城市交通网络分形维数的不确定性估计、控制与分析
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摘要
长度-半径维数模型作为描述城市交通网络复杂不确定性现象的一种分形分维方法,其自身存在的不确定性往往被忽视,且相关研究更是鲜见报道。故针对该模型在分形维数测算全过程中存在的不确定性问题,本文率先开展了系统剖析、定量估计和质量控制研究。首先对数据源、矢量化处理、测算中心、尺度选择、以及分维数模型估计等一系列环节进行了不确定性估计与分析,其中首次给出了分形维数在一定置信水平下的不确定性度量区间,并依据误差传播理论对误差的传递和累积进行了描述;然后着重提出了基于LMed S(Least Median of Squares)的质量控制方法。最后通过对拉萨市的算例实验表明:道路的矢量化过程、测算中心和测算尺度的选择都会导致分维的不确定性;并在对数据质量进行控制的基础上,通过置信区间对长度-半径维数模型的不确定性进行了在一定概率水平下的首次度量;同时结合区域现状对研究结果给出了合乎实际的解释。本文在描述表征不确定性问题的分形几何和分形维数的基础上,系统地揭示了其自身不确定性的本质,不仅进一步丰富了分形分维理论,为控制其质量奠定理论基础,而且可为城市交通网络分形维数的地学应用提供可靠的科学依据。
Finding the ideal fractal of the objective world based on pure mathematics is difficult, but the statistical significance of random fractal is an objective existence such that fractal and fractal dimension has some uncertainty. The length-radius dimension model is a fractal dimension method used to describe the complex and uncertain phenomenon of urban traffic network. However, the uncertainty of the model is frequently neglected, and related research are rarely reported. Therefore, theories and methods of uncertainty for fractal dimension should be developed and improved urgently. Aiming at the uncertainty existing in the measuring process of length-radius dimension model, we first systematically conduct research on the analysis, quantitative estimation, and quality control of fractal dimension for the urban traffic network. The uncertainty estimation and analysis of this model are conducted from the aspects of data source, vector processing, measuring center, scale selection, and fractal dimension estimation. In particular, the uncertainty measurement interval of fractal dimension(i.e., the corresponding regression coefficient) under a certain confidence level is first provided quantitatively. Then, based on the theory of error propagation, we describe the propagation and accumulation of errors. Meanwhile, a method of quality control is proposed using least median of squares(LMed S) to remove the gross error(i.e., outliers) and to determine the scale less range simultaneously. In this study, the experimental data were selected from the traffic network distribution map of Lhasa City in 2011, which is the map of the Lhasa road network with a scale of 1∶370000 published by the China Cartographic Publishing House. The main road includes national,provincial, county, and township roads. The Lhasa City traffic network distribution map was acquired by registration and vectorization using Arc GIS. Experimental results show that vectorization of the traffic network and selection of the measuring center and scale cause uncertainty in the fractal dimension. The road is vectorized under different scale environments, e.g., 1∶1000, 1∶10000, cdots,1∶500000. Thus,the uncertainty for roads can be obviously observed. Transportation hub and geometric centers are employed to verify the uncertainty of fractal dimension. The uncertainty of the length-radius dimension model is controlled at a certain level of probability, and the uncertainty of this model is measured by calculating the confidence interval for the first time. To be exact, the confidence interval of the fractal dimension is(1.633, 1.707) under the confidence level of 95%. Furthermore, the corresponding table of the original data and data processed by LMed S proves that the proposed method is reliable, that is, the coefficient of determination R2 is improved from 0.9931 to 0.9989. From the description of the uncertainty of fractal geometry and fractal dimension, this study systematically reveals the uncertain essence of the model. The proposed methods of uncertainty, quality control, and analysis are not only applicable to the length-radius dimension model but also to the branch dimension that is similar to the calculation method. The model can also provide references for the statistical significance of all random fractals in nature. The proposed model not only further enriches fractal dimension theory and establishes the theoretical foundation of its quality control but also provides reliable scientific basis for the geoscience applications of fractal dimension for the urban traffic network.
