# Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks
## 3 Empirical Analysis
### Cases of Power Spectral Scaling Analysis  
In geographical analysis, there are corresponding and conversion relationships between space and time. For comparison and reference, two power spectrum analysis cases of temporal signals can be presented. Complicated signal analysis, based on time series, can be characterized by self-affine record fractal dimension. Despite the similarity in mathematical essence, the form differs from spatial signal analysis. Replacing wave number \(k\) and wave spectral density \(W(k)\) with frequency \(f\) and power spectral density \(P(f)\), we can turn Equation (10) into a frequency-spectrum relation as below \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Position** & **City** & **Coefficient \(F_{0}\)******* & \begin{tabular}{c} **Spectral** \\ **Exponent \(\alpha\)** \\ \end{tabular} & \begin{tabular}{c} **Goodness of** \\ **Fit \(R^{2}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Fractal** \\ **Dimension \(D_{f}\)** \\ \end{tabular} &
\begin{tabular}{c} **Fractal** \\ **Dimension \(D_{s}\)** \\ \end{tabular} \\ \hline North & Beijing & 0.0002092 & 0.9635 & 0.8543 & 1.9635 & 1.5365 \\  & Tianjin & 0.0005071 & 0.7567 & 0.9236 & 1.7567 & 1.7433 \\  & Guangzhou & 0.0003429 & 0.9107 & 0.9152 & 1.9107 & 1.5893 \\  & Shenzhen & 0.0003437 & 0.9838 & 0.8525 & 1.9838 & 1.5162 \\  & Wuhan & 0.0002729 & 0.8884 & 0.9402 & 1.8884 & 1.6116 \\  & Nanjing & 0.0005069 & 0.7234 & 0.9104 & 1.7234 & 1.7766 \\  & Shanghai & 0.0005876 & 0.7707 & 0.8570 & 1.7707 & 1.7293 \\  & Hangzhou & 0.0003653 & 0.8730 & 0.9291 & 1.8730 & 1.6270 \\  & Chongqing & 0.0003146 & 0.8115 & 0.8455 & 1.8115 & 1.6885 \\  & Chengdu & 0.0003350 & 0.8697 & 0.8206 & 1.8697 & 1.6303 \\ \hline \hline \multicolumn{7}{l}{Notes: Based on power law distribution, the self-similar fractal dimension Df can be converted into self-affine fractal dimension Ds, and vice versa. * \(p\) < 0.05.} \\ \end{tabular}
\end{table} 
Table 2: Wave spectrum exponents of urban density and corresponding fractal dimension values (2016). \[P(f)=P_{0}f^{-\beta} \tag{23}\]  
in which \(P_{0}\) denotes the proportionality coefficient. Equation (23) can be used to analyze temporal signals through fractal parameters. For example, applying Equation (23) to the time series of urbanization level of China yields a model as follows  
\[\hat{P}(f)=1.7995f^{-1.7367} \tag{24}\]  
The goodness of fit is about \(R^{2}\) = 0.9806, the power spectral exponent is about \(\beta\) = 1.7367 (Figure 6). According to Equation (11), the fractal dimension of self-affine record is about \(D_{s}\) = (\(5-1.7367\))/2 = 1.6316. Thus the Hurst exponent is \(H\) = 2\(-D_{s}\) = 0.3684. This suggests an anti-persistence process of urbanization. Urbanization contains a fluctuation process or negative feedback process. Figure 5: The wave-spectrum scaling relations between wave numbers and wave spectrums (2016). (**a**) Beijing; (**b**) Tanjin; (**c**) Guangzhou; (**d**) Shenzhen; (**e**) Wuhan; (**f**) Nanjing; (**g**) Shanghai; (**h**) Hangzhou; (**i**) Chongqing; (**j**) Chengdu. Note: All the spectral exponents were estimated by Equation (17), which is based on Equation (6). By using Equation (8), we can easily turn the spectral exponent values into fractal dimension values. The dynamics of urbanization involves both temporal and spatial processes. In fact, urbanization is associated with urban systems and urban form [44]. The frequency-spectral scaling can be applied to urban form and growth in an urban systems by both temporal and spatial signals. For instance, based on the sample paths of nighttime light (NTL) time series (1992-2013) (see File S2), we can calculate the spectral exponent of urban growth of the cities in Beijing-Tianjin-Hebei region of China (Figure 7, Table 3). The NTL data were originally processed for multi-scaling allometric analyses [45]. The spectral exponents can be converted into self-affine record dimension and self-similar trail dimension, and associated with the Hurst exponent based on R/S analysis [30; 46].