  The direction of outlyingness is crucial to describing the centrality of
multivariate functional data. Motivated by this idea, we generalize classical
depth to directional outlyingness for functional data. We investigate
theoretical properties of functional directional outlyingness and find that the
total outlyingness can be naturally decomposed into two parts: magnitude
outlyingness and shape outlyingness which represent the centrality of a curve
for magnitude and shape, respectively. Using this decomposition, we provide a
visualization tool for the centrality of curves. Furthermore, we design an
outlier detection procedure based on functional directional outlyingness. This
criterion applies to both univariate and multivariate curves and simulation
studies show that it outperforms competing methods. Weather and
electrocardiogram data demonstrate the practical application of our proposed
framework.
