Exploring the Fractal Parameters of Urban Growth and Form with Wave-Spectrum Analysis

Table 2

Comparison between the fractal dimension values of an imaginary city form and its estimated results from the spectral exponent.

Length of sample path (N)	Radial dimension (𝐷𝑓)	Spectral exponent (β)	Goodness of fit (𝑅^2)	Estimation of 𝐷𝑓 (𝐷^∗𝑓)	Estimation of 𝐷𝑠 (𝐷^∗𝑠)	Square error (𝐸^2)
256	1.0500	0.8684	0.9919	1.4342	2.0658	0.1476
	1.2500	1.0044	0.9943	1.5022	1.9978	0.0636
	1.5000	1.1904	0.9950	1.5952	1.9048	0.0091
	1.6536	1.3072	0.9946	1.6536	1.8464	0.0000
	1.7000	1.3417	0.9944	1.6709	1.8292	0.0008
	1.7500	1.3783	0.9942	1.6892	1.8109	0.0037
	1.9500	1.5126	0.9933	1.7563	1.7437	0.0375
512	1.0500	0.8612	0.9903	1.4306	2.0694	0.1449
	1.2500	1.0020	0.9938	1.5010	1.9990	0.0630
	1.5000	1.1974	0.9950	1.5987	1.9013	0.0097
	1.6654	1.3308	0.9947	1.6654	1.8346	0.0000
	1.7000	1.3582	0.9946	1.6791	1.8209	0.0004
	1.7500	1.3970	0.9944	1.6985	1.8015	0.0027
	1.9500	1.5386	0.9934	1.7693	1.7307	0.0327
1024	1.0500	0.8557	0.9889	1.4279	2.0722	0.1428
	1.2500	0.9998	0.9933	1.4999	2.0001	0.0625
	1.5000	1.2026	0.9951	1.6013	1.8987	0.0103
	1.6756	1.3512	0.9947	1.6756	1.8244	0.0000
	1.7000	1.3715	0.9946	1.6858	1.8143	0.0002
	1.7500	1.4124	0.9944	1.7062	1.7938	0.0019
	1.9500	1.5605	0.9934	1.7803	1.7198	0.0288
2048	1.0500	0.8517	0.9877	1.4259	2.0742	0.1413
	1.2500	0.9981	0.9929	1.4991	2.0010	0.0620
	1.5000	1.2066	0.9951	1.6033	1.8967	0.0107
	1.6846	1.3691	0.9947	1.6846	1.8155	0.0000
	1.7000	1.3825	0.9946	1.6913	1.8088	0.0001
	1.7500	1.4152	0.9944	1.7076	1.7924	0.0018
	1.9500	1.5790	0.9933	1.7895	1.7105	0.0258