论文标题
广义空间的基于多变量的相关维度分析
Multivariable-based correlation dimension analysis for generalized space
论文作者
论文摘要
事实证明，分形几何形状是一种有效的数学工具，用于探索基于数字地图和遥感图像的真实地理空间。尚未报道分形理论工具是否可以应用于抽象地理空间。抽象空间可以通过多变量距离指标来定义，该指标在科学研究中经常满足。根据分形的思想，本文致力于通过数学推导和经验分析来开发广义地理空间的相关维度分析方法。定义数学距离或统计距离，我们可以构建广义相关函数。如果相关函数与相关长度之间的关系遵循幂定律，则可以证明幂指数与相关维度相关联。因此，可以使用分形维度来分析广义地理空间的结构和性质。这表明可以将分形几何形状推广到探索无规模的抽象地理空间。数学上证明了理论模型，并通过使用观察数据来说明分析方法。这项研究有助于扩大分形理论在地理分析中的应用，结果和结论可以扩展到其他科学领域。
Fractal geometry proved to be an effective mathematical tool for exploring real geographical space based on digital maps and remote sensing images. Whether the fractal theory tool can be applied to abstract geographical space has not been reported. An abstract space can be defined by multivariable distance metrics, which is frequently met in scientific research. Based on the ideas from fractals, this paper is devoted to developing correlation dimension analysis method for generalized geographical space by means of mathematical derivation and empirical analysis. Defining a mathematical distance or statistical distance, we can construct a generalized correlation function. If the relationship between correlation function and correlation lengths follows a power law, the power exponent can be demonstrated to associate with correlation dimension. Thus fractal dimension can be employed to analyze the structure and nature of generalized geographical space. This suggests that fractal geometry can be generalized to explore scale-free abstract geographical space. The theoretical model was proved mathematically, and the analytical method was illustrated by using observational data. This research is helpful to expand the application of fractal theory in geographical analysis, and the results and conclusions can be extended to other scientific fields.
