The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion of ranking of observations which is absent in more than one dimension. Motivated by the rapid development of technology, in particular the advent of ‘Big Data’, we extend here that concept to general metric spaces, propose a natural depth measure and explore its properties as a statistical depth function. Working in a general metric space allows the depth to be tailored to the data at hand and to the ultimate goal of the analysis, a very desirable property given the polymorphic nature of modern data sets. This flexibility is thoroughly illustrated by several real data analyses.

Functional data analysis and artificial neural networks are the building blocks of the proposed methodology that distinguishes the movement patterns among c’s patients on different stages of the disease and classifies new patients to their appropriate stage of the disease. The movement patterns are obtained by the accelerometer device of android smartphones that the patients carry while moving freely. The proposed methodology is relevant in that it is flexible on the type of data to which it is applied. To exemplify that, it is analyzed a novel real three-dimensional functional dataset where each datum is observed in a different time domain. Not only is it observed on a difference frequency but also the domain of each datum has different length. The obtained classification success rate of $83\%$ indicates the potential of the proposed methodology.

Black-box techniques have been applied with outstanding results to classify, in a supervised manner, the movement patterns of Alzheimer’s patients according to their stage of the disease. However, these techniques do not provide information on the difference of the patterns among the stages. We make use of functional data analysis to provide insight on the nature of these differences. In particular, we calculate the center of symmetry of the underlying distribution at each stage and use it to compute the functional depth of the movements of each patient. This results in an ordering of the data to which we apply nonparametric permutation tests to check on the differences in the distribution, median and deviance from the median. We consistently obtain that the movement pattern at each stage is significantly different to that of the prior and posterior stage in terms of the deviance from the median applied to the depth. The approach is validated by simulation.