User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/04/11

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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

[math]\displaystyle{ x_{t+1} = x_t \left(1- \frac{\kappa}{\gamma} \Delta t \right) + \Delta t \sqrt{\frac{2 K_B T}{\gamma} } \xi_t }[/math]

Whose correlation function is given by

[math]\displaystyle{ \frac{K_BT}{\gamma} e^{-\kappa t / \gamma} }[/math]