城市形态的分维估算与分形判定

doi:10.18306/dlkxjz.2017.05.001

摘要/Abstract

摘要：

城市形态的分形是城市发育到一定阶段涌现出的有序格局和复杂结构,其基本特征是空间分布的无尺度性质。当研究者基于某个显著性水平推断城市分维存在时,实际上就是基于相应的置信度判断分形特征。虽然分形城市研究已经多年,但大量有关维数测算的基础问题依然悬而未决。本文根据分形几何学的基本思想论证城市形态分维测算的若干问题。分维测量的准则是最佳覆盖——不多不少、恰到好处的覆盖。盒子覆盖是最容易理解的测量方法。采用盒子覆盖法测量城市形态分维时,应考虑三个标准：一是快速逼近,二是简便操作,三是稳定拟合。直观估计分维的办法是利用双对数坐标图。由于城市形态不是严格意义的分形,而是类似于文献中的“前分形”,测量尺度与相应测度的幂律关系通常仅在一定尺度范围内有效,从而形成所谓标度区。本文围绕城市形态的分维测量和分形判断开展一系列讨论,包括尺度选取、标度区识别和统计标准等问题,对今后城市分形研究具有理论启示和方法论的参考价值。

关键词: 城市形态, 分形, 分维, 自仿射, 盒子计数法, 多分形, 标度区

Abstract:

Fractal cities and city fractals have been studied for about 30 years, but many basic problems have not yet been fully solved. Among the various basic fractal problems about cities, the most important are how to estimate fractal dimension and how to identify fractal nature of a city as a system or a system of cities in an effective way. Based on urban form and the box-counting method, this article discusses the approaches to calculating fractal dimension and determining fractal property of cities. The theoretical foundation of fractal dimension computation is the concept of perfect covering. In other words, in the process of fractal dimension measurement, a fractal object should be covered with boxes in the best way: nothing more, nothing less. In practice, it is hard to find the optimal way of box covering. Therefore, three rules should be followed. First, the sequence of measuring scales should be consistent with the cascade structure of a fractal city so that the fractal details can be captured in a reasonable way. Second, the operation of obtaining observational data should be simple and convenient to use so that the method can be applied by beginners. Third, the size of the dataset for fractal dimension estimation should be large enough so that the result of fractal parameters is stable. A conclusion can be reached that the geometric scale rather than the arithmetic scale should be employed to make a measurement because the fractal structure can be abstracted as geometric sequences instead of arithmetic sequences, and the measuring scale sequence should comply with the 1/2ⁿ rule (that is, 1, 1/2, 1/4, …). After estimating the fractal dimension of a city, the fractal property can be identified by the statistics from fractal dimension calculation. The good way of identifying the fractal nature of urban form is to use confidence statement, which consists of fractal dimension values, margin of error, and level of confidence. Given a level of significance (for example, α=0.05), we can draw an inference that a city's form is of fractal structure because it has a fractal dimension with a determinate level of confidence (for example, (1-α)×100%=95%). Using statistical analysis, however, one can never have full (100%) assurance that a city has a fractal form.

Key words: urban form, fractals, fractal dimension, self-affinity, box-counting method, multifractals, scaling range

陈彦光. 城市形态的分维估算与分形判定[J]. 地理科学进展, 2017, 36(5): 529-539.

Yanguang CHEN. Approaches to estimating fractal dimension and identifying fractals of urban form[J]. PROGRESS IN GEOGRAPHY, 2017, 36(5): 529-539.

图/表 2

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