A Fractal-based Approach For Desciption Of The Spatial Distribution Pattern Of Point Sets

Spatial point pattern analysis is an important part of spatial statistical analysis.It is widely used in commercial site selection,criminology,disease and epidemiology,and human mobility model analysis.The spatial point model aims to reveal the spatial distribution of point sets and the correlation between points(sets).Among them,the analysis of the spatial distribution characteristics of point sets focuses on the identification of aggregation patterns,the measurement of the degree of aggregation,and the detection of clusters.The current identification of aggregation patterns relies on density and distance analysis of point sets,which can only roughly distinguish between aggregated,random,and evenly distributed patterns.Lack of indicators and methods for quantitative characterization of point set aggregation.It is well known that point set distribution patterns are scale dependent.Point set spatial analysis mode analysis must consider the self-similarity between local and global.Fractal is unique in quantitatively depicting infinitely nested self-similar structures.In theory,fractals can be used in point-pattern spatial distribution features for point pattern recognition and aggregation degree measures.The thesis first reviews the traditional point pattern analysis methods,the concepts and calculation methods of fractal and fractal dimensions,and points out the shortcomings of current research.Then based on fractal theory and complex network method,the basic idea of fractal distribution based on point distribution is designed.The geometrical fractal dimension of fixed radius method,geometric fractal dimension of box covering method and fractal dimension of box covering structure are given.The three fractal dimensions,as well as the traditional point set distribution and the G/F/K function method are programmed.The thesis uses ArcGIS software to generate 62 sets of point sets,among which 31 sets are randomly distributed,30 sets of aggregated distribution points with different aggregation degrees,and 1 set of evenly distributed points.The spatial distribution characteristics of the simulated point set are analyzed by the sample counting method,the G/F/K function method and the fractal method respectively.The experimental results show that the traditional method only considers the characteristics of density or distance,and shows certain limitations in the degree of concentration or randomization of quantitative point sets.Both the fixed radius method and the box cover method geometric fractal dimension can well reveal the spatial distribution pattern of the point set,and can quantitatively measure the degree of agglomeration of the point set.The fractal calculation results of 62 sets of simulation data show that:(1)The geometrical fractal dimension of point set based on fixed radius method is between 1 and 2,and uniform>random>aggregate.Specifically,the fixed radius dimension of the random distribution and the aggregate distribution point set is between [1.7907,1.8104] and [1.5603,1.762],and the uniform distribution point set has a fixed radius dimension of 1.8363.Moreover,for a set of points of the same size aggregate distribution,as the number of clusters increases,the fractal dimension of the fixed radius of the corresponding point set is larger.(2)The geometrical fractal dimension of the box cover method shows a completely consistent law,uniform > random > aggregate.The box cover geometric fractal dimensions of the random distribution and the aggregate distribution point set are [1.6624,1.7357] and [1.3489,1.6534],respectively,and the uniform distribution point set box cover dimension is 1.8836.Similarly,as the number of clusters increases,the value of the box cover geometric fractal dimension increases.(3)The fractal dimension of the box cover structure shows a completely opposite trend.That is,aggregation > random > uniform,wherein the fractal dimension of the box cover structure of the random distribution and the aggregate distribution point set are [2.3781,2.4763] and [2.4932,2.5482],respectively,and the fractal dimension of the uniformly distributed point set box cover structure is 2.0205.In order to further verify the effectiveness of the proposed method,the point set fractal dimension analysis method is applied to the spatial distribution pattern analysis of bus stations in Chengdu,revealing the spatial heterogeneity degree and compactness difference of bus stations in different regions.This method provides an important reference for spatial point pattern analysis,point element synthesis,criminology,disease science and other research.