Impact on Computational Geometry on Depth-Based Statistics

今天的现实生活实验产生了大量的多变量数据集。需要有效和准确地捕捉此类实验的多变量特征的统计分析工具。经典的统计分析要求对数据的潜在概率分布有一个初步假设。这一初步假设影响了分析。在多元数据分析中，越来越多的统计学家提倡数据深度的几何概念，因为它不需要关于数据的概率分布的先验假设，并且可以处理异常值。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)主要研究几何问题的复杂性分析和有效算法的设计。该项目应用CG技术来开发更高效的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。本研究致力于开发有效的数据挖掘实用算法中潜在的CG问题，并承担以下主要任务：解决与半空间深度和单纯深度等值线相关的计算问题；评估深度等值线对数据集的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；探索新的离群点检测方法并评估其有效性。智力上的优势来自于数据挖掘的实际实施依赖于复杂几何问题的解决方案，应用科学家对新的统计分析和可视化方法的需求，以及以前记录的可比研究的成功。科学家将数据挖掘应用于更大的真实世界数据集并提取新见解的实用算法，用于增强理解的新鲜可视化工具，统计学家基于新的计算见解对数据挖掘公式进行的改进，以及对不同的预科、大学和研究生的培训，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.