论文标题
缩放指数与标准偏差比之间关系的推导
Derivation of Relations between Scaling Exponents and Standard Deviation Ratios
论文作者
论文摘要
异态增长定律是理解城市进化的基本规则之一。该法律的一般形式是异形缩放定律。但是，缩放指数的深层含义和基本原理仍有待揭示。在本文中，采用线性代数和回归分析的理论来揭示异量级缩放指数的数学和统计本质。假设城市系统中一组元素之间的几何关系遵循异形生长定律。从理论上讲，一个对数度量与另一个对数度量的标准偏差的标准偏差的比率在理论上被证明是平等的。在基于观察数据的经验分析中，缩放指数等于标准偏差比与相应的Pearson相关系数之间的产物。数学推导结果可以通过经验分析来验证：基于标准偏差比的缩放指数值与基于常规方法的比例完全相同。这一发现可以推广到城市分形和城市规模分布，以解释城市空间的分形维度和城市层次结构的ZIPF缩放指数。可以得出一个结论，即缩放指数反映了特征长度的比率。这项研究可能有助于从新的角度理解缩放，以及缩放和特征尺度之间的联系和区分。
The law of allometric growth is one of basic rules for understanding urban evolution. The general form of this law is allometric scaling law. However, the deep meaning and underlying rationale of the scaling exponents remain to be brought to light. In this paper, the theories of linear algebra and regression analysis are employed to reveal the mathematical and statistic essence of allometric scaling exponents. Suppose that the geometric measure relations between a set of elements in an urban system follow the allometric growth law. An allometric scaling exponent is proved to equal in theory to the ratio of the standard deviation of one logarithmic measure to the standard deviation of another logarithmic measure. In empirical analyses based on observational data, the scaling exponent is equal to the product between the standard deviation ratio and the corresponding Pearson correlation coefficient. The mathematical derivation results can be verified by empirical analysis: the scaling exponent values based on the standard deviation ratios are completely identical to those based on the conventional method. This finding can be generalized to city fractals and city size distribution to explain fractal dimensions of urban space and Zipf scaling exponent of urban hierarchy. A conclusion can be reached that scaling exponents reflect the ratios of characteristic lengths. This study may be helpful for comprehending scaling from a new perspective and the connections and distinctions between scaling and characteristic scales.
