Bayesian Nonlinear Regression with Multivariate Linear Splines

提案ID：DMS-0203215PI：Bani Mallick标题：使用多变量线性样条的贝叶斯非线性回归研究人员和他的同事考虑对单变量和多变量非高斯响应数据进行新颖的非参数建模。通过对均值函数的灵活建模，将通常的广义线性模型推广到广义非线性模型。该方法采用数据自适应的多元光滑样条线，节点的数目和位置被视为随机的。使用可逆跳跃马尔可夫链蒙特卡罗(MCMC)采样器来探索后验模型空间。通过在模型中引入残差效应，使模型参数的许多后验分布保持标准形式，计算困难得到了部分缓解。潜在残差效应的使用为多变量响应数据中的相关性建模提供了一种方便的工具，因此可以看出该方法将看似不相关的回归模型推广到非高斯数据。在该项目的下一部分，研究人员和他的同事们为广义非线性模型开发了半参数贝叶斯方法，其中预报器的测量要么是经典的，要么是Berkson误差的。在存在协变量测量误差的情况下，通常的回归函数的非参数估计是非常困难的，这涉及到反卷积问题。在广义线性模型的情况下，这是比较困难的。函数估计是一种重要的统计工具，它试图根据数据准确地理解变量之间的函数关系，它在许多学科中都有应用，成功地解决了科学问题。大多数灵活的、非线性回归问题都是在响应是连续变量的情况下提出的。在重要的应用问题中，响应可能是计数变量或指示变量，在这些情况下灵活的函数估计要困难得多。在这项建议中，研究人员打算开发在这些更复杂的情况下自适应地估计函数关系的方法。生物技术领域是这些方法的一个特别应用。科学家们现在有了同时测量数千个基因的基因表达水平的技术，这使得确定哪些人类基因与癌症和心脏病等疾病有关成为可能。这些方法将有助于探索基因表达水平与患病几率之间的非线性关系。其他可能的应用是对相关的多变量疾病或事故计数数据进行建模，这里开发的方法将改进(具有不确定性的)疾病或事故地图的建模，这将对疾病或交通风险评估有用。

Proposal ID: DMS-0203215PI: Bani MallickTitle: Bayesian nonlinear regression with multivariate linear splinesThe investigator and his colleagues consider novel, nonparametric modeling of univariate and multivariate non-Gaussian response data. The usual generalized linear models are extended to generalized nonlinear models by modeling the mean function in a flexible way. Data adaptive multivariate smoothing splines are employed to do this, where the number and location of the knot points are treated as random. The posterior model space is explored using a reversible jump Markov chain Monte Carlo (MCMC) sampler. Computational difficulties are partly alleviated by introducing a residual effect in the model that leaves many of the posterior distributions of the model parameters in standard form. The use of the latent residual effect provides a convenient vehicle for modeling correlation in multivariate response data and as such the method can be seen to generalize the seemingly unrelated regression model to non-Gaussian data. In the next part of the project the investigator and his colleagues develop semiparametric Bayesian methods for generalized non-linear models where a predictor is measured with either classical or Berkson error. In the presence of covariate measurement error, estimating usual regression function nonparametrically is extremely difficult, the problem being related todeconvolution. In the case of generalized linear model it is more difficult. Again combinations of spline regression and MCMC techniques are used to handle the problem.Function estimation is an important statistical tool that tries to understand accurately the functional relationships between variables based on data and it has applications in many disciplines for successfully addressing scientific questions. Most of the flexible, nonlinear regression problems are developed when the response is a continuous variable. In important applied problems the response may be count or indicator variable and flexible function estimation is much more harder in these situations. In this proposal the investigator intend to develop the methods that adaptively estimate the functional relationships in these more complicated situations. The area of biotechnology is an especially application for these methods. Scientists now have techniques for measuring gene expression levels for thousands of genes at the same time, allowing the exciting possibility of determining which human genes are involved in a disease such as cancer and heart disease. These methods will be useful to explore nonlinear relationship between gene expression levels and the chance of the disease. Other possible applications are to model correlated multivariate disease or accident counts data where the methods being developed here will improve modeling disease or accidents maps (with uncertainties) which will be useful for disease or transportation risk assessments.