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The Mahalanobis distance for functional data with applications to classification

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  • Esdras Joseph

    ()

  • Pedro Galeano

    ()

  • Rosa E. Lillo

    ()

Abstract

This paper presents a general notion of Mahalanobis distance for functional data that extends the classical multivariate concept to situations where the observed data are points belonging to curves generated by a stochastic process. More precisely, a new semi-distance for functional observations that generalize the usual Mahalanobis distance for multivariate datasets is introduced. For that, the development uses a regularized square root inverse operator in Hilbert spaces. Some of the main characteristics of the functional Mahalanobis semi-distance are shown. Afterwards, new versions of several well known functional classification procedures are developed using the Mahalanobis distance for functional data as a measure of proximity between functional observations. The performance of several well known functional classification procedures are compared with those methods used in conjunction with the Mahalanobis distance for functional data, with positive results, through a Monte Carlo study and the analysis of two real data examples

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Bibliographic Info

Paper provided by Universidad Carlos III, Departamento de Estadística y Econometría in its series Statistics and Econometrics Working Papers with number ws131312.

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Date of creation: May 2013
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Handle: RePEc:cte:wsrepe:ws131312

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Related research

Keywords: Classification methods Functional data analysis; Functional Mahalanobis semi-distance; Functional principal components;

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References

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  1. Mas, André, 2007. "Weak convergence in the functional autoregressive model," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1231-1261, July.
  2. Li, Bin & Yu, Qingzhao, 2008. "Classification of functional data: A segmentation approach," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4790-4800, June.
  3. Aurore Delaigle & Peter Hall, 2012. "Achieving near perfect classification for functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(2), pages 267-286, 03.
  4. Shin, Hyejin, 2008. "An extension of Fisher's discriminant analysis for stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1191-1216, July.
  5. Alonso, Andrés M. & Casado, David & Romo, Juan, 2012. "Supervised classification for functional data: A weighted distance approach," Computational Statistics & Data Analysis, Elsevier, vol. 56(7), pages 2334-2346.
  6. Peter Hall & Mohammad Hosseini-Nasab, 2006. "On properties of functional principal components analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 109-126.
  7. Amparo Baíllo & Antonio Cuevas & Juan Antonio Cuesta‐Albertos, 2011. "Supervised Classification for a Family of Gaussian Functional Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 38(3), pages 480-498, 09.
  8. Gareth M. James & Trevor J. Hastie, 2001. "Functional linear discriminant analysis for irregularly sampled curves," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(3), pages 533-550.
  9. Wang, Xiaohui & Ray, Shubhankar & Mallick, Bani K., 2007. "Bayesian Curve Classification Using Wavelets," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 962-973, September.
  10. Cristian Preda & Gilbert Saporta & Caroline Lévéder, 2007. "PLS classification of functional data," Computational Statistics, Springer, vol. 22(2), pages 223-235, July.
  11. Ferraty, F. & Vieu, P., 2003. "Curves discrimination: a nonparametric functional approach," Computational Statistics & Data Analysis, Elsevier, vol. 44(1-2), pages 161-173, October.
  12. Carlo Sguera & Pedro Galeano & Rosa E. Lillo, 2012. "Spatial depth-based classification for functional data," Statistics and Econometrics Working Papers ws120906, Universidad Carlos III, Departamento de Estadística y Econometría.
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