# Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks
## 5 Conclusions  
The Fourier transform and wave-spectrum scaling relation can be applied to fractal-based spatial signal analyses. The academic contribution of this article to science includes two aspects. One is to propose an analytical framework based on wave-spectrum scaling for geographical spatial signals, and the other is to demonstrate that traffic network density distribution follow wave-spectrum scaling law. The first contribution is the writing goal of this paper, and the second contribution is the by-product of this study. The complete analytical processes and cases are presented. Where wave-spectrum scaling analysis method is concerned, the main points of this work are as follows: Firstly, if the self-similar fractal structure of spatial signals is concealed by random noises, or spatial patterns are not incomplete for signal extraction, the wave-spectrum scaling relation can be utilized to estimate fractal dimension. By changing spatial signals to density distribution, we can model the distribution with scale-free correlation function, then the correlation function can be converted into wave-spectrum density by the Fourier transform. Further, using the power law relation between wave number and wave spectrum, we can calculate spectral scaling exponent. The spectral exponent can be turned into the fractal dimension, which can be employed to make spatial analyses for geographical signals. Secondly, if spatial signals bear self-affine fractal property, we can use wave-spectrum scaling to compute the self-affine record dimension. Then, the anisotropic fractal process can be treated as isotropic fractal process approximately. Thus, the fractal dimension of self-affine records can be converted to the fractal dimension of self-similar trails. Combining the self-affine fractal parameter and the self-similar fractal parameter, we can make spatial analysis for geographical signals. Thirdly, the wave-spectrum scaling can be naturally applied to temporal signals of geographical systems. If we have a time series representing temporal signals of geographical systems, e.g., the level of urbanization, we can apply the wave-spectrum scaling to the dataset. If the temporal signal bears fractal property, we will be able to work out the self-affine record dimension. By means of the fractal dimension of self-affine record, we can make time series analysis for the temporal signal of urban evolution.  
The following are available online at [[https://www.mdpicom/2076-341](https://www.mdpicom/2076-341)]([https://www.mdpicom/2076-341](https://www.mdpicom/2076-341)) 7/11/1/87/s1. File S1: Datasets of 10 Chinese urban traffic networks for fractal-based wave spectral analysis (Excel). File S2: Datasets of nighttime lights of cities in Beijing-Tianjin-Hebei region of China (1992-2013).  
Conceptualization, Y.C. and Y.L.; methodology, Y.C.; software, Y.L.; validation, Y.C.; formal analysis, Y.C.; investigation, Y.C.; resources, Y.L.; data curation, Y.L.; writing--original draft preparation, Y.C.; writing--review and editing, Y.C.; visualization, Y.L.; supervision, Y.C.; project administration, Y.C.; funding acquisition, Y.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundations of China, grant numbers 41590843 & 41671167.  
The nighttime light dataset (1992-2014) come from American NOAA National Centers for Environmental Information (NCEI). We would like to thank the two anonymous reviewers whose constructive comments were helpful in improving the quality of this paper.  
Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.