922 research outputs found
Modeling fractal structure of city-size distributions using correlation function
Zipf's law is one the most conspicuous empirical facts for cities, however,
there is no convincing explanation for the scaling relation between rank and
size and its scaling exponent. Based on the idea from general fractals and
scaling, this paper proposes a dual competition hypothesis of city develop to
explain the value intervals and the special value, 1, of the power exponent.
Zipf's law and Pareto's law can be mathematically transformed into one another.
Based on the Pareto distribution, a frequency correlation function can be
constructed. By scaling analysis and multifractals spectrum, the parameter
interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf
distribution, a size correlation function can be built, and it is opposite to
the first one. By the second correlation function and multifractals notion, the
Pareto exponent interval is derived as [1, 2). Thus the process of urban
evolution falls into two effects: one is Pareto effect indicating city number
increase (external complexity), and the other Zipf effect indicating city size
growth (internal complexity). Because of struggle of the two effects, the
scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium
with each other, the scaling exponent approaches 1. A series of mathematical
experiments on hierarchical correlation are employed to verify the models and a
conclusion can be drawn that if cities in a given region follow Zipf's law, the
frequency and size correlations will follow the scaling law. This theory can be
generalized to interpret the inverse power-law distributions in various fields
of physical and social sciences.Comment: 18 pages, 3 figures, 3 table
Fractal Systems of Central Places Based on Intermittency of Space-filling
The central place models are fundamentally important in theoretical geography
and city planning theory. The texture and structure of central place networks
have been demonstrated to be self-similar in both theoretical and empirical
studies. However, the underlying rationale of central place fractals in the
real world has not yet been revealed so far. This paper is devoted to
illustrating the mechanisms by which the fractal patterns can be generated from
central place systems. The structural dimension of the traditional central
place models is d=2 indicating no intermittency in the spatial distribution of
human settlements. This dimension value is inconsistent with empirical
observations. Substituting the complete space filling with the incomplete space
filling, we can obtain central place models with fractional dimension D<d=2
indicative of spatial intermittency. Thus the conventional central place models
are converted into fractal central place models. If we further integrate the
chance factors into the improved central place fractals, the theory will be
able to well explain the real patterns of urban places. As empirical analyses,
the US cities and towns are employed to verify the fractal-based models of
central places.Comment: 30 pages, 8 figures, 5 table
Spatial autocorrelation approaches to testing residuals from least squares regression
In statistics, the Durbin-Watson test is always employed to detect the
presence of serial correlation of residuals from a least squares regression
analysis. However, the Durbin-Watson statistic is only suitable for ordered
time or spatial series. If the variables comprise cross-sectional data coming
from spatial random sampling, the Durbin-Watson will be ineffectual because the
value of Durbin-Watson's statistic depends on the sequences of data point
arrangement. Based on the ideas from spatial autocorrelation, this paper
presents two new statistics for testing serial correlation of residuals from
least squares regression based on spatial samples. By analogy with the new form
of Moran's index, an autocorrelation coefficient is defined with a standardized
residual vector and a normalized spatial weight matrix. Then on the analogy of
the Durbin-Watson statistic, a serial correlation index is constructed. As a
case, the two statistics are applied to the spatial sample of 29 China's
regions. These results show that the new spatial autocorrelation model can be
used to test the serial correlation of residuals from regression analysis. In
practice, the new statistics can make up for the deficiency of the
Durbin-Watson test.Comment: 27 pages, 4 figures, 5 tables, 2 appendice
Defining urban and rural regions by multifractal spectrums of urbanization
The spatial pattern of urban-rural regional system is associated with the
dynamic process of urbanization. How to characterize the urban-rural terrain
using quantitative measurement is a difficult problem remaining to be solved.
This paper is devoted to defining urban and rural regions using ideas from
fractals. A basic postulate is that human geographical systems are of
self-similar patterns associated with recursive processes. Then multifractal
geometry can be employed to describe or define the urban and rural terrain with
the level of urbanization. A space-filling index of urban-rural region based on
the generalized correlation dimension is presented to reflect the degree of
geo-spatial utilization in terms of urbanization. The census data of America
and China are adopted to show how to make empirical analyses of urban-rural
multifractals. This work is not so much a positive analysis as a normative
study, but it proposes a new way of investigating urban and rural regional
systems using fractal theory.Comment: 21 pages, 4 figures, 6 table
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