Figure 8

Orientation decomposition of the fBm trajectory shown in Fig. 7. (a) The two components \(x^{(1)}[i]\) and \(x^{(2)}[i]\) of the mixed fBm trajectory. (b) \(\theta \)-dependent heterogeneity in \(F^{(\theta )}(s)\), indicated by the angle dependence of \(\log _{10}^{(\theta )}F^{(\theta )}(\tilde{s})\) vs. \(\log _{10}\tilde{s}\), where \(\tilde{s}\sim s/1.93\) in the second order DDMA. (c) \(\theta \)-dependence of the local slopes of \(\log _{10}^{(\theta )}F^{(\theta )}(\tilde{s})\) vs. \(\log _{10}\tilde{s}\), indicating the spatial distribution of temporal correlations. (d) \(\theta \)-dependence of the slope in the range of \(1.2<\log _{10}{\tilde{s}}<2.5\). (e) Reconstructed original components \(\hat{\epsilon }_{1}[i],\hat{\epsilon }_{2}[i]\). (f) Fluctuation functions of the reconstructed original components \(\hat{\epsilon }_{1}\) with \(\hat{\theta }_{1}=14^{\circ }\) and \(\hat{\epsilon }_{2}\) with \(\hat{\theta }_{2}=74^{\circ }\).