Fast DD-classification of functional data

A fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional hypercube. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the $DD$-plot, which is a subset of the unit hypercube. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification on $[0,1]^q$. The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik-Cervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as a benchmark study

Adaptivity to Noise Parameters in Nonparametric Active Learning

This work addresses various open questions in the theory of active learning for nonparametric classification. Our contributions are both statistical and algorithmic: -We establish new minimax-rates for active learning under common \textit{noise conditions}. These rates display interesting transitions -- due to the interaction between noise \textit{smoothness and margin} -- not present in the passive setting. Some such transitions were previously conjectured, but remained unconfirmed. -We present a generic algorithmic strategy for adaptivity to unknown noise smoothness and margin; our strategy achieves optimal rates in many general situations; furthermore, unlike in previous work, we avoid the need for \textit{adaptive confidence sets}, resulting in strictly milder distributional requirements

The DDG^G-classifier in the functional setting

The Maximum Depth was the first attempt to use data depths instead of multivariate raw data to construct a classification rule. Recently, the DD-classifier has solved several serious limitations of the Maximum Depth classifier but some issues still remain. This paper is devoted to extending the DD-classifier in the following ways: first, to surpass the limitation of the DD-classifier when more than two groups are involved. Second to apply regular classification methods (like $k$NN, linear or quadratic classifiers, recursive partitioning,...) to DD-plots to obtain useful insights through the diagnostics of these methods. And third, to integrate different sources of information (data depths or multivariate functional data) in a unified way in the classification procedure. Besides, as the DD-classifier trick is especially useful in the functional framework, an enhanced revision of several functional data depths is done in the paper. A simulation study and applications to some classical real datasets are also provided showing the power of the new proposal.Comment: 29 pages, 6 figures, 6 tables, Supplemental R Code and Dat

Nonparametrically consistent depth-based classifiers

We introduce a class of depth-based classification procedures that are of a nearest-neighbor nature. Depth, after symmetrization, indeed provides the center-outward ordering that is necessary and sufficient to define nearest neighbors. Like all their depth-based competitors, the resulting classifiers are affine-invariant, hence in particular are insensitive to unit changes. Unlike the former, however, the latter achieve Bayes consistency under virtually any absolutely continuous distributions - a concept we call nonparametric consistency, to stress the difference with the stronger universal consistency of the standard $k$NN classifiers. We investigate the finite-sample performances of the proposed classifiers through simulations and show that they outperform affine-invariant nearest-neighbor classifiers obtained through an obvious standardization construction. We illustrate the practical value of our classifiers on two real data examples. Finally, we shortly discuss the possible uses of our depth-based neighbors in other inference problems.Comment: Published at http://dx.doi.org/10.3150/13-BEJ561 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm