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Title: On depths for noisy functional data: The quantile Integrated depth Authors:  Sara Lopez Pintado - Northeastern University (United States) [presenting]
Abstract: Functional data analysis is an exciting field of statistics where the basic unit of observation is a function or image. The development of robust exploratory tools and inferential methods for functional data is very much needed. Data depth is a well-known and useful non-parametric tool for analyzing functional data. It provides a way of ranking a sample of curves from the centre outwards and of defining robust statistics. Several notions of depth for functional data were introduced in the literature in the last few decades. These functional depths usually satisfy desirable properties, such as some type of invariance, maximality at the center and monotonicity with respect to the deepest point. We develop a new family of depths denoted by quantile integrated depth (QID) based on integrating up to the K-th percentile/quantile of the univariate depths. The theoretical properties of this new family of depths are studied. In practice, functional data are often observed with noise. We explore the effect of noise on different notions of depth and propose the SAD (Spearman Agreement Depth) plot to compare the performance of these functional depths in the presence of noise. The proposed quantile integrated depths are shown to be robust to noisy functional data outperforming alternative functional depths. Procedures for choosing optimal tuning parameter K in QID based on the SAD plot are discussed.