通過對抗性位翻轉實現信號源信道編碼（CS LG）
- 2020 年 4 月 6 日
- 筆記
異速生長規律是理解城市演化的基本規律之一。 這個定律的一般形式是異速比例定律。 然而，標度指數的深層含義和基本原理仍有待揭示。 本文應用線性代數和回歸分析理論，揭示了異速標度指數的數學和統計本質。 假設城市系統中一組元素之間的幾何度量關係遵循異速生長定律。 理論上證明了異速比例指數等於一個對數測度的標準差與另一個對數測度的標準差之比。 在基於觀測數據的實證分析中，標度指數等於標準差與相應皮爾遜相關係數的乘積。 這一發現可以推廣到城市分形和城市規模分佈來解釋城市空間的分形維數和城市層次的 zipf 標度指數。 結果表明，標度指數反映了特徵長度的比值。 這項研究可能有助於從一個新的角度理解尺度，以及尺度和特徵尺度之間的聯繫和區別。
原文題目：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
