Estimating Intrinsic Dimensionality

瓦尔特9704557 本研究的目标是通过估计数据集的内在维数来检测多变量数据中的非线性结构。 一个多元数据集，除了噪声，福尔斯落入一个低维光滑子流形被称为具有内在维数等于这样的子流形的最小维度。 数据集的内在维度的知识或估计有助于解决多变量统计和模式分析中的两个重要问题：找到适当数量的参数来表示数据的问题，以及决定数据是否存在二维或三维表示的问题，然后可以进行可视化分析。 本研究发展了一种新的方法来估计内在维数，承诺是上级现有的方法的启发式的原因，并调查新的估计和它的竞争对手的统计特性。 新的估计器使用一种方法，以非线性的方式，这是基于数学形态学领域的工具来平滑的形状的多变量数据集。 本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。在复杂的高维数据集中，往往存在一些简单的、低维的几何结构，为了有效地从这些数据中提取信息，必须找到这些结构，并以简单易行的方式描述它们，以便于计算机处理。本研究使用某些几何工具来开发这样的处理系统，其中要找到的图案被噪声破坏，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.