Functional depth is the functional data analysis technique that orders a functional data set. Unlike the case of data on the real line, defining this order is non-trivial, and particularly, with functional data, there are a number of properties that any depth should satisfy. We propose a new depth which both satisfies the properties required of a functional depth but also one which can be used in the case where there are a very large number of functional observations or in the case where the observations are functions of several continuous variables (such as images, for example). We give theoretical justification for our choice, and evaluate our proposed depth through simulation. We finally apply the proposed depth to the problem of yielding a completely non-parametric deconvolution of Positron Emission Tomography (PET) data for a very large number of curves across the image, as well as to the problem of finding a representative subject from a set of PET scans.
翻译:功能深度是功能性数据分析技术,它要求一个功能性数据集。与实际线上的数据不同,确定这一顺序并非三重数据,特别是,与功能性数据不同,任何深度都应满足若干特性。我们提议新的深度,既满足功能深度要求的特性,也满足功能性深度要求的特性,在观测功能性观测非常多的情况下,或在观测是若干连续变量(例如图像)的功能的情况下,可以使用这种深度。我们从理论上为我们的选择提供理由,并通过模拟评估我们提议的深度。我们最后将提议的深度应用于对图像上大量曲线产生完全非参数性分解活光学(PET)数据的问题,以及从一系列PET扫描中找到具有代表性的课题的问题。