Finding the ideal fractal of the objective world based on pure mathematics is difficult, but the statistical significance of random fractal is an objective existence such that fractal and fractal dimension has some uncertainty. The length-radius dimension model is a fractal dimension method used to describe the complex and uncertain phenomenon of urban traffic network. However, the uncertainty of the model is frequently neglected, and related research are rarely reported. Therefore, theories and methods of uncertainty for fractal dimension should be developed and improved urgently. Aiming at the uncertainty existing in the measuring process of length-radius dimension model, we first systematically conduct research on the analysis, quantitative estimation, and quality control of fractal dimension for the urban traffic network. The uncertainty estimation and analysis of this model are conducted from the aspects of data source, vector processing, measuring center, scale selection, and fractal dimension estimation. In particular, the uncertainty measurement interval of fractal dimension(i.e., the corresponding regression coefficient) under a certain confidence level is first provided quantitatively. Then, based on the theory of error propagation, we describe the propagation and accumulation of errors. Meanwhile, a method of quality control is proposed using least median of squares(LMed S) to remove the gross error(i.e., outliers) and to determine the scale less range simultaneously. In this study, the experimental data were selected from the traffic network distribution map of Lhasa City in 2011, which is the map of the Lhasa road network with a scale of 1∶370000 published by the China Cartographic Publishing House. The main road includes national,provincial, county, and township roads. The Lhasa City traffic network distribution map was acquired by registration and vectorization using Arc GIS. Experimental results show that vectorization of the traffic network and selection of the measuring center and scale cause uncertainty in the fractal dimension. The road is vectorized under different scale environments, e.g., 1∶1000, 1∶10000, cdots,1∶500000. Thus,the uncertainty for roads can be obviously observed. Transportation hub and geometric centers are employed to verify the uncertainty of fractal dimension. The uncertainty of the length-radius dimension model is controlled at a certain level of probability, and the uncertainty of this model is measured by calculating the confidence interval for the first time. To be exact, the confidence interval of the fractal dimension is(1.633, 1.707) under the confidence level of 95%. Furthermore, the corresponding table of the original data and data processed by LMed S proves that the proposed method is reliable, that is, the coefficient of determination R2 is improved from 0.9931 to 0.9989. From the description of the uncertainty of fractal geometry and fractal dimension, this study systematically reveals the uncertain essence of the model. The proposed methods of uncertainty, quality control, and analysis are not only applicable to the length-radius dimension model but also to the branch dimension that is similar to the calculation method. The model can also provide references for the statistical significance of all random fractals in nature. The proposed model not only further enriches fractal dimension theory and establishes the theoretical foundation of its quality control but also provides reliable scientific basis for the geoscience applications of fractal dimension for the urban traffic network.
引文
Ai T H,Liu Y L and Huang Y F.2007.The hierarchical watershed par titioning and generalization of river network.Acta Geodaetica e Cartographica Sinica,36(2):231-236(艾廷华,刘耀林,黄亚锋2007.河网汇水区域的层次化剖分与地图综合.测绘学报36(2):231-236)[DOI:10.3321/j.issn:1001-1595.2007.02.020]
Bai C G and Cai X H.2008.Fractal characteristics of transportation network of Nanjing city.Geographical Research,27(6):1419-1426(柏春广,蔡先华.2008.南京市交通网络的分形特征.地理研究27(6):1419-1426)[DOI:10.11821/yj2008060021]
