Impact on Computational Geometry on Depth-Based Statistics

计算几何对基于深度的统计的影响

• 批准号: 0431027
• 负责人: Diane Souvaine
• 金额: --
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-15 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0431027&HistoricalAwards=false
• 关键词: Impact; Computational; Geometry; Depth; Based

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.

今天现实生活中的实验产生了大量的多元数据集。统计分析工具，有效和准确地捕捉这些实验的多元特征是必要的。经典的统计分析需要对数据的潜在概率分布有一个初步的假设。这个初步的假设影响了分析。越来越多的统计学家在多变量数据分析中提倡数据深度（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，并且可以处理异常值。基于数据深度的分析方法已经存在，但大多数方法对于处理大型数据集还不够有效。计算几何（CG）侧重于几何问题的复杂性分析和有效求解算法的设计。这个项目应用CG技术来开发更有效的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。本研究解决了在开发高效实用的DD算法中潜在的CG问题，并承担了以下主要任务：解决与半空间深度和简单深度轮廓相关的计算问题；评估深度轮廓在数据集多变量特征量化及其可视化中的适用性；扩展二维算法，评估高维近似算法；探索异常值检测的新方法并评估其有效性。智力价值来自于DD的实际实施对复杂几何问题解决方案的依赖，应用科学家对统计分析和可视化新方法的需求，以及先前记录的类似调查的成功。科学家将DD应用于更大的现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，基于新计算见解的统计学家DD公式的改进，以及对不同的大学预科，大学和研究生的培训提供了更广泛的影响。