Local half-region depth for functional data

Data depth has proved successful in the analysis of multivariate data sets, e.g., for deriving an overall center and assigning ranks to the observed units. Two key features are the directions of the ordering, from the center towards the outside, and the recognition of a unique center irrespective of the distribution being unimodal or multimodal. These properties derive from the monotonicity of the ranks that decrease along any ray from the deepest point. Recently, a wider framework allowing for the identification of partial centers was suggested in Agostinelli (2011). The corresponding generalized depth functions, called local depth functions, can record local fluctuations and be used for mode detection, identification of components in mixture models, and cluster analysis. As functional data are becoming more common, López-Pintado and Romo (2011) recently proposed a notion of half-region depth suited for functional data and for high-dimensional data. Here, we propose a local version of this concept, we study its theoretical properties, we define new similarity measures based on it, and we illustrate its behavior with examples based on real data sets.