通过对抗性位翻转实现信号源信道编码（CS LG）
- 2020 年 4 月 6 日
- 筆記
异速生长规律是理解城市演化的基本规律之一。 这个定律的一般形式是异速比例定律。 然而，标度指数的深层含义和基本原理仍有待揭示。 本文应用线性代数和回归分析理论，揭示了异速标度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度量关系遵循异速生长定律。 理论上证明了异速比例指数等于一个对数测度的标准差与另一个对数测度的标准差之比。 在基于观测数据的实证分析中，标度指数等于标准差与相应皮尔逊相关系数的乘积。 这一发现可以推广到城市分形和城市规模分布来解释城市空间的分形维数和城市层次的 zipf 标度指数。 结果表明，标度指数反映了特征长度的比值。 这项研究可能有助于从一个新的角度理解尺度，以及尺度和特征尺度之间的联系和区别。
原文题目：Derivation of Relations between Scaling Exponents and Standard Deviation Ratios
原文：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. 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 understanding scaling from a new perspective and the connections and distinctions between scaling and characteristic scales.
原文作者：Yanguang Chen
原文地址：https://arxiv.org/abs/2004.01385
