关于点态维数的估计
On the Estimation of Pointwise Dimension
摘要：
本文的目的是发展一个有效的分形维数估计。我们调查现有的想法，在维估计，重点是目前流行的方法Grassberger和Procaccia的相关维数的估计。基于该方法的估计存在两个主要困难。首先是关联维数本身对数据的维数差异不敏感，我们称之为“维数盲”。第二个问题来自于该方法依赖于从有限数据中推断极限行为。针对关联维数的维数盲性，提出以点态维数作为估计对象。点态维数是一个局部量，点态维数在数据上的分布包含了关联维数不可见的信息。我们使用逐点维数的“无极限”描述来开发一个新的估计量。最后，我们讨论了我们的估计器的潜在应用以及它提出的一些挑战。
英文摘要：
Our goal in this paper is to develop an effective estimator of fractal dimension. We survey existing ideas in dimension estimation, with a focus on the currently popular method of Grassberger and Procaccia for the estimation of correlation dimension. There are two major difficulties in estimation based on this method. The first is the insensitivity of correlation dimension itself to differences in dimensionality over data, which we term "dimension blindness". The second comes from the reliance of the method on the inference of limiting behavior from finite data. We propose pointwise dimension as an object for estimation in response to the dimension blindness of correlation dimension. Pointwise dimension is a local quantity, and the distribution of pointwise dimensions over the data contains the information to which correlation dimension is blind. We use a "limit-free" description of pointwise dimension to develop a new estimator. We conclude by discussing potential applications of our estimator as well as some challenges it raises.
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