An empirical comparison of global and local functional depths

A functional data depth provides a center-outward ordering criterion which allows the definition of measures such as median, trimmed means, central regions or ranks in a functional framework. A functional data depth can be global or local. With global depths, the degree of centrality of a curve $x$ depends equally on the rest of the sample observations, while with local depths, the contribution of each observation in defining the degree of centrality of $x$ decreases as the distance from $x$ increases. We empirically compare the global and the local approaches to the functional depth problem focusing on three global and two local functional depths. First, we consider two real data sets and show that global and local depths may provide different insights. Second, we use simulated data to show when we should expect differences between a global and a local approach to the functional depth problem.