User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/08/07

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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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Reading
Trying to make this show reading of the day, the script below shows only current reading:  

Friendfeed discussion on solving this: 

Warning Signals

 * Implementing likelihood-based methods for early warning signals. Trying three models:

$$ \begin{align} dX =& (\theta_1 + \theta_2 X) dt + \theta_3 dW_t \\ dX =&(\alpha_0+\beta t) (\theta + X) dt + \sigma dW_t \\ dX =& \alpha(t) (\theta + X) dt + \sigma dW_t \end{align} $$


 * Model 3 is intended to be a changepoint analysis, where $$ \alpha(t) $$ is piecewise constant.
 * I solved analytically the transition probability density in Model 2 on ../../05/06, see this and the following entry.
 * Code implementations are in Structured Populations repository

To Do

 * 1) Simulation routine for custom equations (2, 3)
 * 2) Neyman-Pearson Likelihood ratio test method between models
 * 3) Simulation from actual population dynamics -- update warning signals method to return individual runs


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