Abstract

Partially observed functional data are frequently encountered in applications and are the object of an increasing interest by the literature. We here address the problem of measuring the centrality of a datum in a partially observed functional sample. We propose an integrated functional depth for partially observed functional data, dealing with the very challenging case where partial observability can occur systematically on any observation of the functional dataset. In particular, differently from many techniques for partially observed functional data, we do not request that some functional datum is fully observed, nor we require that a common domain exist, where all of the functional data are recorded. Because of this, our proposal can also be used in those frequent situations where reconstructions methods and other techniques for partially observed functional data are inapplicable. By means of simulation studies, we demonstrate the very good performances of the proposed depth on finite samples. Our proposal enables the use of benchmark methods based on depths, originally introduced for fully observed data, in the case of partially observed functional data. This includes the functional boxplot, the outliergram and the depth versus depth classifiers. We illustrate our proposal on two case studies, the first concerning a problem of outlier detection in German electricity supply functions, the second regarding a classification problem with data obtained from medical imaging. Supplementary materials for this article are available online.

摘要

部分观察到的函数数据在应用中经常遇到，并且是文献日益关注的对象。我们在这里解决了测量部分观察到的功能样本中数据的中心性的问题。我们提出了针对部分观察到的功能数据的集成功能深度，以处理非常具有挑战性的情况，即部分可观察性可以在功能数据集的任何观察中系统地发生。特别是，与部分观察功能数据的许多技术不同，我们不要求完全观察某些功能数据，也不要求存在记录所有功能数据的公共域。因为这，我们的建议也可以用于那些对部分观测的功能数据进行重建方法和其他技术不适用的常见情况。通过模拟研究，我们证明了所提出的深度在有限样本上的非常好的性能。我们的建议允许在部分观察到的功能数据的情况下使用基于深度的基准方法，该方法最初是针对完全观察到的数据引入的。这包括函数箱线图、离群值图和深度与深度分类器。我们通过两个案例研究来说明我们的建议，第一个案例涉及德国供电功能中的异常值检测问题，第二个案例涉及从医学成像获得的数据的分类问题。本文的补充材料可在线获取。