Effective warning signals

  • Predicting variance of variance by direct calculation – still need to crunch some math for the expected convergence.
    • Still, the approach should be able to do more than describe single points as unexpected deviates.
    • Still need to address gradual change vs change point analysis.
    • Essentially the same as the phyolgenentic problem – one rate vs two rates. Model selection approaches?
    • So far theory is essentially built on a model selection between linear models.
  • Calculate the eigenvalue directly rather than ratio of eigenvalue to noise:
    • Estimate the eigenvalue from the correlation function and from power spectrum, rather than the lag-1 autocorrelation, or variance.
    • Proper signal processing techniques for detecting bifurcations?

Coding Progress

  • Added a proper autocorrelation function calculation, log transform and linear regression gives the eigenvalue and the variance.
  • Tested using the Langevin model
x_{t+1} = x_t \left(1- \frac{\kappa}{\gamma} \Delta t \right) + \Delta t \sqrt{\frac{2 K_B T}{\gamma} } \xi_t
x_{t+1} = x_t \left(1- \frac{\kappa}{\gamma} \Delta t \right) + \Delta t \sqrt{\frac{2 K_B T}{\gamma} } \xi_t

Whose correlation function is given by

\frac{K_BT}{\gamma} e^{-\kappa t / \gamma}
\frac{K_BT}{\gamma} e^{-\kappa t / \gamma}