On visual distances for spectrum-type functional data

A functional distance ${\mathbb H}$, based on the Hausdorff metric between the function hypographs, is proposed for the space ${\mathcal E}$ of non-negative real upper semicontinuous functions on a compact interval. The main goal of the paper is to show that the space $({\mathcal E},{\mathbb H})$ is particularly suitable in some statistical problems with functional data which involve functions with very wiggly graphs and narrow, sharp peaks. A typical example is given by spectrograms, either obtained by magnetic resonance or by mass spectrometry. On the theoretical side, we show that $({\mathcal E},{\mathbb H})$ is a complete, separable locally compact space and that the ${\mathbb H}$-convergence of a sequence of functions implies the convergence of the respective maximum values of these functions. The probabilistic and statistical implications of these results are discussed in particular, regarding the consistency of $k$-NN classifiers for supervised classification problems with functional data in ${\mathbb H}$. On the practical side, we provide the results of a small simulation study and check also the performance of our method in two real data problems of supervised classification involving mass spectra.