Estimation of instrinsic dimension via clustering

The problem of estimating the intrinsic dimension of a set of points in high dimensional space is a critical issue for a wide range of disciplines, including genomics, finance, and networking. Current estimation techniques are dependent on either the ambient or intrinsic dimension in terms of computational complexity, which may cause these methods to become intractable for large data sets. In this paper, we present a clustering-based methodology that exploits the inherent self-similarity of data to efficiently estimate the intrinsic dimension of a set of points. When the data satisfies a specified general clustering condition, we prove that the estimated dimension approaches the true Hausdorff dimension. Experiments show that the clustering-based approach allows for more efficient and accurate intrinsic dimension estimation compared with all prior techniques, even when the data does not conform to obvious self-similarity structure. Finally, we present empirical results which show the clustering-based estimation allows for a natural partitioning of the data points that lie on separate manifolds of varying intrinsic dimension

A Box-Counting Method with Adaptable Box Height for Measuring the Fractal Feature of Images

Most of the existing box-counting methods for measuring fractal features are only applicable to square images or images with each dimension equal to the power of 2 and require that the box at the top of the box stack of each image block is of the same height as that of other boxes in the same stack, which gives rise to inaccurate estimation of fractal dimension. In this paper, we propose a more accurate box-counting method for images of arbitrary size, which allows the height of the box at the top of each grid block to be adaptable to the maximum and minimum gray-scales of that block so as to circumvent the common limitations of existing box-counting methods

The solutions to uncertainty problem of urban fractal dimension calculation

Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure, multi-scaling fractal patterns, and self-affine fractal growth. The solution to the problem is to substitute the real fractal dimension values with comparable fractal dimensions. The main measures are as follows: First, select a proper method for a special fractal study. Second, define a proper study area for a city according to a study aim, or define comparable study areas for different cities. These suggestions may be helpful for the students who takes interest in or even have already participated in the studies of fractal cities.Comment: 27 pages, 3 figures, 8 table