地理科学进展 ›› 2015, Vol. 34 ›› Issue (3): 321-329.doi: 10.11820/dlkxjz.2015.03.007

• 理论模型与GIS 应用 • 上一篇    下一篇

简单、复杂与地理分布模型的选择

陈彦光   

  1. 北京大学城市与环境学院,北京 100871
  • 收稿日期:2014-10-01 修回日期:2015-02-01 出版日期:2015-03-25 发布日期:2015-03-25
  • 作者简介:

    作者简介:陈彦光(1965-),男,河南罗山人,博士,教授,主要从事城市和理论地理学研究,E-mail: chenyg@pku.edu.cn

  • 基金资助:
    国家自然科学基金项目(41171129)

Simplicity, complexity, and mathematical modeling of geographical distributions

Yanguang CHEN   

  1. Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China
  • Received:2014-10-01 Revised:2015-02-01 Online:2015-03-25 Published:2015-03-25

摘要:

地理分布的数学建模是空间分析的基本途径之一,但空间维度建模素为科学研究的难题。由于数学新方法的发展和复杂性研究的兴起,地理空间建模的一些传统困难有望解决。本文通过两类地理分布的对比分析,论述地理建模的关键在于简单分布的特征尺度和复杂分布的标度。地理分布包括空间分布和规模分布,其本质均为概率分布和广义的数学空间分布,而概率分布可以分为简单分布和复杂分布。简单分布具有特征尺度,平均值有效,概率结构清楚;复杂分布没有特征尺度,平均值无效,概率结构不明确。对于简单分布,应该采用有尺度分布函数开展尺度分析;对于复杂分布,理当采用无尺度分布函数开展标度分析。分形几何学、异速生长理论和无尺度网络理论都是复杂系统分析的定量方法,这些方法的综合集成,可望为地理分布建模和地理系统的空间分析提供有效的数理工具。

关键词: 复杂系统, 无尺度分布, 特征尺度, 标度, 分形, 异速, 地理建模, 空间分析

Abstract:

Because of quantitative revolution, geography evolved from a discipline of spatial description into a science of distributions. Accordingly, qualitative methods were replaced by the integrated methods of quantitative analysis and qualitative analysis. One of the important approaches to spatial analysis is to characterize geographical distributions. Geographical distributions fall into spatial distributions and size distributions, both of which can be divided into simple distributions and complex distributions. A simple distribution has a characteristic scale (represented by a characteristic length), while a complex distribution has no characteristic scale but bears a property of scaling invariance (represented by a scaling exponent). The key step of studying a simple distribution is to find its characteristic scale, while the basic way of research for a complex distribution is to make a scaling analysis. For simple distributions, traditional methods based on advanced mathematics are effective, but for complex distributions, the old-fashioned mathematical tools are ineffective. However, due to the lack of understanding of characteristic scale and scaling, geographers often failed to distinguish between simple distributions and complex distributions. As a result, many complex systems such as cities and systems of cities were mistaken for simple systems. Consequently, geography did not succeed in theorization in the 1960s-1970s after quantification in the 1950s-1960s. Geographical distributions can be mathematically abstracted as probability distributions. For a simple distribution, its characteristic scale can be determined. A typical characteristic scale is a mean (average value). Based on means, we can compute a variance and a covariance. Thus we have a clear probability structure comprising means, variances, and covariances, which explain the pattern of the geographical system and predict its process of evolution. In the case of a complex distribution, an effective mean cannot be determined, and thus little is known about its probability structure. In this situation, characteristic scale analysis should be substituted with scaling analysis. Quantitative methods of scaling analysis have emerged from interdisciplinary studies. A new integrated theory based on concepts from fractals, allometry, and complex network has been developing for geographical modeling and analysis.

Key words: complex system, scale-free distribution, characteristic scale, scaling, fractals, allometry, geographical modeling, spatial analysis