Qratio

JumpDiff.Qratio.Qratio(lag: numpy.ndarray, timeseries: numpy.ndarray, loc: int = None, correction: bool = False) → numpy.ndarray

Qratio method to distinguish pure diffusion from jump-diffusion timeseries, Given by the relation of the 4th and 6th Kramers─Moyal coefficient with increasing lag

\[\begin{split}Q(x,\tau) = \frac{D_6(x,\tau)}{5 D_4(x,\tau)} = \left\{\begin{array}{ll} b(x)^2 \tau, & \text{diffusive} \\ \sigma_\xi^2(x), & \text{jumpy} \end{array}\right.\end{split}\]
Parameters:
  • lag (np.ndarray of ints) – An array with the time-lag to extract the Kramers–Moyal coefficient for different lags.
  • timeseries (np.ndarray) – A 1-dimensional timeseries.
  • loc (float (defaul None)) – Use a particular point in space to calculate the ratio. If None given, the maximum of the probability density function is taken.
  • corrections (bool (defaul False)) – Select whether to use corrective terms.
Returns:

  • lag (np.ndarray of ints) – Same as input, but only lag > 0 and as ints.
  • ratio (np.ndarray of len(lag)) – Ratio of the sixth-order over forth-order Kramers–Moyal coefficient.

References

Anvari, M., Tabar, M. R. R., Peinke, J., Lehnertz, K., ‘Disentangling the stochastic behavior of complex time series.’ Scientific Reports, 6, 35435, 2016. doi: 10.1038/srep35435.

Lehnertz, K., Zabawa, L., and Tabar, M. R. R., ‘Characterizing abrupt transitions in stochastic dynamics.’ New Journal of Physics, 20(11):113043, 2018. doi: 10.1088/1367-2630/aaf0d7.