Achieving near perfect classification for functional data

• Published in: Journal of the Royal Statistical Society. Series B, Statistical methodology Vol. 74; no. 2; pp. 267 - 286
• Main Authors: Delaigle, Aurore, Hall, Peter
• Format: Journal Article
• Language: English
• Published: Oxford, UK Blackwell Publishing Ltd 01.03.2012; Wiley-Blackwell; Royal Statistical Society; Oxford University Press
• Series: Journal of the Royal Statistical Society Series B
• Subjects: Asymptotic properties; Bayes classifier; Centroid method classifier; Centroids; Classification; Consistency; Covariance; Cross-validation; Data analysis; Dimension reduction; Eigenvalues; Error rate; Error rates; Estimators; Functional analysis; Gaussian process; Information classification; Least squares; Least squares method; Linear model; Linear models; Linear regression; Mathematical analysis; Multivariate analysis; Projection; Samples; Standard deviation; Statistical analysis; Statistical methods; Statistics; Studies
• ISSN: 1369-7412; 1467-9868
• DOI: 10.1111/j.1467-9868.2011.01003.x

We show that, in functional data classification problems, perfect asymptotic classification is often possible, making use of the intrinsic very high dimensional nature of functional data. This performance is often achieved by linear methods, which are optimal in important cases. These results point to a marked contrast between classification for functional data and its counterpart in conventional multivahate analysis, where the dimension is kept fixed as the sample size diverges. In the latter setting, linear methods can sometimes be quite inefficient, and there are no prospects for asymptotically perfect classification, except in pathological cases where, for example, a variance vanishes. By way of contrast, in finite samples of functional data, good performance can be achieved by truncated versions of linear methods. Truncation can be implemented by partial least squares or projection onto a finite number of principal components, using, in both cases, cross-validation to determine the truncation point. We establish consistency of the cross-validation procedure.