Optimal estimation and confidence sets for discontinuities in noisy, blurred regression functions

噪声、模糊回归函数中不连续性的最佳估计和置信度集

基本信息

批准号：
263784853
负责人：
Professor Dr. Hajo Holzmann
金额：
--
依托单位：
Lehrstuhl Stochastik, insbesondere Statistik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2015
资助国家：
德国
起止时间：
2014-12-31 至 2017-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/263784853?language=en
关键词：
Optimal estimation confidence sets discontinuities

项目摘要

Nonparametric estimation of smooth univariate (1d) and bivariate (2d) regression functions is a very well-developed theory. However, in the presence of a discontinuity (jump) in a 1d regression, or of a discontinuity curve (edge) in a 2d image function, standard smoothing methods like local polynomial estimation do no longer work without modifications. Further, the discontinuity point and height in 1d as well as the location of the edge and its contrast in 2d may be of substantial independent scientific interest. Currently, no generally appropriate methods are available neither for the construction of confidence intervals for jumps in 1d regression curves which are observed under blurring, that is, convolution with a point-spread function, nor for uniform confidence sets for edges in 2d images. Developing and investigating confidence intervals and sets in these problems is therefore the first main aim of the project. For 2d image functions, often an additional blurring occurs which must be taken into account in reconstructions of the image itself as well as of edges contained in the image. The systematic treatment of deblurring (in addition to denoising) in edge estimation is therefore the second main aim of the project. Both the theoretical optimality properties of edge estimators and the construction and implementation of more practical recovering methods shall be investigated.

平滑单变量（1D）和双变量（2D）回归函数的非参数估计是一个非常发达的理论。但是，在1D回归中存在不连续性（跳跃）或2D图像函数中的不连续性曲线（边缘）的情况下，诸如局部多项式估计之类的标准平滑方法在没有修改的情况下不再工作。此外，1D以及边缘的位置及其在2D中的对比度可能具有很大的独立科学利益。当前，在1D回归曲线中跳跃的置信区间尚无通常适当的方法，而在模糊下观察到的跳跃曲线，即具有点传播功能的卷积，也不适合2D图像中边缘的均匀置信度集。因此，开发和调查这些问题的置信区间和设定是该项目的第一个主要目的。对于2D图像函数，通常会发生其他模糊，必须在图像本身以及图像中包含的边缘的重建中考虑到。因此，在边缘估计中对脱毛的系统处理是该项目的第二个主要目的。均应研究边缘估计器的理论最优性能以及更实用的恢复方法的构建和实施。