稀疏函数数据的深度概念
A notion of depth for sparse functional data
摘要：
数据深度是一个众所周知的和有用的非参数工具，用于分析功能数据。它提供了一种从中心向外排列曲线样本的新方法，并定义了稳健的统计数据，如中位数或修剪均值。它也被用作功能异常检测方法和分类的构建块。在过去的几十年里，在文献中介绍了几个概念的深度功能数据。这些功能深度只能直接应用于在精细和常见网格上测量的曲线样本。在实践中，情况并非总是如此，曲线经常在稀疏和主题相关的网格中观察到。在这些情况下，通常的方法包括在一个共同的密集网格上估计的轨迹，并在深度分析中使用的估计。该方法忽略了与曲线估计步骤相关联的不确定性。我们的目标是扩展深度的概念，以便考虑这种不确定性。使用这两个功能估计和相关的置信区间，我们提出了一种新的方法，允许曲线估计的不确定性被纳入深度分析。我们描述了新的方法，使用修改后的带深度，虽然可以使用任何其他功能的深度。所提出的方法的性能示出在不同的设置，我们控制的稀疏度的模拟曲线。也被认为是一个真正的数据集组成的女性medflies产卵轨迹。结果显示了在计算稀疏函数数据的深度时使用不确定性的好处。
英文摘要：
Data depth is a well-known and useful nonparametric tool for analyzing functional data. It provides a novel way of ranking a sample of curves from the center outwards and defining robust statistics, such as the median or trimmed means. It has also been used as a building block for functional outlier detection methods and classification. Several notions of depth for functional data were introduced in the literature in the last few decades. These functional depths can only be directly applied to samples of curves measured on a fine and common grid. In practice, this is not always the case, and curves are often observed at sparse and subject dependent grids. In these scenarios the usual approach consists in estimating the trajectories on a common dense grid, and using the estimates in the depth analysis. This approach ignores the uncertainty associated with the curves estimation step. Our goal is to extend the notion of depth so that it takes into account this uncertainty. Using both functional estimates and their associated confidence intervals, we propose a new method that allows the curve estimation uncertainty to be incorporated into the depth analysis. We describe the new approach using the modified band depth although any other functional depth could be used. The performance of the proposed methodology is illustrated using simulated curves in different settings where we control the degree of sparsity. Also a real data set consisting of female medflies egg-laying trajectories is considered. The results show the benefits of using uncertainty when computing depth for sparse functional data.
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