Depth-based nonparametric description of functional data, with emphasis on use of spatial depth

•Methods for displaying sample quantile curves for a functional data set.•Methods for computing and displaying confidence bands for population quantile curves for a functional data model.•Formulation of double-fence functional boxplot to better identify shape outliers.•Validation of the spatial depth approach for nonparametric description of functional data.

Statistical depth and related quantile functions, originally introduced for nonparametric description and analysis of multivariate data in a way sensitive to inherent geometry, are in active development for functional data and in this setting offer special options since the data may be visualized regardless of dimension. This paper provides depth-based methods for revealing the structure of a functional data set in terms of relevant sample quantile curves displayed at selected levels, and for constructing and displaying confidence bands for corresponding “population” versions. Also, the usual functional boxplot is enhanced, by adding inner fences to flag possible shape outliers, along with the outer fences that flag location outliers. This enables the boxplot to serve as a stand-alone tool for functional data, as with univariate and multivariate data. Further, the spatial depth approach, well-established for multivariate data, is investigated for nonparametric description of functional data along these lines. In comparison with four other commonly used depth approaches for functional data, over a range of actual and simulated data sets, the spatial depth approach is seen to offer a very competitive combination of robustness, efficiency, computational ease, simplicity, and versatility.