Achieving near Perfect Classification for Functional Data

Abstract

Summary. 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 multivariate 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.

Selected statistical methods of data analysis for multivariate functional data

“…Methods of functional data analysis are becoming increasingly popular, e.g. in the cluster analysis (Jacques and Preda 2013;James and Sugar 2003;Peng and Müller 2008), classification (Chamroukhi et al 2013;Delaigle and Hall 2012;Mosler and Mozharovskyi 2015;Rossi and Villa 2006) and regression (Ferraty et al 2012;Goia and Vieu 2014;Kudraszow and Vieu 2013;Peng et al 2015;Rachdi and Vieu 2006;Wang et al 2015). Unfortunately, multivariate data methods cannot be directly used for functional data, because of the problem of dimensionality and difficulty in putting functional data into order.…”

The Mahalanobis Distance for Functional Data With Applications to Classification

“…Note that any functional distance can be used to implement the procedure. In particular, Delaigle and Hall (2012) considered the case of G = 2 classes that have different mean and a common covariance operator and proposed to project the functions into a given direction and then compute the squared Euclidean distance between the observations. More precisely, Delaigle and Hall (2012) proposed to use the centroid classifier with the distance between χ 0 and the sample functional mean µ χg , for g = 1, 2, denoted by DH, and given by:…”

“…As it can be seen in the figure, the four scenarios appear to be complicated scenarios for classification purposes. For each generated dataset, the functional observations in the test sample are classified using the following procedures: (1) the kNN procedure with seven different functional distances, the L 1 , L 2 and L ∞ distances as proposed by Baíllo et al (2011), the functional principal components (FPC) semi-distance assuming either a common or a different covariance operator, denoted by F P C C and F P C D , respectively, and the functional Mahalanobis (FM) semi-distance assuming either a common or a different covariance operator, denoted by F M C and F M D , respectively, as proposed in Section 3; (2) the centroid procedure with eight different functional distances, the first seven as in the kNN procedure and the distance proposed by Delaigle and Hall (2012) given in (17) and denoted by DH; (3) the linear and quadratic Bayes classification rules as proposed in Section 3, denoted by F LBCR and F QBCR, respectively; and (4) the multivariate linear and quadratic Bayes classification rules applied on the coefficients of the B-splines basis representation, denoted by LBCR Coef. and QBCR Coef., respectively.…”

“…The centroid procedure for functional datasets, proposed by Delaigle and Hall (2012), is probably the fastest and simplest classification method for functional observations. The centroid method consists in assigning a new function χ 0 to the class with closer mean.…”

“…For instance, Biau et al (2005) proposed to filter the training samples in the Fourier basis and to apply the kNN method to the first Fourier coefficients of the expansion, while Baíllo et al (2011) derived several consistency results of the kNN procedure for a particular type of Gaussian processes. Additionally, the centroid method based on assign the function to the group with closer mean has been adapted to the functional framework by Delaigle and Hall (2012). Alternatively, several papers have extended the Fisher's discriminant analysis to the functional framework.…”