Mathematical Sciences: Multivariate Polynomial Splines for Function Estimation

在许多情况下，观测数据是由某种(未知)机制生成的，其中感兴趣的是估计与数据的模型相关的函数。利用多项式样条函数将未知函数建模为线性空间。采用逐步加法和逐步删除基函数的算法使得自适应地确定该空间成为可能。用极大似然法估计各基函数的系数。这一建议讨论的问题是风险回归，包括存在时间相关协变量或区间删失数据的情况；双变量生存估计；双变量生存回归；大地震余震的建模；多点回归和分类；旋转不变回归和密度估计。这个项目的主要目的是开发在各种问题中使用多项式样条来估计未知函数的方法。多项式样条是一种可用于对函数进行建模而无需对其形式进行假设的石块。该提案讨论的问题包括：生存分析，包括存在依赖于时间的解释变量的情况；已知存活时间仅在一个区间内的情况以及为每个单位测量一次以上生存时间的情况；大地震余震的模拟；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。

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.