IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v69y1999i1p135-153.html
   My bibliography  Save this article

Halfspace Depth and Regression Depth Characterize the Empirical Distribution

Author

Listed:
  • Struyf, Anja J.
  • Rousseeuw, Peter J.

Abstract

For multivariate data, the halfspace depth function can be seen as a natural and affine equivariant generalization of the univariate empirical cdf. For any multivariate data set, we show that the resulting halfspace depth function completely determines the empirical distribution. We do this by actually reconstructing the data points from the depth contours. The data need not be in general position. Moreover, we prove the same property for regression depth.

Suggested Citation

  • Struyf, Anja J. & Rousseeuw, Peter J., 1999. "Halfspace Depth and Regression Depth Characterize the Empirical Distribution," Journal of Multivariate Analysis, Elsevier, vol. 69(1), pages 135-153, April.
    • Handle: RePEc:eee:jmvana:v:69:y:1999:i:1:p:135-153
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(98)91804-8
    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. E. Elisabet RutstrÃm, 1998. "Home-grown values and incentive compatible auction design," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(3), pages 427-441.
    2. Ruts, Ida & Rousseeuw, Peter J., 1996. "Computing depth contours of bivariate point clouds," Computational Statistics & Data Analysis, Elsevier, vol. 23(1), pages 153-168, November.
    3. Masse, J. C. & Theodorescu, R., 1994. "Halfplane Trimming for Bivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 48(2), pages 188-202, February.
    4. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    5. Peter J. Rousseeuw & Ida Ruts, 1996. "Bivariate Location Depth," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 45(4), pages 516-526, December.
    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. Hassairi, Abdelhamid & Regaieg, Ons, 2008. "On the Tukey depth of a continuous probability distribution," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2308-2313, October.
    2. Wei, Bei & Lee, Stephen M.S., 2012. "Second-order accuracy of depth-based bootstrap confidence regions," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 112-123.
    3. Christmann, Andreas, 2004. "Regression depth and support vector machine," Technical Reports 2004,54, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    4. Yi He & John H. J. Einmahl, 2017. "Estimation of extreme depth-based quantile regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 449-461, March.
    5. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2015. "Multivariate functional outlier detection," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 177-202, July.
    6. Christmann, Andreas & Steinwart, Ingo, 2005. "Consistency and robustness of kernel based regression," Technical Reports 2005,01, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    7. Gather, Ursula & Davies, P. Laurie, 2004. "Robust Statistics," Papers 2004,20, Humboldt University of Berlin, Center for Applied Statistics and Economics (CASE).
    8. Koshevoy, Gleb A., 2002. "The Tukey Depth Characterizes the Atomic Measure," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 360-364, November.
    9. Bernholt, Thorsten & Nunkesser, Robin & Schettlinger, Karen, 2005. "Computing the Least Quartile Difference Estimator in the Plane," Technical Reports 2005,51, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    10. Kong, Linglong & Zuo, Yijun, 2010. "Smooth depth contours characterize the underlying distribution," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2222-2226, October.
    11. Jonas Baillien & Irène Gijbels & Anneleen Verhasselt, 2023. "Flexible asymmetric multivariate distributions based on two-piece univariate distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(1), pages 159-200, February.
    12. Hamel, Andreas H. & Kostner, Daniel, 2018. "Cone distribution functions and quantiles for multivariate random variables," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 97-113.
    13. Marc Hallin & Zudi Lu & Davy Paindaveine & Miroslav Siman, 2012. "Local Constant and Local Bilinear Multiple-Output Quantile Regression," Working Papers ECARES ECARES 2012-003, ULB -- Universite Libre de Bruxelles.
    14. Christmann, Andreas & Steinwart, Ingo & Hubert, Mia, 2006. "Robust Learning from Bites for Data Mining," Technical Reports 2006,03, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    15. Stanislav Nagy, 2021. "Halfspace depth does not characterize probability distributions," Statistical Papers, Springer, vol. 62(3), pages 1135-1139, June.
    16. Laketa, Petra & Nagy, Stanislav, 2021. "Reconstruction of atomic measures from their halfspace depth," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    17. Xiaohui Liu & Karl Mosler & Pavlo Mozharovskyi, 2017. "Fast computation of Tukey trimmed regions and median in dimension p > 2," Working Papers 2017-71, Center for Research in Economics and Statistics.
    18. Christmann, Andreas & Steinwart, Ingo, 2003. "On robustness properties of convex risk minimization methods for pattern recognition," Technical Reports 2003,15, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    19. Gather, Ursula & Fried, Roland & Lanius, Vivian, 2005. "Robust detail-preserving signal extraction," Technical Reports 2005,54, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    20. Schettlinger, Karen & Fried, Roland & Gather, Ursula, 2006. "Robust Filters for Intensive Care Monitoring: Beyond the Running Median," Technical Reports 2006,23, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    21. Petra Laketa & Stanislav Nagy, 2022. "Halfspace depth for general measures: the ray basis theorem and its consequences," Statistical Papers, Springer, vol. 63(3), pages 849-883, June.
    22. Lanius, Vivian & Gather, Ursula, 2007. "Robust online signal extraction from multivariate time series," Technical Reports 2007,38, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.

