Regularized halfspace depth for functional data
针对传统半空间深度在函数型数据中退化的问题，提出一种通过正则化避免退化的新深度，能按形状或幅度排序数据并检测异常值，适用于平方可积函数空间。
Abstract Data depth is a powerful tool originally proposed to rank multivariate data from centre outward. In this context, one of the most archetypical depth notions is Tukey’s halfspace depth. In the last few decades, notions of depth have also been proposed for functional data. However, a naive extension of Tukey’s depth cannot handle functional data because of its degeneracy. Here, we propose a new halfspace depth for functional data, which avoids degeneracy by regularization. The halfspace projection directions are constrained to have a small reproducing kernel Hilbert space norm. Desirable theoretical properties of the proposed depth, such as isometry invariance, maximality at centre, monotonicity relative to a deepest point, upper semi-continuity, and consistency are established. Moreover, the regularized halfspace depth can rank functional data with varying emphasis in shape or magnitude, depending on the regularization. A new outlier detection approach is also proposed, which is capable of detecting both shape and magnitude outliers. It is applicable to trajectories in the space of all square-integrable functions, a very general space of functions that include nonsmooth trajectories. Based on extensive numerical studies, our methods are shown to perform well in detecting outliers of different types. Real data examples showcase the proposed depth.