Estimating Intrinsic Dimensionality

Walther 9704557本研究的目标是通过估计数据集的内在维度来检测多变量数据中的非线性结构。除噪声外，属于低维光滑子流形的多变量数据集被称为具有等于此子流形的最小维度的本征维度。对数据集的内在维度的了解或估计有助于解决多元统计和模式分析中的两个重要问题：找到用于表示数据的适当数目的参数的问题，以及确定是否存在数据的二维或三维表示的问题，然后可以对其进行可视分析。这项研究开发了一种新的估计内在维度的方法，由于启发式的原因，这种方法有望优于现有的方法，并研究了新估计器及其竞争对手的统计特性。新的估计器使用了一种以非线性方式平滑多变量数据集形状的方法，该方法基于数学形态学领域的工具。这项研究的目标是检测数据中的某些模式，例如医学图像中的结构化部分。相当复杂的高维数据集通常具有某些简单的、低维的几何结构。为了以有效的方式从这些数据中提取信息，重要的是找到这样的结构，并以简单、易于计算机处理的方式描述它们。这项研究使用某些几何工具来开发这样的处理系统，在要发现的图案被噪声破坏的背景下，例如图像中的传输错误或模糊。

Walther 9704557 The goal of this research is to detect nonlinear structure in multivariate data by estimating the intrinsic dimensionality of a data set. A multivariate data set that, apart from noise, falls into a lower-dimensional smooth submanifold is said to have intrinsic dimensionality equal to the smallest dimension of such a submanifold. A knowledge or estimate of the intrinsic dimensionality of a data set contributes to the solution of two important problems in multivariate statistics and pattern analysis: The problem of finding an appropriate number of parameters for representing the data, and the problem of deciding whether a two- or three-dimensional representation of the data exists, which may then be analyzed visually. This research develops a new way to estimate intrinsic dimensionality that promises be superior to existing methods for heuristic reasons, and investigates the statistical properties of the new estimator and its competitors. The new estimator uses a method to smooth the shape of a multivariate data set in a nonlinear way which is based on tools from the field of mathematical morphology. The goal of this research is to detect certain patterns in data, such as structured parts in medical images. Quite complicated and high-dimensional data sets have often certain simple, low-dimensional geometric structures in it. To extract information from these data in an efficient way it is important to find such structures and describe them in ways that are simple and and amenable for the processing by computers. This research uses certain geometric tools to develop such a processing system in a context where the patterns to be found are corrupted by noise, e.g. transmission errors or blurring in images.