Table 1 This table makes easy reference to the scaling index \(\delta\) from the above homogeneous scaling relation for the scaled variable X(t); relates it to the PSD \(S_{p}(f)\) index \(\beta\) through the waiting-time PDF \(\psi (t)\) index \(\mu\) in the two asymptotic regimes. The value \(\mu =2\) is the boundary between the underlying process having a finite (\(\mu > 2\)) or an infinite (\(\mu < 2\)) average waiting time and is also the point at which \(\beta = 1\) where the process is that of 1/f-noise.
From: Complexity synchronization: a measure of interaction between the brain, heart and lungs
| Â | Scaled functions | Parameter relations | Parameter range | Â |
|---|---|---|---|---|
Waiting-time PDF | \(\psi (t) \propto t^{-\mu }\) | Â | 1 \(\leqslant\) \(\mu\) \(\leqslant\) 3 | Â |
Power spectrum | \(S(f) \propto f^{-\beta }\) | \(\mu = 3 - \beta\) | Â | Â |
Scale variable | \(X(t) \propto t^{\delta }\) | \(\mu = 1 + \delta\) | 1 \(\leqslant\) \(\mu\) \(\leqslant\) 2 | Non-ergodic |
| Â | Â | \(\mu = 1 + 1/\delta\) | 2 \(\leqslant\) \(\mu\) \(\leqslant\) 3 | Ergodic |
| Â | Â | \(\delta = 0.5\) | \(\mu\) \(\ge\) 3 | Â |
