• P-ISSN 0974-6846 E-ISSN 0974-5645

Indian Journal of Science and Technology


Indian Journal of Science and Technology

Year: 2023, Volume: 16, Issue: 26, Pages: 1927-1934

Original Article

Computing Robust Measure of Location on Multivariate Statistical Data Using Euclidean Depth Procedures

Received Date:11 April 2023, Accepted Date:05 June 2023, Published Date:27 June 2023


Objectives: To suggest reliable location parameters (central value) in multivariate datasets using data depth procedures in order to reduce the presence of outliers. Methods: Applying depth techniques in both outlier-free and outliercontaining scenarios, the data sets starsCYG and delivery time data are utilized to determine the measure of location. Various classical and robust data depth procedures are used to find the location parameters, namely Mahalanobis depth, Tukey’s half space depth, Projection depth, Zonoid depth, Spatial depth, and L2 Depth (Euclidean Depth). Distance-Distance plot is used for identifying the outliers. Further, it has been researched how well the data depth processes work by computing the parameters under actual and simulation environments, with and without outliers by considering different levels of contaminations (0%, 1%, 2%, 3%, 5%, 10%, 20%, 30%, 40%). Findings: From the two data sets studied, Halfspace depth and Euclidean (using MCD estimator) give the same location parameters if the anomalies are present. These two procedures work equally well and more effectively than the others. The robust depth procedures work well if the outliers are present in the datasets and from the simulation study it can handle a certain level of contamination present in the data set. Novelty: Any dataset that contains outliers makes analysis results risky. Robust statistical techniques can tolerate some getting contaminated. According to the study, even if the data contains outliers, the Depth processes employing robust estimators can withstand a certain amount of contamination and still produce accurate findings.

Keywords: Location; Data Depth; Outliers; Mahalanobis Distance; Robust


  1. Geenens G, Nieto-Reyes A, Francisci G. Statistical depth in abstract metric spaces. Statistics and Computing. 2023;33(2). Available from: https://doi.org/10.1007/s11222-023-10216-4
  2. Mosler K, Mozharovskyi P. Choosing Among Notions of Multivariate Depth Statistics. arXiv. 2022;37(3):37. Available from: https://doi.org/10.48550/arXiv.2004.01927
  3. Dai X, Lopez-Pintado S. Tukey’s Depth for Object Data. Journal of the American Statistical Association. 2021;2022. Available from: https://doi.org/10.48550/arXiv.2109.00493
  4. Liu X, Li Y. General Notions of Regression Depth Functions. Statistics Methodology. 2020. Available from: https://arxiv.org/abs/1710.03904
  5. Priya M, Karthikeyan M. Outlier Detection Technique for Multivariate Data: Using Classical and Robust Mahalanobis Distance Method. Journal of Emerging Technologies and InnovativeResearch. 2019(6):766–769. Available from: https://www.jetir.org/papers/JETIR1906J50.pdf
  6. Muthukrishnan R, Gowri D, Ramkumar N. Measure of Location using Data Depth Procedures. International Journal of Scientific Research in Mathematical and Statistical Sciences. 2018(5):273–277. Available from: https://doi.org/10.26438/ijsrmss/v5i6.273277
  7. Pawar SD, Digambar T, Shirke. Data Depth-Based Nonparametric Test for Multivariate Scales. Journal of Statistical Theory and Practice. 2022;p. 16. Available from: https://doi.org/10.1007/s42519-021-00236-6
  8. Vardi Y, Zhang CH. The multivariate L 1-median and associated data depth. Proceedings of the National Academy of Sciences. 2000;97(4):1423–1426. Available from: http://dx.doi.org/10.1073/pnas.97.4.1423
  9. Mahalanobis P. On the generalized distance in statistics. Proceedings of the National Academy India. 1936;p. 49–55. Available from: https://www.scirp.org/(S(i43dyn45teexjx455qlt3d2q))/reference/ReferencesPapers.aspx?ReferenceID=1649365
  10. Liu X, Zuo Y. Computing projection depth and its associated estimators. Statistics and Computing. 2014;24(1):51–63. Available from: https://doi.org/10.1007/s11222-012-9352-6
  11. Muthukrishnan R, Ramkumar N. Computing Adjusted Projection depth using GSO algorithm. AIP Conference Proceedings. 2020. Available from: https://doi.org/10.1063/5.0016972
  12. Koshevoy G, Mosler K. Zonoid trimming for multivariate distributions. The Annals of Statistics. 1997;25(5). Available from: https://www.jstor.org/stable/2959013
  13. Chaudhuri P. On a geometric notion of quantiles for multivariate data. Journal of the American Statistical Association. 1996;(434) 862–872. Available from: https://doi.org/10.2307/2291681
  14. Rousseeuw PJ, Leroy AM. Robust Regression and Outlier Detection. New York. Wiley. 1987.
  15. Lopuhaa HP, Rousseeuw PJ. Breakdown Points of Affine Equivariant Estimators of Multivariate Location and Covariance Matrices. The Annals of Statistics. 1991;19(1):229–248. Available from: https://doi.org/10.124/AOS/1176347978
  16. Zuo YJ, Serfling R. General notions of statistical depth function. The Annals of Statistics. 2000;28(2):461–482. Available from: https://doi.org/10.124/aos/1016218226


© 2023 R Muthukrishnan & Nair. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee)


Subscribe now for latest articles and news.