On the Concept of Depth for Functional Data
On the Concept of Depth for Functional Data
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DOI：
10.1198/jasa.2009.0108
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发表时间：
2009-06-01
影响因子：
3.7
通讯作者：
Romo, Juan
中科院分区：
文献类型：
--
作者：
Lopez-Pintado, Sara;Romo, Juan
许多研究领域对功能数据的统计分析的需求日益增长。特别是，稳健的方法对于研究曲线非常重要，曲线是应用统计学中许多实验的输出。作为这一稳健分析的起点。我们提出建议、分析。并根据曲线的图形观察结果应用新的功能深度定义。给定一个函数集合。它建立了观察的“中心性”并提供样本曲线的自然中心向外排序。稳健的统计数据。可以根据该深度定义来定义诸如中值函数或修剪平均函数之类的函数。其有限维版本为多元数据提供了新的深度，这在计算上是可行的，并且对于研究高维观测很有用。因此。这种新的深度也适用于复杂的观察，例如微阵列数据、图像以及最近一些营销和金融研究中出现的观察。建立了这些新概念的自然属性并证明了样本深度的均匀一致性。仿真结果表明，对于某些污染模型，相应的基于深度的截尾均值比文献中提出的其他可能的位置估计器表现出更好的性能。数据深度也可用于筛选异常值。提出了新的深度概念检测“形状”异常值的能力。我们考虑了几个真实的数据集来说明这个新的深度概念。包括微阵列观测、天气数据的应用。和生长曲线。最后，通过这个深度，我们将 Wilcoxon 秩和检验推广到函数。它允许测试两组曲线是否来自同一群体。当应用于儿童生长曲线时，该功能等级测试显示男孩和女孩的不同生长模式。
The statistical analysis of functional data is a growing need in many research areas. In particular, a robust methodology is important to study curves, which are the output of many experiments in applied statistics. As a starting point for this robust analysis. we propose, analyze. and apply a new definition of depth for functional based on the graphic observations based presentation of the curves. Given a collection of functions. it establishes the "centrality" of an observation and provides a natural center-outward ordering of the sample curves. Robust statistics. such as the median function or a trimmed mean function, can be defined from this depth definition. Its finite-dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high-dimensional observations. Thus. this new depth is also suitable for complex observations such as microarray data, images, and those arising in some recent marketing and financial studies. Natural properties of these new concepts are established and the uniform consistency of the sample depth is proved. Simulation results show that the corresponding depth based trimmed mean presents better performance than other possible location estimators proposed in the literature for some contaminated models. Data depth can be also used to screen for outliers. The ability of the new notions of depth to detect "shape" outliers is presented. Several real datasets are considered to illustrate this new concept of depth. including applications to microarray observations, weather data. and growth curves. Finally, through this depth, we generalize to functions the Wilcoxon rank sum test. It allows testing, whether two groups of curves come from the same population. This functional rank test when applied to children growth curves shows different growth patterns for boys and girls.