Depth and depth-based classification with R-package ddalpha

Following the seminal idea of Tukey, data depth is a function that measures how close an arbitrary point of the space is located to an implicitly defined center of a data cloud. Having undergone theoretical and computational developments, it is now employed in numerous applications with classification being the most popular one. The R-package ddalpha is a software directed to fuse experience of the applicant with recent achievements in the area of data depth and depth-based classification. ddalpha provides an implementation for exact and approximate computation of most reasonable and widely applied notions of data depth. These can be further used in the depth-based multivariate and functional classifiers implemented in the package, where the $DD\alpha$-procedure is in the main focus. The package is expandable with user-defined custom depth methods and separators. The implemented functions for depth visualization and the built-in benchmark procedures may also serve to provide insights into the geometry of the data and the quality of pattern recognition.

Classification with the pot-pot plot

We propose a procedure for supervised classification that is based on potential functions. The potential of a class is defined as a kernel density estimate multiplied by the class's prior probability. The method transforms the data to a potential-potential (pot-pot) plot, where each data point is mapped to a vector of potentials. Separation of the classes, as well as classification of new data points, is performed on this plot. For this, either the $\alpha$-procedure ($\alpha$-P) or $k$-nearest neighbors ($k$-NN) are employed. For data that are generated from continuous distributions, these classifiers prove to be strongly Bayes-consistent. The potentials depend on the kernel and its bandwidth used in the density estimate. We investigate several variants of bandwidth selection, including joint and separate pre-scaling and a bandwidth regression approach. The new method is applied to benchmark data from the literature, including simulated data sets as well as 50 sets of real data. It compares favorably to known classification methods such as LDA, QDA, max kernel density estimates, $k$-NN, and $DD$-plot classification using depth functions.