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处理等应用中，曲线和曲面的光滑和非光滑部分同时存在。在高维的情况下，在其他光滑的表面上出现的孤立的不连续对应于边界或“断裂曲线”。将利用曲线估计、最小二乘和(局部)最大似然技术，开发分析断裂曲线和估计光滑-非光滑曲面和回归函数的方法。应用包括断裂曲线和等值线的置信度区域、图像分析中的边缘检测、泊松森林问题以及用于高维回归的降维技术。生物、环境和经济科学中的许多现象可以用发生急剧变化来描述，无论是时间还是地点都不同。将开发统计工具来分析描述这种变化的数据。这对于检测图像中的边缘和对高维数据的简约描述特别有意义。在这种情况下，将建立置信度区域，例如，允许找到污染物水平低于临界水平的“安全”区域。拟议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.