Estimating Intrinsic Dimensionality

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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.