IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v100y2009i6p1198-1218.html
   My bibliography  Save this article

Multivariate mode hunting: Data analytic tools with measures of significance

Author

Listed:
  • 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.

Suggested Citation

  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(08)00235-2
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Alexander Estes & David J. Lovell & Michael O. Ball, 2019. "Unsupervised prototype reduction for data exploration and an application to air traffic management initiatives," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 8(5), pages 467-510, December.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Álvarez, Adolfo, 2013. "Recombining partitions via unimodality tests," DES - Working Papers. Statistics and Econometrics. WS ws130706, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Baíllo, Amparo, 2003. "Total error in a plug-in estimator of level sets," DES - Working Papers. Statistics and Econometrics. WS ws032806, Universidad Carlos III de Madrid. Departamento de Estadística.
    3. Bock, Hans H., 1996. "Probabilistic models in cluster analysis," Computational Statistics & Data Analysis, Elsevier, vol. 23(1), pages 5-28, November.
    4. Mammen, Enno & Polonik, Wolfgang, 2013. "Confidence regions for level sets," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 202-214.
    5. Baíllo, Amparo, 2003. "Total error in a plug-in estimator of level sets," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 411-417, December.
    6. Dau, Hai Dang & Laloë, Thomas & Servien, Rémi, 2020. "Exact asymptotic limit for kernel estimation of regression level sets," Statistics & Probability Letters, Elsevier, vol. 161(C).
    7. Pavlides, Marios G. & Wellner, Jon A., 2012. "Nonparametric estimation of multivariate scale mixtures of uniform densities," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 71-89.
    8. Di, J. & Kolaczyk, E., 2010. "Complexity-penalized estimation of minimum volume sets for dependent data," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 1910-1926, October.
    9. Giovanni Paolo Crespi & Elisa Mastrogiacomo, 2020. "Qualitative robustness of set-valued value-at-risk," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 25-54, February.
    10. Paula Saavedra-Nieves & Rosa M. Crujeiras, 2022. "Nonparametric estimation of directional highest density regions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 16(3), pages 761-796, September.
    11. Kaido, Hiroaki & Molinari, Francesca & Stoye, Jörg, 2022. "Constraint Qualifications In Partial Identification," Econometric Theory, Cambridge University Press, vol. 38(3), pages 596-619, June.
    12. Francesca Molinari, 2020. "Microeconometrics with Partial Identi?cation," CeMMAP working papers CWP15/20, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    13. Victor Chernozhukov & Emre Kocatulum & Konrad Menzel, 2015. "Inference on sets in finance," Quantitative Economics, Econometric Society, vol. 6(2), pages 309-358, July.
    14. Yen-Chi Chen & Christopher R. Genovese & Larry Wasserman, 2017. "Density Level Sets: Asymptotics, Inference, and Visualization," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(520), pages 1684-1696, October.
    15. 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.
    16. W. M. Schaffer, 2009. "A Surfeit of Cycles," Energy & Environment, , vol. 20(6), pages 985-996, October.
    17. Moritz Herrmann & Fabian Scheipl, 2021. "A Geometric Perspective on Functional Outlier Detection," Stats, MDPI, vol. 4(4), pages 1-41, November.
    18. Burman, Prabir, 2002. "Estimation of equifrequency histograms," Statistics & Probability Letters, Elsevier, vol. 56(3), pages 227-238, February.
    19. Pascal Yiou & Julien Cattiaux & Aurélien Ribes & Robert Vautard & Mathieu Vrac, 2018. "Recent Trends in the Recurrence of North Atlantic Atmospheric Circulation Patterns," Complexity, Hindawi, vol. 2018, pages 1-8, February.
    20. Daniel Hlubinka & Miroslav Šiman, 2015. "On generalized elliptical quantiles in the nonlinear quantile regression setup," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 249-264, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:100:y:2009:i:6:p:1198-1218. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.