Mathematical Sciences: Multivariate Polynomial Splines for Function Estimation

在许多情况下，观察到的数据是由某些（未知的）机制生成的，其中的兴趣在于估计与数据模型相关的函数。利用多项式样条，将一个未知函数在线性空间中建模。采用基函数的逐步加法和逐步删除算法，使该空间的自适应确定成为可能。基函数的系数用极大似然法估计。本建议中讨论的问题是风险回归，包括存在时间相关协变量或间隔审查数据的情况；双变量生存估计；双变量生存回归；大地震余震的模拟；多元回归与分类；旋转不变回归与密度估计。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估计未知函数的方法。多项式样条是可以用来对函数建模而无需对其形式进行假设的基石。本提案中讨论的问题是：生存分析，包括存在时间相关解释变量的情况，生存时间仅在一个间隔内已知的情况以及为每个单位测量多个生存时间的情况；大地震余震的模拟；根据解释变量对观测值进行分类；旋转不变回归与密度估计。对于我们提出的每一个问题，都有大量的数据，这些数据的改进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.