Inference for Contour Sets

这个项目的中心对象是轮廓集或水平集。这些是函数（如回归函数或概率密度）超过给定阈值的集合。开发允许对轮廓集进行统计推断的方法是本项目的主要目标。在其中一个子项目中，研究者正在使用基于核估计的插件估计开发轮廓集的置信区域。这些方法学的发展得到了大样本理论的支持，该理论表明，所提出的置信区域是（渐近）有效的，这意味着它们保持预先指定的置信水平。本文将考虑建立信任区域的两种方法。一种是基于自举方法，另一种是基于插件水平集估计的大样本分布理论。对于后者，当适当标准化时，表明插件估计与其理论对应的l1 -距离是渐近正态的。在另一个子项目中，研究者正在分析最近在文献中开发的高维轮廓集估计计算的相关新算法。这里的重点是实用性。在科学中，等高线集通过几乎每个科学软件包附带的等高线图而闻名。在许多应用领域，这样的轮廓集对于得出科学结论至关重要。这些领域包括天文天空调查、流式细胞仪、雷区检测、地震数据分析、图像分割以及异常或新颖性检测，包括入侵检测、异常喷气发动机振动检测、医学成像和基于脑电图的癫痫发作分析。这些应用中使用的等高线集通常取决于观测数据。换句话说，这些集合是随机对象，因此，为了量化科学结论，对这些集合进行统计分析是可取的，甚至是必要的。这种统计分析方法的发展是这个项目的主要议题之一。到目前为止还没有这样的方法存在，尽管它的可用性显示出在上述许多应用领域具有直接影响的明显潜力。最近这一领域的活动急剧增加，强调了对等高线集估计的统计理解的重要性。然而，到目前为止，所有现有的工作，虽然从各个角度来看都很重要，但都不允许量化伴随轮廓集估计而来的统计不确定性。在这个项目中开发的统计方法以及这些发展背后的具有挑战性的理论是新颖的，并为现代统计领域增加了重要的洞察力。该项目还通过研究生的支持和他们在现代统计领域的教育对统计领域产生影响。

The central objects of this project are contour sets or level sets. These are sets on which a function, such as a regression function or a probability density, exceeds a given threshold. The development of methodology allowing to draw statistical inference about contour sets is the main objective of this project. In one of the subprojects the investigator is developing confidence regions for contour sets using plug-in estimates based on kernel estimation. These methodological developments are supported by large sample theory showing that the proposed confidence regions are (asymptotically) valid, meaning that they hold the pre-specified confidence level. Two approaches for the construction of confidence regions will be considered. One is based on the bootstrap methodology, and the other is based on large sample distribution theory for plug-in level set estimates. As for the latter it is shown that the L1-distance between the plug-in estimate and their theoretical counterpart is asymptotically normal when standardized appropriately. In another subproject the investigator is analyzing related novel algorithms for the computation of contour set estimates in high dimensions that have been developed in the literature recently. The focus here is practical applicability.In the sciences, contour sets are well-known via contour plots that come with almost every scientific software package. Such contour sets are crucial for drawing scientific conclusions in many fields of application. These fields include astronomical sky surveys, flow cytometrie, detection of minefields, analysis of seismic data, image segmentation, as well as anomaly or novelty detection including intrusion detection, detection of anomalous jet engine vibration, medical imaging and EEG-based seizure analysis. The contour sets used in these applications usually depend on observed data. In other words, these sets are random objects, and consequently a statistical analysis of these sets is desirable or even necessary, in order to quantify scientific conclusions. The development of methodology for such a statistical analysis is one of the main topics of this project. No such methodology exists so far, although its availability shows the clear potential to have an immediate impact in many of the fields of application mentioned above. The importance of a statistical understanding of contour set estimates is underlined by a recent sharp increase in activity in this field. However, so far all the existing work, while important from various points of view, does not allow for quantifying the statistical uncertainty that goes along with estimation of the contour sets. The statistical methodology developed in this project as well as the challenging theory underlying these developments is novel and adds significant insight to a modern field of statistics. The project also impacts the field of statistics via the support of graduate students and their education in a modern field of statistics.