Impact on Computational Geometry on Depth-Based Statistics

今天的现实生活实验产生了大量的多元数据集。统计分析工具，有效地和准确地捕捉这些实验的多变量特征是必要的。经典的统计分析需要对数据的潜在概率分布进行初步假设。这个初步假设影响了分析。越来越多的统计学家提倡多元数据分析的数据深度（DD）的几何概念，因为它不需要对数据的概率分布进行预先假设，并处理离群值。基于数据深度的分析方法是存在的，但它们中的大多数还不足以有效地处理大型数据集。计算几何（CG）专注于几何问题的复杂性分析和有效算法解决方案的设计。本项目应用CG技术开发更有效的DD分析工具。航空安全分析、生物信息学、临床数据挖掘和统计过程控制是潜在的应用。本研究的主要任务是：解决半空间深度和简单深度等值线的计算问题;评估深度等值线对数据集多变量特征量化及其可视化的适用性;扩展二维算法并评估高维近似算法;探索离群值检测的新方法并评估其有效性。智力的优点来自于DD的实际实施对复杂几何问题的解决方案的依赖，应用科学家对统计分析和可视化新方法的需求，以及先前记录的可比调查的成功。科学家将DD应用于更大的真实世界数据集并提取新见解的实用算法，用于增强理解的新可视化工具，统计学家基于新计算见解的DD公式的改进，以及对不同大学预科生，大学生和研究生的培训提供了更广泛的影响。

Today's real-life experiments generate massive multivariate datasets. Statistical analysis tools that efficiently and accurately capture the multivariate features of such experiments are needed. Classical statistical analysis requires a preliminary assumption as to the underlying probability distribution of the data. This preliminary assumption affects the analysis. Increasingly, statisticians are advocating the geometric notion of data depth (DD) for multivariate data analysis as it requires no prior assumptions on the probability distribution of data and handles outliers. Data-depth-based analysis methods exist, but most of them are not yet sufficiently efficient to handle large datasets. Computational Geometry (CG) focuses on the complexity analysis of geometric problems and the design of effective algorithmic solutions. This project applies CG techniques to develop more efficient tools for DD analysis. Aviation safety analysis, bioinformatics, clinical data mining and statistical process control are potential applications. This research addresses underlying CG issues in the development of efficient practical algorithms for DD and undertakes these major tasks: resolve computational problems related to half-space-depth and simplicial-depth contours; evaluate the applicability of depth contours to quantification of multivariate features of dataset and to their visualization; expand two-dimensional algorithms and assess approximation algorithms for high dimensions; explore new approaches for outlier detection and assess their validity. Intellectual merit derives from the dependence of practical implementations of DD on solutions to complex geometric questions, the demand from applied scientists for new methods of statistical analysis and visualization, and the prior recorded success of comparable investigations. The practical algorithms for scientists to apply DD to larger real-world datasets and extract fresh insights, the fresh visualization tools for enhanced understanding, the refinements in the statisticians' DD formulations based on new computational insights, and the training of diverse pre-college, college, and graduate students provide the broader impact.