Mathematical Sciences: Multivariate Polynomial Splines for Function Estimation

数学科学：用于函数估计的多元多项式样条

• 批准号: 9403371
• 负责人: Charles Kooperberg
• 金额: $ 6万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-15 至 1998-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403371&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Multivariate; Polynomial; Splines

There are numerous situations in which observed data is generated by some (unknown) mechanism, where interest lies in estimating a function that is related to a model for the data. Using polynomial splines, an unknown function is modeled to be in a linear space. An algorithm employing stepwise addition and stepwise deletion of basis functions makes it possible to determine this space adaptively. The coefficients of the basis functions are estimated by the maximum likelihood method. The problems that are discussed in this proposal are hazard regression, including situations in which there are time-dependent covariates or interval censored data; bivariate survival estimation; bivariate survival regression; modeling of aftershocks of a major earthquake; polychotomous regression and classification; rotation invariant regression and density estimation. The primary aim of this project is to develop methodologies for the estimation of unknown functions using polynomial splines in a variety of problems. Polynomial splines are building stones that can be used to model functions without making assumptions of their form. The problems that are discussed in this proposal are: survival analysis, including situations in which there are time-dependent explanatory variables, situations in which the survival times are only known to be in an interval and situations in which more than one survival time is measured for each unit; modeling of aftershocks of a major earthquake; classification of observations based upon explanatory variables; rotation invariant regression and density estimation. For each of the problems that we propose, there are vast amounts of data whose improved analysis would be of much scientific interest.

在许多情况下，观测数据是由某种（未知）机制生成的，其中兴趣在于估计与数据模型相关的函数。使用多项式样条，未知函数被建模为线性空间。采用逐步增加和逐步删除的基函数的算法，使它能够确定这个空间自适应。基函数的系数由最大似然法估计。在这个建议中讨论的问题是风险回归，包括情况下，有时间依赖的协变量或区间删失数据;双变量生存估计;双变量生存回归;建模的余震的大地震;多元回归和分类;旋转不变回归和密度估计。这个项目的主要目的是开发方法的估计未知函数使用多项式样条在各种问题。多项式样条是基石，可用于建模函数，而无需对其形式进行假设。该建议中讨论的问题是：生存分析，包括存在时间依赖性解释变量的情况、生存时间仅已知在一个区间内的情况和每个单元测量到一个以上生存时间的情况;大地震余震的建模;基于解释变量的观测分类;旋转不变回归和密度估计。对于我们提出的每一个问题，都有大量的数据，其改进的分析将具有很大的科学意义。