Mathematical Sciences: Break Curves and Isoklines in Curves Estimation Models

在图像分析、信号检测和生态空间数据等应用中，曲线和曲面的光滑和非光滑部分同时出现。在高维情况下，孤立的不连续性发生在其他光滑的表面对应的边界或“中断曲线”。将使用曲线估计、最小二乘法和（局部）最大似然技术，开发分析突变曲线和估计光滑-非光滑表面和回归函数的方法。 应用包括置信区域的中断曲线和isoklines，边缘检测图像分析中，泊松森林问题，高维回归和降维技术。 生物、环境和经济科学中的许多现象都可以用时间或地点变化时发生的急剧变化来描述。将开发统计工具，用于分析描述这些变化的数据。这些将是特别感兴趣的图像中的边缘检测和简约的高维数据的描述。 在这方面，将建立置信区，以便例如找到污染物水平低于临界水平的“安全”地区。所提出的方法将在对数据性质的最小假设下工作。

In applications like image analysis, signal detection and ecological spatial data smooth and non-smooth parts for curves and surfaces occur simultaneously. In the higher dimensional case, isolated discontinuities occurring in otherwise smooth surfaces correspond to boundaries or "break curves". Methods for analyzing break curves and to estimate smooth-non-smooth surfaces and regression functions will be developed, using curve estimation, least squares and (local) maximum likelihood techniques. Applications include confidence regions for break curves and isoklines, edge detection in image analysis, the Poisson forest problem, and dimension reduction techniques for high dimensional regression. Many phenomena in biological, environmental and economic sciences can be described by the occurrence of sharp changes when either time or location varies. Statistical tools will be developed for the analysis of data describing such changes. These will be of particular interest for the detection of edges in images and for parsimonious descriptions of high-dimensional data. In this context, confidence regions will be constructed which allow for instance to find "safe" areas where pollutant levels are below critical levels. The proposed methods will work under minimal assumptions on the nature of the data.