Spatial autocorrelation plays an important role in geographical analysis; however, there is still room for improvement of this method. The formula for Moran’s index is complicated, and several basic problems remain to be solved. Therefore, I will reconstruct its mathematical framework using mathematical derivation based on linear algebra and present four simple approaches to calculating Moran’s index. Moran’s scatterplot will be ameliorated, and new test methods will be proposed. The relationship between the global Moran’s index and Geary’s coefficient will be discussed from two different vantage points: spatial population and spatial sample. The sphere of applications for both Moran’s index and Geary’s coefficient will be clarified and defined. One of theoretical findings is that Moran’s index is a characteristic parameter of spatial weight matrices, so the selection of weight functions is very significant for autocorrelation analysis of geographical systems. A case study of 29 Chinese cities in 2000 will be employed to validate the innovatory models and methods. This work is a methodological study, which will simplify the process of autocorrelation analysis. The results of this study will lay the foundation for the scaling analysis of spatial autocorrelation.
This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann's equation. For the normalized data, Boltzmann's equation is equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is made that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise, and the city system will fall into disorder. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos seem to be related with some kind of replacement process.
Urban form takes on properties similar to random growing fractals and can be described in terms of fractal geometry. However, a model of simple fractals is not effectual enough to characterize both the global and local features of urban patterns. In this paper multifractal measurements are employed to model urban form and analyze urban growth. The capacity dimension Da, information dimension D\, and correlation dimension Di of a city's pattern can be estimated utilizing tbe box-counting method. If Da> D\> D2 significantly, the city can be treated as a system of multifractals, and two sets of fractal parameters, including global and local parameters, can be used to spatially analyze urban growth. In tbis case study, multifractal geometry was applied to Beijing city, China. The results based on the remote-sensing images taken in 1988, 1992, 1999, 2006, and 2009 show that the urban landscape of Beijing bears multiscaling fractal attributes. The dimension spectrum curves show several abnormal aspects, especially the upper limit of the global dimension breaks through the Euclidean dimension of embedding space and the local dimension fails to converge in a proper way. The geographical features of Beijing's spatiotemporal evolution are discussed, and the conclusions may be instructive for spatial optimization and city planning in the future.
Using fractal theory and urban land-use maps for 1949, 1959, 1980, and 1996, this study is devoted to analyzing the evolutionary features of urban form and land-use structure in Hangzhou, China. We find that self-similarity exists in both the built-up area and the municipal area, and fractal properties tend to become better defined with time. The fractal dimension of each type of land use is less than that of all land use. From 1980 to 1996, the fractal dimensions of residential, industrial, and external transport increased while those of educational and virescent land use decreased, indicating that partial degradation accompanied holistic optimization during the process of Hangzhou's self-organizing evolvement. The mechanisms of the spatiotemporal evolution of urban form in Hangzhou are discussed, including socioeconomic development and the dispersal or concentration of activities. The Tandabing approach to urban spatial expansion, the Jianfengchazhen approach to urban construction, and a top-down urban management style led to differences in fractal evolution between Chinese and Western cities.
Abstract:The rank-size regularity known as Zipf's law is one of scaling laws and frequently observed within the natural living world and in social institutions. Many scientists tried to derive the rank-size scaling relation by entropy-maximizing methods, but the problem failed to be resolved thoroughly. By introducing a pivotal constraint condition, I present here a set of new derivations based on the self-similar hierarchy of cities. First, I derive a pair of exponent laws by postulating local entropy maximizing. From the two exponential laws follows a general hierarchical scaling law, which implies general Zipf's law. Second, I derive a special hierarchical scaling law with exponent equal to 1 by postulating global entropy maximizing, and this implies the strong form of Zipf's law. The rank-size scaling law proved to be one of the special cases of the hierarchical law, and the derivation suggests a certain scaling range with the first or last data point as an outlier. The entropy maximization of social systems differs from the notion of entropy increase in thermodynamics. For urban systems, entropy maximizing suggests the best equilibrium state of equity for parts/individuals and efficiency for the whole.
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