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 12

Fractal dimension 𝐷_𝐻𝑚 against the amplitude multiplier 𝑚; 𝐷_𝐻𝑚max=𝐷_𝐻𝑚 with the highest value (at 24.2s±127 data of Figure 13, window width of 1s=255data); 𝐷_𝐻𝑚min=𝐷_𝐻𝑚 with the smallest value (at 5.5s±127 data of Figure 13); 𝐷_𝐻𝑚 average (avg)=𝐷_𝐻 averaged over 60 s; max−min, max−avg, and avg−min=𝐷_𝐻𝑚 differentials; (max−1)/(avg−1)=𝐷_𝐻𝑚 ratio.