Benguigui L.1992.The fractal dimension of some railway networks Journal De Physique I,2(4):385-388[DOI:10.1051/jp11992151]
Benguigui L and Daoud M.1991.Is the suburban railway system a fractal?Geographical Analysis,23(4):362-368[DOI:10.1111/j1538-4632.1991.tb00245.x]
Chen J and Ma S Y.2012.Research on fractal dimension of road net in China at different map scales.Geomatics&Spatial Information Technology,35(8):145-147(陈杰,马素媛.2012.顾及尺度全国主要省市道路网分形维数的研究.测绘与空间地理信息,35(8)145-147)[DOI:10.3969/j.issn.1672-5867.2012.08.047]
Chen Y G.2008.Fractal Urban Systems:Scaling,Symmetry,Spatia Complexity.Beijing:Science Press(陈彦光.2008.分形城市系统:标度·对称·空间复杂性.北京:科学出版社)
Frankhauser P.1990.Aspects fractals des structures urbaines.Lespace Géographique,19(1):45-69[DOI:10.3406/spgeo.1990.2943]
Li B J,Gu H H and Ji Y Z.2012.Study on fractal features of transport ation network in Xuzhou City//Proceedings of the 2nd Interna tional Conference on Remote Sensing,Environment and Trans portation Engineering(RSETE 2012).Nanjing:IEEE:1-4.[DOI10.1109/RSETE.2012.6260745]
Li D Y,Liu C Y,Du Y and Han X.2004.Artificial intelligence with uncertainty.Journal of Software,15(11):1583-1594(李德毅,刘常昱,杜鹢,韩旭.2004.不确定性人工智能.软件学报,15(11)1583-1594)
Li J J,Chen J and Zhu J L.2010.Scale transferring and uncertainty analysis of river length based on DEM.Yangtze River,41(8)55-58(李静静,陈健,朱金玲.2010.基于DEM的河流长度尺度转换与不确定性分析.人民长江,41(8):55-58)[DOI:10.3969/j issn.1001-4179.2010.08.015]
Li Z L.2007.Digital map generalization at the age of enlightenment:a review of the first forty years.The Cartographic Journal,44(1)80-93[DOI:10.1179/000870407X173913]
Liang D F,Li Y L and Jiang C B.2002.Research on the box counting algorithm in fractal dimension measurement.Journal of Image and Graphics,7(3):246-250(梁东方,李玉梁,江春波.2002.测量分维的“数盒子”算法研究.中国图象图形学报,7(3):246-250[DOI:10.3969/j.issn.1006-8961.2002.03.007]
Liu C L,Duan D Z,Yu R L and Luo J.2013.Multi-scale analysis about urban-rural road network of four major metropolitan area in China based on fractal theory.Economic Geography,33(3):52-58(刘承良,段德忠,余瑞林,罗静.2013.中国四大都市圈城乡道路网分形的多尺度比较分析.经济地理,33(3):52-58)
Liu M L and Huang P B.2003.Application of fractal theory to re search on the tempo-spatial changes of urban traffic network-tak ing Shanghai as a case.Geomatics and Information Science o Wuhan University,28(6):749-753(刘妙龙,黄佩蓓.2003.分形理论在城市交通网络时空演变特征研究中的应用-以上海市为例.武汉大学学报(信息科学版),28(6):749-753)
Liu M L and Huang B P.2004.Spatial-temporal evolution of fracta feature in traffic network of Shanghai metropolis.Scientia Geo graphica Sinica,24(2):144-149(刘妙龙,黄蓓佩.2004.上海大都市交通网络分形的时空特征演变研究.地理科学,24(2)144-149)[DOI:10.3969/j.issn.1000-0690.2004.02.003]
Liu Z W,Wang X Z and Chen S Q.2002.Uncertainties of data in urb an census GIS.Geomatics and Information Science of Wuhan University,27(6):627-631(柳宗伟,王新洲,陈顺清.2002.城市人口GIS中数据的不确定性研究.武汉大学学报(信息科学版)27(6):627-631)
Ma J H,Liu D X and Chen Y Q.2015.Random prefractal dimension and length uncertainty of the continental coastline of China.Geo graphical Research,34(2):319-327(马建华,刘德新,陈衍球2015.中国大陆海岸线随机前分形分维及其长度不确定性探讨地理研究,34(2):319-327)[DOI:10.11821/dlyj201502011]
Mandelbrot B B and Aizenman M.1979.Fractals:form,chance,and dimension.Physics Today,32(5):65-66[DOI:10.1063/12995555]
Rousseeuw P J.1984.Least median of squares regression.Journal o the American Statistical Association,79(388):871-880[DOI10.1080/01621459.1984.10477105]
Shi W Z.2009.Principles of Modeling Uncertainties in Spatial Data and Spatial Analyses.Boca Raton,FL:CRC Press
Wang Q,Mao F and Wu J T.1998.Research on geographical informa tion generalization in GIS.Journal of Remote sensing,2(2)155-160(王桥,毛锋,吴纪桃.1998.GIS中的地理信息综合.遥感学报,2(2):155-160)[DOI:10.11834/jrs.19980214]
Yang Y X.2012.Some notes on uncertainty,uncertainty measure and accuracy in satellite navigation.Acta Geodaetica et Cartographica Sinica,41(5):646-650(杨元喜.2012.卫星导航的不确定性、不确定度与精度若干注记.测绘学报,41(5):646-650)
