# Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks
## 3 Empirical Analysis
### Results and Analysis  
The extraction method of spatial signal depends on the research objective. Suppose that we want to investigate urban growth by using traffic networks comprising streets and roads, the system of concentric circles can be employed to measure and calculate the total lengths of routes within certain zones (see File S1). The formula is Equation (13). The length series of streets and roads form the origin spatial signals of urban traffic networks, where trend is concerned, these signals take on unimodal curves (Figure 3 shows two examples). However, we cannot get too more geographical information from these curves. Utilizing Equation (14), we can turn the network length into network density between two adjacent circles. Traffic network density is associated with geospatial diffusion, and spatial diffusion is often based on distance-decay scaling law [19; 43]. If the traffic network density follows standard power law, then we can estimate fractal dimension directly. However, due to incomplete patterns of urban network based on administrative boundaries, as well as self-affine growth, the network density decay comes between exponential distribution and logarithmic distribution (Figure 4 shows two examples). In other words, the density distributions of traffic networks do not follow power law in a strict sense, therefore [PERSON]'s model cannot be used to estimate fractal dimension effectively. It is necessary to make wave-spectral analysis using Equations (15)-(18). Figure 3: The spatial signals for traffic network of Beijing and Shanghai cities (2016). (**a**) Beijing; (**b**) Shanghai. Note: These spatial signals represent urban traffic lines extracted through concentric circles and Equation (13). Each value reflects a total length of streets and roads falling in two adjacent circles. Figure 2: A flow chart of fractal-based spatial signal analysis for geographical systems. Note: The wave spectral series is based on Equations (4) and (6) as well as Equations (16) and (17), the wave-spectrum density is based on Equations (9) and (10). Based on spatial series of traffic network density, wave-spectrum scaling relations can be examined easily. Using mathematical software Matlab or even MS Excel, we can make Fourier transform for each density series. The wave number can be defined as  
\[k=\frac{j}{N} \tag{19}\]  
where \(j=0\), \(1\), \(2\),..., \(N-1\). According to FFT, the number \(N\) must satisfy the following condition  
\[N=2^{j+1} \tag{20}\]  
Otherwise, we have to remove a number of numbers at the beginning or at the end of a sequence, or add a set of 0 to the end, so that the length of the sequence becomes the integer power of 2. The circle number of each city is 100. We, either, add 28 zeros to the end of the sequence so that the length of sample path become 128 (\(N=2^{7}\)), or remove the last 36 data points, so that the sample path length become 64 (\(N=2^{6}\)). The former way (adding 0) is better than the latter way (removing partial data). Compared with deleting 36 data points, adding 28 zeros to the end of the sequence results in less geospatial information loss. After the FFT of density sequence, we can compute wave spectrum by means of Equation (16). Then, by using the least square regression based on Equations (17) and (18), we can make models for wave spectrum scaling relation. For example, for Beijing's traffic network, the model is  
\[\hat{F}^{*}(k)=0.0002092k^{-0.9635} \tag{21}\]  
The goodness of fit is about R2 = 0.8543, the spectral exponent is \(\alpha\) = 0.9635. Thus the fractal dimension is estimated as Df = 1.9635. For Shanghai's network density, the model is  
\[\hat{F}^{*}(k)=0.0005876k^{-0.7707} \tag{22}\]  
The goodness of fit is about R2 = 0.8570, the spectral exponent is \(\alpha\) = 0.7707. Consequently, the fractal dimension is estimated as Df = 1.7707. The traffic networks of other cities can be dealt with by analogy (Table 2, Figure 5). Figure 4: The patterns of traffic network density distributions of Beijing and Shanghai cities (2016).