非线性系统中结构突变的稳健检测与统计模型的稀疏重构

The problems about structural abrupt changes in nonlinear system are driven by applications and theories, and becomes one of important issues, however, existing researches focus on dectection which relies much on preassumptions and model reconstruction which is simply due to change-point estimation. The project plans to introduce robust dectection method in abrupt changes and sparse ideas in model reconstruction. The specific route includes two aspects. Firstly,based estimating equations and classical residuals under constant structure assumption, together with existing experience we construct weighted residuals to avoid error of "model misspecification", and furtherly we construct robust functionals especially including ratio-type ones to avoid estimation errors, and get the final test statistics. Secondly, to adapt data features, we choose appropriate threshold as the the boudary between sigificant abrupt changes and jump by chance, and to select response factors sparsely, we add an appropriate penalty on the other parameters different from change points. Research in structural abrupt changes has important applications in climate，process control and image processing etc., and richens statistical diagnosis methods and statistical modelling theory under the condition of discontiuity and nonsmoothing, and provides powerful tool to uncover the potential systematic structure and build the response mechanism of exotic shock.

非线性系统中的结构突变问题受到应用与理论的驱动，成为现代统计学的一个热点问题，但已有研究常会在突变检测方面过多依赖于预设条件且在突变结构建模方面简单归结为变点估计。本项目拟采用具有"稳健性"的突变检测手段和具有"稀疏性"的模型重构方法，具体研究内容为：①基于"未突变结构"下的估计方程或残差，结合先验知识构造稳健的加权残差以避免"模型误设"风险,在此基础上选择稳健泛函如"比型"泛函以避免方差估计误差，获得最终检测统计量；②选定合适的"阈值"作为显著突变与偶然跳跃之间的分界，从而能适应数据特点自动选择变点参数值，对非变点参数加合适的"罚"以稀疏选定突变响应因素，最终建立稀疏突变模型。非线性系统突变结构的稳健检测与稀疏重构在气候水文、过程控制、图像信息处理等领域有重要应用，丰富了统计诊断方法和非连续、非光滑条件下的统计建模理论，对于揭示系统结构、勾勒外在冲击在系统内的突变响应机制提供有力支撑。

结构突变研究在气候水文、过程控制、图像信息处理等领域受到极大的应用驱动，丰富了统计诊断方法和非连续、非光滑条件下的统计建模理论，值得深入探索。本项目采用了具有“稳健性”的突变检测手段和具有“稀疏性”的模型重构思想。其基本研究内容有为：1）、引入了突变检测的小波方法、基于加权残差序列选择稳健泛函如“比型”泛函做检测手段以避免估计误差，同时将这些思想应用到工业质量控制领域；2）、经典重构估计困难，本项目引入“稀疏性”思想，①选定合适的“阈值”作为显著突变与偶然跳跃之间的分界，从而能适应数据特点自动选择变点参数值，②多未知多个变点引入罚方法，且将变点有无(检验问题)以及变点个数及跃度结合起来进行研究。