Trajectory functional boxplots

With the development of data-monitoring techniques in various fields of science, multivariate functional data are often observed. Consequently, an increasing number of methods have appeared to extend the general summary statistics of multivariate functional data. However, trajectory functional data, as an important subtype, have not been studied very well. This article proposes two informative exploratory tools: the trajectory functional boxplot and the modified simplicial band depth (MSBD) versus wiggliness of directional outlyingness (WO) plot, to visualize the centrality of trajectory functional data. The newly defined WO index effectively measures the shape variation of curves and hence serves as a detector for shape outliers; additionally, MSBD provides a centre-outward ranking and works as a detector for magnitude outliers. Using these two measures, the functional boxplot of the trajectory reveals centre-outward patterns and potential outliers using the raw curves, whereas the MSBD-WO plot illustrates such patterns and outliers in a space spanned by MSBD and WO. The proposed methods are validated on hurricane path data and migration trace data recorded from two types of birds.