Statistical Methods & Applications

, Volume 24, Issue 2, pp 257–261 | Cite as

Hubert, Rousseeuw and Segaert: multivariate functional outlier detection

  • Aldo Corbellini
  • Marco RianiEmail author
  • Anthony C. Atkinson

As would be expected from these authors, this is an interesting and well-written paper. It provides a stimulating introduction to robustness problems in the analysis of functional data. We have four general comments, none extensive.

Robust bivariate boxplots

The “bag-plot” of Rousseeuw et al. (1990), which you exemplify in your Figure 8, provides a polygonal approximation to the unknown distribution. The robust bivariate boxplot introduced by Zani et al. (1998) provides a smoother approximation to the unknown distribution. We briefly recall some of its properties.

Zani et al. (1998) use the peeling of convex hulls to, in the tradition of very robust statistics, find a region that contains approximately 50 % of the data. Peeling of hulls continues until the first one is obtained which includes not more than 50 % of the data (and asymptotically half the data as the sample size increases). The “50 % hull” so found is smoothed using B-splines, with cubic pieces which use the vertices of...


Mahalanobis Distance Functional Data Kriging Model Unknown Distribution Functional Data Analysis 
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  1. Atkinson AC, Riani M (1997) Bivariate boxplots, multiple outliers, multivariate transformations and discriminant analysis: the 1997 Hunter lecture. Environmetrics 8:583–602CrossRefGoogle Scholar
  2. Atkinson AC, Riani M, Cerioli A (2004) Exploring multivariate data with the forward search. Springer, New YorkCrossRefzbMATHGoogle Scholar
  3. Harvey AC (1989) Forecasting, structural time series models and the Kalman filter. Cambridge University Press, CambridgeGoogle Scholar
  4. Maadooliat M, Huang JZ, Hu J (2015) Integrating data transformation in principal components analysis. J Comput Gr Stat 24:84–103Google Scholar
  5. Riani M, Zani S (1998) Generalized distance measures for asymmetric multivariate distributions. In: Rizzi A, Vichi M, Bock H-H (eds) Advances in data science and classification. Springer, Berlin, pp 503–508CrossRefGoogle Scholar
  6. Riani M, Cerioli A, Atkinson AC, Perrotta D (2014) Monitoring robust regression. Electr J Stat 8:642–673MathSciNetGoogle Scholar
  7. Rousseeuw PJ, Ruts I, Tukey JW (1990) The bagplot: a bivariate boxplot. Am Stat 53:87–88Google Scholar
  8. West M, Harrison PJ (1989) Bayesian forecasting and dynamic models. Springer, New YorkCrossRefzbMATHGoogle Scholar
  9. Yeo I-K, Johnson RA (2000) A new family of power transformations to improve normality or symmetry. Biometrika 87:954–959MathSciNetCrossRefzbMATHGoogle Scholar
  10. Zani S, Riani M, Corbellini A (1998) Robust bivariate boxplots and multiple outlier detection. Comput Stat Data Anal 28:257–270CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Aldo Corbellini
    • 1
  • Marco Riani
    • 1
    Email author
  • Anthony C. Atkinson
    • 2
  1. 1.Dipartimento di EconomiaUniversità di ParmaParmaItaly
  2. 2.Department of StatisticsLondon School of EconomicsLondonUK

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