采样曲线的凸壳面积：一种稳健的函数统计深度度量
The Area of the Convex Hull of Sampled Curves: a Robust Functional Statistical Depth Measure
摘要：
随着物联网时代传感器的普及，统计观测越来越多地以大规模（多变量）时间序列的形式提供。作为无监督的异常检测任务，航空安全管理、复杂基础设施的健康监测或欺诈检测等大量应用现在可以依赖于以更细的粒度获取和存储的此类功能数据。统计深度的概念，它反映了中心的任意观察w.r.t.在这方面，统计总体可能发挥关键作用，异常对应于具有“小”深度的观测。在近几十年来的理论和计算发展的支持下，它已被证明是非常有用的，特别是在函数空间中。然而，文献中记载的大多数方法包括独立评估形成时间序列的每个点的中心性，因此对可能的形状变化表现出一定的不敏感性。在本文中，我们提出了一个新的概念的功能深度的基础上的面积的凸包采样曲线，捕捉逐渐偏离中心，甚至超出了包络线的数据，在一个自然的方式。我们讨论的实际意义的功能深度和调查，其中之一是满意的概念，我们在这里推广的深度。估计和计算问题也得到解决，各种数值实验提供了经验证据的相关性的方法。
英文摘要：
With the ubiquity of sensors in the IoT era, statistical observations are becoming increasingly available in the form of massive (multivariate) time-series. Formulated as unsupervised anomaly detection tasks, an abundance of applications like aviation safety management, the health monitoring of complex infrastructures or fraud detection can now rely on such functional data, acquired and stored with an ever finer granularity. The concept of statistical depth, which reflects centrality of an arbitrary observation w.r.t. a statistical population may play a crucial role in this regard, anomalies corresponding to observations with 'small' depth. Supported by sound theoretical and computational developments in the recent decades, it has proven to be extremely useful, in particular in functional spaces. However, most approaches documented in the literature consist in evaluating independently the centrality of each point forming the time series and consequently exhibit a certain insensitivity to possible shape changes. In this paper, we propose a novel notion of functional depth based on the area of the convex hull of sampled curves, capturing gradual departures from centrality, even beyond the envelope of the data, in a natural fashion. We discuss practical relevance of commonly imposed axioms on functional depths and investigate which of them are satisfied by the notion of depth we promote here. Estimation and computational issues are also addressed and various numerical experiments provide empirical evidence of the relevance of the approach proposed.
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