Inference for Contour Sets

University Of California-Davis, Davis CA

Investigators

Abstract

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.