Estimating Intrinsic Dimensionality

本研究的目的是通过估计数据集的固有维数来检测多变量数据中的非线性结构。一个多变量数据集，除噪声外，属于低维光滑子流形，我们说它的内在维数等于这种子流形的最小维数。了解或估计数据集的内在维度有助于解决多元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，然后可以对其进行可视化分析。本研究开发了一种新的估计内在维数的方法，由于启发式原因，该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性质。该估计器采用了一种基于数学形态学的工具，以非线性方式平滑多元数据集的形状。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往往包含某些简单的、低维的几何结构。为了有效地从这些数据中提取信息，重要的是要找到这样的结构，并以简单和适合计算机处理的方式描述它们。本研究使用特定的几何工具来开发这样一个处理系统，在这种情况下，要发现的模式被噪声破坏，例如传输错误或图像模糊。

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.