximizer of a nonparametric regression function," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 319-334, February.
    5. S. Corti & F. Molteni & T. N. Palmer, 1999. "Signature of recent climate change in frequencies of natural atmospheric circulation regimes," Nature, Nature, vol. 398(6730), pages 799-802, April.
    6. Polonik, Wolfgang & Wang, Zailong, 2005. "Estimation of regression contour clusters--an application of the excess mass approach to regression," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 227-249, June.
    7. Ilya S. Molchanov, 1998. "A Limit Theorem for Solutions of Inequalities," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 235-242, March.
    8. Polonik, Wolfgang, 1997. "Minimum volume sets and generalized quantile processes," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 1-24, July.
    9. J. Hartigan & Surya Mohanty, 1992. "The runt test for multimodality," Journal of Classification, Springer;The Classification Society, vol. 9(1), pages 63-70, January.
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    Cited by:

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    2. Christopher R. Genovese & Marco Perone-Pacifico & Isabella Verdinelli & Larry Wasserman, 2016. "Non-parametric inference for density modes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 99-126, January.
    3. Polonik, Wolfgang & Wang, Zailong, 2010. "PRIM analysis," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 525-540, March.
    4. Yen-Chi Chen, 2017. "Modal Regression using Kernel Density Estimation: a Review," Papers 1710.07004, arXiv.org, revised Dec 2017.
    5. Giovanna Menardi, 2016. "A Review on Modal Clustering," International Statistical Review, International Statistical Institute, vol. 84(3), pages 413-433, December.
    6. Lasse Holmström & Kyösti Karttunen & Jussi Klemelä, 2017. "Estimation of level set trees using adaptive partitions," Computational Statistics, Springer, vol. 32(3), pages 1139-1163, September.
    7. Hsu, Chih-Yuan & Wu, Tiee-Jian, 2013. "Efficient estimation of the mode of continuous multivariate data," Computational Statistics & Data Analysis, Elsevier, vol. 63(C), pages 148-159.
    8. José E. Chacón, 2020. "The Modal Age of Statistics," International Statistical Review, International Statistical Institute, vol. 88(1), pages 122-141, April.
    9. Álvarez, Adolfo, 2013. "Recombining partitions via unimodality tests," DES - Working Papers. Statistics and Econometrics. WS ws130706, Universidad Carlos III de Madrid. Departamento de Estadística.
    10. Konstantin Eckle & Nicolai Bissantz & Holger Dette & Katharina Proksch & Sabrina Einecke, 2018. "Multiscale inference for a multivariate density with applications to X-ray astronomy," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(3), pages 647-689, June.
    11. Alessandro Casa & Giovanna Menardi, 2022. "Nonparametric semi-supervised classification with application to signal detection in high energy physics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(3), pages 531-550, September.

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