扩展带深的泛函分析

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摘要翻译：
数据深度的概念长期以来一直被用于在多变量环境中获得稳健的位置和尺度估计。一个观测的深度是它相对于一个数据集或一个分布的中心性的度量。一组多元观测值的数据深度转化为数据的中心向外排序。因此，数据深度提供了中值到多元设置（最深观测）的泛化，也可以用来筛选极端观测或异常值（数据深度低的观测）。数据深度已经被用于发展广泛的多元数据的稳健和非参数方法，如位置和尺度的非参数检验[Li和Liu(2004)]、多元秩检验[Liu和Singh(1993)]、非参数分类和聚类[Jornsten(2004)]和稳健回归[Rousseeuw和Hubert(1999)]。对于多元数据，人们提出了许多不同的数据深度概念。相反，函数数据的数据深度度量是最近才提出的[Fraiman和Muniz(1999),L\'{o}pez-Pintado和Romo(2006a)]。虽然这两个数据深度度量的定义都是由数据的功能方面驱动的，但这些度量本身实际上是不变的，关于域的排列（即定义函数的紧间隔）。因此，这些度量同样适用于数据维度没有显式排序的多变量数据。在本文中，我们探索了函数数据深度的一些扩展，以便考虑到数据维度的排序。
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英文标题：
《Functional analysis via extensions of the band depth》
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作者：
Sara L\'opez-Pintado, Rebecka Jornsten
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最新提交年份：
2007
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分类信息：

一级分类：Statistics        统计学
二级分类：Methodology        方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
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英文摘要：
  The notion of data depth has long been in use to obtain robust location and scale estimates in a multivariate setting. The depth of an observation is a measure of its centrality, with respect to a data set or a distribution. The data depths of a set of multivariate observations translates to a center-outward ordering of the data. Thus, data depth provides a generalization of the median to a multivariate setting (the deepest observation), and can also be used to screen for extreme observations or outliers (the observations with low data depth). Data depth has been used in the development of a wide range of robust and non-parametric methods for multivariate data, such as non-parametric tests of location and scale [Li and Liu (2004)], multivariate rank-tests [Liu and Singh (1993)], non-parametric classification and clustering [Jornsten (2004)], and robust regression [Rousseeuw and Hubert (1999)]. Many different notions of data depth have been developed for multivariate data. In contrast, data depth measures for functional data have only recently been proposed [Fraiman and Muniz (1999), L\'{o}pez-Pintado and Romo (2006a)]. While the definitions of both of these data depth measures are motivated by the functional aspect of the data, the measures themselves are in fact invariant with respect to permutations of the domain (i.e. the compact interval on which the functions are defined). Thus, these measures are equally applicable to multivariate data where there is no explicit ordering of the data dimensions. In this paper we explore some extensions of functional data depths, so as to take the ordering of the data dimensions into account.
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PDF链接：
https://arxiv.org/pdf/708.1107

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关键词：泛函分析 Multivariate observations distribution multivariat multivariate 函数 位置 用于 中心