Computational Statistics and Data Analysis 56 (2012) 2334–2346 Contents lists available at SciVerse ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Supervised classification for functional data: A weighted distance approach Andrés M. Alonso ∗ , David Casado, Juan Romo Departamento de Estadística, Universidad Carlos III de Madrid, 28903, Spain article info Article history: Received 9 March 2010 Received in revised form 10 January 2012 Accepted 10 January 2012 Available online 24 January 2012 Keywords: Supervised classification Discriminant analysis Functional data Weighted distances abstract A natural methodology for discriminating functional data is based on the distances from the observation or its derivatives to group representative functions (usually the mean) or their derivatives. It is proposed to use a combination of these distances for supervised classification. Simulation studies show that this procedure performs very well, resulting in smaller testing classification errors. Applications to real data show that this technique behaves as well as – and in some cases better than – existing supervised classification methods for functions. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Functional data have great – and growing – importance in Statistics. Most of the classical techniques for the finite- and high-dimensional frameworks have been adapted to cope with the infinite dimensions, but due to the curse of dimensionality, new and specific treatments are still required. As with other types of data, statisticians must put forward different steps – registration, missing data, representation, transformation, typicality – and tackle different tasks—modelization, discrimination or clustering, amongst others. In practice, curves can neither be registered continuously nor at infinitely many points. Thus, techniques dealing with high-dimensional data can sometimes be applied: Hastie et al. (1995), for example, adapt the discriminant analysis to cope with many highly correlated predictors, ‘‘such as those obtained by discretizing a function’’. Among the approaches designed for supervised classification or discrimination of functional data, some project the data onto a finite-dimensional space of functions and work with the coefficients. James and Hastie (2001) model the coefficients with ‘‘Gaussian distribution with common covariance matrix for all classes, by analogy with LDA [linear discriminant analysis]’’; their classification minimizes the distance to the group mean. The discriminant method of Hall et al. (2001) maximizes the likelihood, and although they propose a fully nonparametric density estimation, in practice multivariate Gaussian densities are considered, leading to quadratic discriminant analysis. Biau et al. (2003) apply k-nearest neighbors to the coefficients, while Rossi and Villa (2006) apply support vector machines. Berlinet et al. (2008) consider wavelet bases. The following proposals are designed to make direct use of the continuity of the functional data. Ferraty and Vieu (2003) classify new curves in the group with the highest posterior probability of membership kernel estimate. On the other hand, López-Pintado and Romo (2006) also take into account the continuous feature of the data and propose two classification methods based on the notion of depth for curves; in their first proposal new curves are assigned to the group with the closest trimmed mean, while the second method minimizes a weighted average distance to each element in the group. Nerini and Ghattas (2007) classify density functions with functional regression trees. Baíllo and Cuevas (2008) provide ∗ Corresponding author. E-mail addresses: andres.alonso@uc3m.es (A.M. Alonso), david.casado@uc3m.es (D. Casado), juan.romo@uc3m.es (J. Romo). 0167-9473/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2012.01.013