地理科学 ›› 2003, Vol. 23 ›› Issue (5): 526-534.doi: 10.13249/j.cnki.sgs.2003.05.526

• 论文 • 上一篇    下一篇

城市体系时空演化的广义维数分析——刻划城市资源分享空间的理论基础、计算方法与应用实例

陈彦光1, 刘继生2   

  1. 1. 北京大学地理系, 北京 100871;
    2. 东北师范大学地理系, 吉林 长春 130024
  • 收稿日期:2002-01-25 修回日期:2003-05-20 出版日期:2003-09-20 发布日期:2003-09-20
  • 基金资助:
    国家自然科学基金(项目编号:40071035)

New Methods of Analyzing Spatial-Temporal Evolution of Urban Systems Using Generalized Fractal Dimension:Underlying Rationale, Computational Process, and a Case of Application

CHEN Yan-Guang1, LIU Ji-Sheng2   

  1. 1. Department of Geography, Peking University, Beijing 100871;
    2. Department of Geography, Northeast Normal University, Changchun, Jilin 130024
  • Received:2002-01-25 Revised:2003-05-20 Online:2003-09-20 Published:2003-09-20

摘要: 根据城市组成元素和城市体系构成要素的异速生长性质发展了一套城市体系时空演化的广义维数分析方法。只要城市和城市体系满足异速生长或准异速生长关系,就可以据之建立基于广义维数比值的分享系数矩阵,通过矩阵的特征向量计算出城市元素和各个城市在某种测度下的相对发展速度权重,进而利用先前的结果算出综合权重数值。将有关数值予以适当的处理可以揭示城市体系时空演化的复杂动态,从而借之实现城市体系的功能调控和结构优化。本方法的数学处理过程与Saaty的递阶分析方法(AHP)异曲同工,但其理论前提、建模技巧和应用方向等等都与后者具有严格的本质性区别。

Abstract: A new approach to analyzing the process of spatial-temporal evolution of urban systems was proposed in the paper using the ideas from generalized fractals as well as analytical hierarchical process.Defining an urban dynamic system as dxi/dt=fl(x1,x2,...,xn), we can derive an equation of allometric growth,xi∝xaijj, from which a generalized fractal-dimension equation is found as aij=ai/aj=Di/Dj, where the scale factor αij is called allometric coefficient, or share coefficient by ecologists,ai and aj the relative growth ratios of xi and xj, and Di and Dj the generalized dimension of the elements reflected by measurements xi and xj.Then a share coefficient matrix can be made as M=[αij]n×n=[αi/αj]n×n, which gives MD=nD by multiplying the vector D=[Di] on the left side (i,j=1,2,...,n).Obviously, M is a symmetric matrix since αii=1, αij=1/αji, and αijisjs.D is an eigenvector and the largest eigenvalue λmax=n .Now that both cities as systems and systems of cities are conformable to the law of allometric growth, that is, we can use the power equation given above to characterize the allometric relationships of urban elements such as urban area and population, or to describe the interactive relation between city A and city B based on some measurements such as population size, thus an analytical hierarchical process can be developed to study the spatial-temporal structure of systems of cities and towns.Supposing an urban system with n cities each of which comprises m elements such as population, land, transport network, etc., then according to the measurement related to the kth element, we have a share coefficient matrix of the urban system that yields an eigenvector as follows, Ak=[Wki]1*n(i=1,2,...,n;k=1,2,...,m), thus we can obtain a matrix W=[AkT]n*m=[Wik]n*m; as for the power relationships of different elements, similarly, we can gain a eigenvector B=[Wk]1*m from the share coefficient matrix related; therefore, the share space of different cities can be defined by Sf=ABT=[Wik]n*m[Wk]T1*m=[<Wik·Wk>]n*1, where <.> denotes dot product.The relationship by mathematical marriage is easy to be found between the analytical hierarchical process (AHP) developed by T.L.Saaty and the generalized dimension analysis (GNA) advanced by the authors of this paper, but they are very different at underlying rationale, practical fields, analytical purposes, and some other aspects.GNA is used to deal with the complex geographical systems with multi-elements, multi-classes, multi-variables, uniting cities as systems and systems of cities, temporal dimension as evolution and spatial dimension as interaction.The conclusions of analyses includes both characteristic values reflecting each single element or city and those illustrating the systematic regularity by synthesizing the different parts of the calculated results.Though the share coefficients can only reveal the relative superiority by one-to-one comparison of elements or measurements, but it is not difficult for us to transform the results into another kinds of vectors to show the absolute superiority of each cities or towns.However, where plan or optimization is concerned, the comparative superiority analysis is more important since it's just the ratio of generalized dimension that show us how to improve the structure and function of studied geographical systems.As a case, GNA is applied to analyzing Hangzhou urban systems with eight cities and towns (n=8), two variables being taken to reflect population and industrial output respectively (i.e.m=2).In the context the spatial-emporal regularity of studied system is illustrating while demonstrating how to utilize the new methods, which provides a typical example of GNA for others' imitation or reference in following practice or researches.

中图分类号: 

  • F290