• 论文 •

### 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

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

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.

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