A Robust Algorithm for Optimisation and Customisation of Fractal Dimensions of Time Series Modified by Nonlinearly Scaling Their Time Derivatives: Mathematical Theory and Practical Applications

Figure 11

Fractal dimension 𝐷_𝐻𝑚 against the amplitude multiplier 𝑚; 𝐷_𝐻𝑚max=𝐷_𝐻𝑚 with the highest value (at 371.1 s ±75 data of Figure 10, window width of 2.5s=151data); 𝐷_𝐻𝑚min=𝐷_𝐻𝑚 with the smallest value (at 423.5s±75 data of Figure 10); 𝐷_𝐻𝑚max−𝐷_𝐻𝑚min=Δ𝐷_𝐻𝑚, the 𝐷_𝐻𝑚 differential; optimal amplitude multiplier 𝑚 is defined at the peak of the 𝐷_𝐻𝑚 differential (maximal 𝐷_𝐻𝑚 distance between 𝐷_𝐻𝑚max and 𝐷_𝐻𝑚min).