The Quantile Integrated Depth with Applications to Noisy Functional Data

Functional data analysis involves data for which the basic unit of observation is a function or image. The development of robust exploratory tools and inferential methods is very much needed since few assumptions can be made about the generating process. Data depth, a well-known nonparametric tool for analyzing functional data, provides a rigorous method for ranking a sample of curves from the center outwards, allowing for robust inference and outlier detection. Several notions of depth for functional data have been introduced in the last few decades. Here we develop a new family of depths, termed quantile integrated depth (QID), that are based on integrating up to the Kth quantile of the univariate depths. We show that this new family of depths has desirable properties, including a type of invariance, maximality at the center, and monotonicity with respect to the deepest point. In addition, since functional data are commonly observed with noise, we explore the effect of noise on different notions of depth. Compared to alternatives, the proposed QID is shown to be robust and perform well on noisy functional data. We also illustrate the advantages of using QIDK to identify potential hard-to-detect shape outliers.