Bai C G and Cai X H.2008.Fractal characteristics of transportation network of Nanjing city.Geographical Research,27(6):1419-1426(柏春广,蔡先华.2008.南京市交通网络的分形特征.地理研究27(6):1419-1426)[DOI:10.11821/yj2008060021]
Benguigui L.1992.The fractal dimension of some railway networks Journal De Physique I,2(4):385-388[DOI:10.1051/jp11992151]
Benguigui L and Daoud M.1991.Is the suburban railway system a fractal?Geographical Analysis,23(4):362-368[DOI:10.1111/j1538-4632.1991.tb00245.x]
Chen J and Ma S Y.2012.Research on fractal dimension of road net in China at different map scales.Geomatics&Spatial Information Technology,35(8):145-147(陈杰,马素媛.2012.顾及尺度全国主要省市道路网分形维数的研究.测绘与空间地理信息,35(8)145-147)[DOI:10.3969/j.issn.1672-5867.2012.08.047]
Chen Y G.2008.Fractal Urban Systems:Scaling,Symmetry,Spatia Complexity.Beijing:Science Press(陈彦光.2008.分形城市系统:标度·对称·空间复杂性.北京:科学出版社)
Frankhauser P.1990.Aspects fractals des structures urbaines.Lespace Géographique,19(1):45-69[DOI:10.3406/spgeo.1990.2943]
Li B J,Gu H H and Ji Y Z.2012.Study on fractal features of transport ation network in Xuzhou City//Proceedings of the 2nd Interna tional Conference on Remote Sensing,Environment and Trans portation Engineering(RSETE 2012).Nanjing:IEEE:1-4.[DOI10.1109/RSETE.2012.6260745]
Li D Y,Liu C Y,Du Y and Han X.2004.Artificial intelligence with uncertainty.Journal of Software,15(11):1583-1594(李德毅,刘常昱,杜鹢,韩旭.2004.不确定性人工智能.软件学报,15(11)1583-1594)
Li J J,Chen J and Zhu J L.2010.Scale transferring and uncertainty analysis of river length based on DEM.Yangtze River,41(8)55-58(李静静,陈健,朱金玲.2010.基于DEM的河流长度尺度转换与不确定性分析.人民长江,41(8):55-58)[DOI:10.3969/j issn.1001-4179.2010.08.015]
Li Z L.2007.Digital map generalization at the age of enlightenment:a review of the first forty years.The Cartographic Journal,44(1)80-93[DOI:10.1179/000870407X173913]
Liang D F,Li Y L and Jiang C B.2002.Research on the box counting algorithm in fractal dimension measurement.Journal of Image and Graphics,7(3):246-250(梁东方,李玉梁,江春波.2002.测量分维的“数盒子”算法研究.中国图象图形学报,7(3):246-250[DOI:10.3969/j.issn.1006-8961.2002.03.007]
Liu C L,Duan D Z,Yu R L and Luo J.2013.Multi-scale analysis about urban-rural road network of four major metropolitan area in China based on fractal theory.Economic Geography,33(3):52-58(刘承良,段德忠,余瑞林,罗静.2013.中国四大都市圈城乡道路网分形的多尺度比较分析.经济地理,33(3):52-58)
Liu M L and Huang P B.2003.Application of fractal theory to re search on the tempo-spatial changes of urban traffic network-tak ing Shanghai as a case.Geomatics and Information Science o Wuhan University,28(6):749-753(刘妙龙,黄佩蓓.2003.分形理论在城市交通网络时空演变特征研究中的应用-以上海市为例.武汉大学学报(信息科学版),28(6):749-753)
Liu M L and Huang B P.2004.Spatial-temporal evolution of fracta feature in traffic network of Shanghai metropolis.Scientia Geo graphica Sinica,24(2):144-149(刘妙龙,黄蓓佩.2004.上海大都市交通网络分形的时空特征演变研究.地理科学,24(2)144-149)[DOI:10.3969/j.issn.1000-0690.2004.02.003]
Liu Z W,Wang X Z and Chen S Q.2002.Uncertainties of data in urb an census GIS.Geomatics and Information Science of Wuhan University,27(6):627-631(柳宗伟,王新洲,陈顺清.2002.城市人口GIS中数据的不确定性研究.武汉大学学报(信息科学版)27(6):627-631)
Ma J H,Liu D X and Chen Y Q.2015.Random prefractal dimension and length uncertainty of the continental coastline of China.Geo graphical Research,34(2):319-327(马建华,刘德新,陈衍球2015.中国大陆海岸线随机前分形分维及其长度不确定性探讨地理研究,34(2):319-327)[DOI:10.11821/dlyj201502011]
Mandelbrot B B and Aizenman M.1979.Fractals:form,chance,and dimension.Physics Today,32(5):65-66[DOI:10.1063/12995555]
Rousseeuw P J.1984.Least median of squares regression.Journal o the American Statistical Association,79(388):871-880[DOI10.1080/01621459.1984.10477105]
Shi W Z.2009.Principles of Modeling Uncertainties in Spatial Data and Spatial Analyses.Boca Raton,FL:CRC Press
Wang Q,Mao F and Wu J T.1998.Research on geographical informa tion generalization in GIS.Journal of Remote sensing,2(2)155-160(王桥,毛锋,吴纪桃.1998.GIS中的地理信息综合.遥感学报,2(2):155-160)[DOI:10.11834/jrs.19980214]
Yang Y X.2012.Some notes on uncertainty,uncertainty measure and accuracy in satellite navigation.Acta Geodaetica et Cartographica Sinica,41(5):646-650(杨元喜.2012.卫星导航的不确定性、不确定度与精度若干注记.测绘学报,41(5):646-650)