贝叶斯多维相关曲线非参数模型研究与应用

This project provides a novel method to convert multivariate smoothing spline models to linear mixed models in order to model and predict multivariate nonlinear and nonparametric regression in correlated functional or curve data. Three broad research areas in Bayesian analysis.involving theory, methodology and application will be pursued: A. establish framework of multivariate smoothing spline models, propose and study the generalized Bayesian analysis, all under the fixed smoothing penalty terms. B. Relaxed assumption of fixed smoothing penalty, establish adaptive smoothing spline model for fitting space-time data. C. A novel method using generalized principal components is proposed to simplify the data without loosing related information, to reduce the involved computation significantly, to find the optimal model, and.to increase accuracy of estimation and prediction. One of the crucial technics and tools for the project is to borrow strength from similar data, and analyze the correlated curve simultaneously obviously better in estimation and prediction comparing that based on individual curve fitting. The methods have the potential both to explain underlying relationships and to summarize overall trends with relatively few component curves, that can be used to analyze either the number of curves is much larger then the number of observations or both numbers are large..The results will advance the statistics theory, method and applications of Bayesian procedures, that can be used in practice directly. It can significantly impact other disciplines such as finance, economics, disease mapping, and environmental science. Graduate students and post-doctoral fellows will be intimately involved in the research project, learning the research process and how to operate in an interdisciplinary environment. It will promote the growth of statistics experts in China, and build up the international reputation of Chinese researchers in Bayesian analysis.

本项目将多维平滑样条模型转化为线性混合模型，从而实现多维非线性、非参数相关函数型数据的建模与预测。主要包含以下三组研究课题。A.提出多元平滑样条模型，开创固定粗糙惩罚参数下的广义贝叶斯分析框架。B.放宽固定粗糙惩罚参数的限制，建立时空自适应的多元平滑样条模型。C.研究曲线的主成分分析，实现数据压缩和化简，降低计算量，找出最优模型，提高推断和预测的精度。项目采用的关键技术，一是利用贝叶斯理论“借力”的优点，实现相关曲线的同时分析，在参数估计和趋势预测方面显著优于单独使用各类数据。二是采用广义主成分方法引出一个不损失有用信息的维数压缩策略，适用于维数远大于样本量或两者都很大的情形。相关成果将显著推动贝叶斯技术在统计理论、方法和应用等方面的发展，并可迅速转化为实际生产力，对金融、经济、医学、环境等众多学科产生重要影响。同时还将促进我国统计专业人才的培养和成长，提升相关领域的国际学术影响。

本项目致力于研究贝叶斯多维相关曲线非参数模型的研究与应用，主要包含以下三组研究课题：A.提出多元平滑样条模型，开创固定粗糙惩罚参数下的广义贝叶斯分析框架。B.放宽固定粗糙惩罚参数的限制，建立时空自适应的多元平滑样条模型。C.研究曲线的主成分分析，实现数据压缩和化简，降低计算量，找出最优模型，提高推断和预测的精度。经过团队通力合作，项目组建立起时空自适应的多元平滑样条模型及相应的广义贝叶斯分析体系，系统化地实现了不同曲线之间的“借力”，并对先验选取、后验计算和模型选择等关键问题给出科学、高效的解决方案，促进了贝叶斯统计理论和方法的发展。应用实践方面，本项目成果获得业界认可，主要算法已被国内知名金融服务机构用于债券收益率曲线的日常生产，大幅提高了相关数据指标的生产效率，为我国金融市场基础信息服务水平的提升贡献了力量。在研究后期还结合业界最新需求，探索了一类更精细的曲线结构，是理论和应用的重要发展方向。