# Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks
## 3 Empirical Analysis  
### Data and Methods  
The methods illustrated above can be applied to the traffic network of 10 cities in China to make case studies. The spatial patterns of traffic networks of cities proved to bear a fractal nature [23; 34; 35; 36; 37; 38; 39]. In this study, the urban road network dataset is derived from the Chinese digital navigation map in 2016 ([[http://geodata.pku.edu.cn](http://geodata.pku.edu.cn)]([http://geodata.pku.edu.cn](http://geodata.pku.edu.cn))), including freeways, arterials, and collectors. Traffic networks in Chinese cities bear a fractal nature, and the fractal dimension can be estimated by means of the box-counting method [40]. The box dimension is a kind of global parameter [24], which is strongly linked with spatial entropy [7; 10; 41]. Now, we want to know the local fractal parameter, i.e., radial dimension. Taking the central business district (CBD) of each city as the center, we create 0.5-km to 50-km buffer rings in an increasing, step-wise manner. The ring areas outside the city boundary were excluded, so the spatial patterns are incomplete. Then, road lengths in each concentric ring are measured by ArcGIS 10.2 (Figure 1). As indicated above, to extract spatial signals of urban traffic networks, we can draw concentric circles with the urban center as the center. The circle is numbered as j = 0, 1, 2,..., N\(-\)1, where j = 0 represents the center of a city, and N refers to the total number of circles. In our examples, the number of circles is 100. The spatial signal of urban street lines can be expressed as  
\[x(r)=\Delta L(r)=L(r_{j+1})-L(r_{j}) \tag{13}\]  
where x(r) denotes spatial signals, \(\Delta\) is difference operator, L(r) is the cumulative length of street lines around the city center, L(r) is the cumulative length of street lines within the jth circle. Therefore, \(\Delta\)L(r) refers to the total length of street lines between the jth circle and the (j + 1)th circle. The zone between any two adjacent circles can be regarded as a ring. The average density of each ring can be defined as  
\[\rho(r)=\frac{\Delta L(r)}{\Delta A(r)}=\frac{L(r_{j+1})-L(r_{j})}{\pi r_{j+1} ^{2}-\pi r_{j}^{2}} \tag{14}\]  
where \(A(r)=\pi r^{2}\) denotes the area within the circle with radius \(r\). The parameter estimation of any mathematical model needs certain algorithms. The wave-spectral analysis needs two algorithms: one is fast Fourier transform (FFT), and the other is ordinary least squares (OLS) regression method. The former is used to obtain wave spectrums, and the latter is used to estimate spectral exponent. The application of a mathematical methods to concrete problems involves what is called three worlds: real world, mathematical world, and computational world [29; 42]. In fact, the computational world can be treated as the bridge between the mathematical world and the real world. Equation (4) describes Fourier transform in pure theoretical form, that is, it is defined in the mathematical world. For the observational data of cities in the real world, the Fourier transform should be replaced by an algorithm termed FFT. Data processing and algorithm application is implemented in the computational world [29]. Applying FFT to the urban density series, we have \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & **Data** & **Object** & **Parameter** & **Complexity** & **Geographical Space** \\ \hline Temporal signal & Time series data & Dynamic evolution & Hurst exponent & Time lag & Phase space \\ \hline Spatial signal & Spatial data & Spatial distribution & Fractal dimension & Spatial dimension & Real space \\ \hline Hierarchical signal & Cross-sectional data & Rank-size distribution & Zipf exponent & Interaction & Order space \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Three types of signals and uses for scientific studies.