A Topologically Valid Definition of Depth for Functional Data
A Topologically Valid Definition of Depth for Functional Data
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DOI：
10.1214/15-sts532
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发表时间：
2016-02-01
影响因子：
5.7
通讯作者：
Battey, Heather
中科院分区：
文献类型：
--
作者：
Nieto-Reyes, Alicia;Battey, Heather
这项工作的主要重点是在六个属性的基础上提供功能数据统计深度的正式定义，识别连续性、平滑性和邻接性等拓扑特征。我们的深度定义属性之一是解决功能数据固有的部分可观察性这一微妙挑战的属性，其实现对经验深度的性能提供了最低限度的保证，超出了理想化且实际上不可行的完全可观察性的情况。作为一个偶然的产品，满足我们定义的功能深度实现了通常归因于深度的鲁棒性，尽管深度的多元定义中缺乏正式的保证。我们证明了六个广泛使用的函数深度建议的属性是否满足，从而为深度函数的选择提供了系统的基础。
The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfillment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve a robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfillment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function.