Optimal Bayes classifiers for functional data and density ratios

SummaryBayes classifiers for functional data pose a challenge. One difficulty is that probability density functions do not exist for functional data, so the classical Bayes classifier using density quotients needs to be modified. We propose to use density ratios of projections onto a sequence of eigenfunctions that are common to the groups to be classified. The density ratios are then factorized into density ratios of individual projection scores, reducing the classification problem to obtaining a series of one-dimensional nonparametric density estimates. The proposed classifiers can be viewed as an extension to functional data of some of the earliest nonparametric Bayes classifiers that were based on simple density ratios in the one-dimensional case. By means of the factorization of the density quotients, the curse of dimensionality that would otherwise severely affect Bayes classifiers for functional data can be avoided. We demonstrate that in the case of Gaussian functional data, the proposed functional Bayes classifier reduces to a functional version of the classical quadratic discriminant. A study of the asymptotic behaviour of the proposed classifiers in the large-sample limit shows that under certain conditions the misclassification rate converges to zero, a phenomenon that has been referred to as perfect classification. The proposed classifiers also perform favourably in finite-sample settings, as we demonstrate through comparisons with other functional classifiers in simulations and various data applications, including spectral data, functional magnetic resonance imaging data from attention deficit hyperactivity disorder patients, and yeast gene expression data.