Abstract
A number of mathematical methods have been developed to make temporal signal analyses based on time series. However, no effective method for spatial signal analysis, which are as important as temporal signal analyses for geographical systems, has been devised. Nonstationary spatial and temporal processes are associated with nonlinearity, and cannot be effectively analyzed by conventional analytical approaches. Fractal theory provides a powerful tool for exploring spatial complexity and is helpful for spatio-temporal signal analysis. This paper is devoted to developing an approach for analyzing spatial signals of geographical systems by means of wave-spectrum scaling. The traffic networks of 10 Chinese cities are taken as cases for positive studies. Fast Fourier transform (FFT) and ordinary least squares (OLS) regression methods are employed to calculate spectral exponents. The results show that the wave-spectrum density distribution of all these urban traffic networks follows scaling law, and that the spectral scaling exponents can be converted into fractal dimension values. Using the fractal parameters, we can make spatial analyses for the geographical signals. The wave-spectrum scaling methods can be applied to both self-similar fractal signals and self-affine fractal signals in the geographical world. This study has implications for the further development of fractal-based spatiotemporal signal analysis in the future.
Highlights
Signal processing and analysis is important for exploring complex systems, especially when the systems are treated as black boxes
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)
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
Summary
Signal processing and analysis is important for exploring complex systems, especially when the systems are treated as black boxes. Many mathematical methods can be employed to make signal analyses. The Hausdorff dimension proved to be equivalent to information entropy [5]. A number of mathematical and empirical relationships have been found between entropy and fractal dimension [6,7,8,9]. Based on box-counting method, normalized fractal dimension was proved to equal normalized information entropy [10]. Fractal dimension can be employed to reveal hidden information in spatial signals. Among various fractal-based signal processing and analyses, the most significant ones include reconstructing phase space, rescaled range analysis (R/S analysis), wavelet analysis, and spectral analysis. Spectral exponents has been demonstrated to connect fractal dimension, and are associated with the Hurst exponent in R/S analysis [12,13,14]
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