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

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 * style="background-color: #EEE"|[[Image:owwnotebook_icon.png|128px]] Stochastic Population Dynamics
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 * style="background-color: #F2F2F2" align="center"|  |Main project page


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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

$$ x_{t+1} = x_t \left(1- \frac{\kappa}{\gamma} \Delta t \right) + \Delta t \sqrt{\frac{2 K_B T}{\gamma} } \xi_t $$

Whose correlation function is given by

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


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