Exploring the Spectrum Power Fractal Scaling Parameters by Hurst Range Increment - Second Order Moment Generation Techniques

Muhammad Ilyas; Shaheen Abbas; Kamran Malik

Authors

Muhammad Ilyas Department of Mathematics Government College University Hyderabad, Sindh. Pakistan
Shaheen Abbas Laboratory of Applied Mathematics and Data Analysis (LAMDA, Al-Khwarazm DIUI Unit)
Kamran Malik Department of Mathematics Government College University Hyderabad, Sindh. Pakistan

Keywords:

Fractal Dimension (FD), Mann-Kendell test, Spectrum power fractal scaling, Hurst Range increment (HRange), Hurst Second Order Moment Generation (H2ndM)

Abstract

Spectral power analysis was employed to assess the Fractal Dimension (FD) and explore fractal scaling using Hurst increment ranges and second-order moment relations in the context of urban population trends. This research aimed to scrutinize population trends in Karachi over both uneven periods (1729 to 1946) and even periods (1951 to 2020) using non-parametric Mann-Kendall tests and Hurst error accuracy testing. The primary focus was on analyzing spectrum power fractal scaling through Hurst exponent ranges and second-order moment generation. The FD results indicated irregular (1.371) and regular (1.058) intervals within the inequality range of 1 < D < 1.5. The log-population trend cumulatively increased from 3.0 in 1729 to 5.72 in 1946, and from 6.05 in 1951 to 7.36 in 2020, suggesting that the fractal dimension is more appropriately fitted for total regular intervals. The second-order and range exponents were H2ndM (0.60 ± 0.09) and H-Range (0.83 ± 0.05) for the uneven period (1729 to 1946), and H2ndM (0.85 ± 0.06) and H-Range (0.93 ± 0.02) for the even period (1951 to 2020). The study's results demonstrate that the range increment method is suitable and consistent across both long and short intervals. For regular intervals, the Hurst exponents show a linear relationship, indicating stability in the population trend analysis.

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