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LMedS（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 (LMedS) 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 ArcGIS. 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 LMedS 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.