    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. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2015. "Multivariate functional outlier detection," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 177-202, July.
    2. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2017. "Multivariate and functional classification using depth and distance," 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. 11(3), pages 445-466, September.
    3. Chakraborty, Biman & Chaudhuri, Probal, 1999. "A note on the robustness of multivariate medians," Statistics & Probability Letters, Elsevier, vol. 45(3), pages 269-276, November.
    4. Struyf, Anja & Rousseeuw, Peter J., 2000. "High-dimensional computation of the deepest location," Computational Statistics & Data Analysis, Elsevier, vol. 34(4), pages 415-426, October.
    5. Małgorzata Kobylińska, 2018. "Concept of Observation Depth Measure in the Statistical Analysis of E-Commerce Data in Enterprises," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 49, pages 515-526.
    6. Zuo, Yijun & Serfling, Robert, 2000. "Nonparametric Notions of Multivariate "Scatter Measure" and "More Scattered" Based on Statistical Depth Functions," Journal of Multivariate Analysis, Elsevier, vol. 75(1), pages 62-78, October.
    7. Romanazzi, Mario, 2001. "Influence Function of Halfspace Depth," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 138-161, April.
    8. Cascos Fernández, Ignacio & Ochoa Arellano, Maicol Jesús, 2019. "Multivariate expectile trimming and the BExPlot," DES - Working Papers. Statistics and Econometrics. WS 28434, Universidad Carlos III de Madrid. Departamento de Estadística.
    9. Averous, Jean & Meste, Michel, 1997. "Median Balls: An Extension of the Interquantile Intervals to Multivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 63(2), pages 222-241, November.
    10. Dyckerhoff, Rainer & Mozharovskyi, Pavlo, 2016. "Exact computation of the halfspace depth," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 19-30.
    11. Nolan, D., 1999. "On min-max majority and deepest points," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 325-333, July.
    12. Hamel, Andreas H. & Kostner, Daniel, 2022. "Computation of quantile sets for bivariate ordered data," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    13. Ochoa Arellano, Maicol Jesús & Cascos Fernández, Ignacio, 2022. "Data depth and multiple output regression, the distorted M-quantiles approach," DES - Working Papers. Statistics and Econometrics. WS 35465, Universidad Carlos III de Madrid. Departamento de Estadística.
    14. Mosler, Karl & Lange, Tatjana & Bazovkin, Pavel, 2009. "Computing zonoid trimmed regions of dimension d>2," Computational Statistics & Data Analysis, Elsevier, vol. 53(7), pages 2500-2510, May.
    15. Małgorzata Kobylińska, 2021. "Spatial Diversity of Organic Farming in Poland," Sustainability, MDPI, vol. 13(16), pages 1-19, August.
    16. Masse, Jean-Claude & Plante, Jean-Francois, 2003. "A Monte Carlo study of the accuracy and robustness of ten bivariate location estimators," Computational Statistics & Data Analysis, Elsevier, vol. 42(1-2), pages 1-26, February.
    17. Cascos Fernández, Ignacio, 2006. "The expected convex hull trimmed regions of a sample," DES - Working Papers. Statistics and Econometrics. WS ws066919, Universidad Carlos III de Madrid. Departamento de Estadística.
    18. López Pintado, Sara & Romo, Juan, 2005. "Depth-based classification for functional data," DES - Working Papers. Statistics and Econometrics. WS ws055611, Universidad Carlos III de Madrid. Departamento de Estadística.
    19. Belzunce, F. & Castano, A. & Olvera-Cervantes, A. & Suarez-Llorens, A., 2007. "Quantile curves and dependence structure for bivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5112-5129, June.
    20. Abellanas, Manuel & Claverol, Merce & Hurtado, Ferran, 2007. "Point set stratification and Delaunay depth," Computational Statistics & Data Analysis, Elsevier, vol. 51(5), pages 2513-2530, February.

    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:69:y:1999:i:1:p:135-153. 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.