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

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Reading

Trying to make this show reading of the day, the script below shows only current reading: <html> <script type="text/javascript" src="http://www.google.com/reader/ui/publisher-en.js"></script> <script type="text/javascript" src="http://www.google.com/reader/public/javascript/user/16488617539907435411/state/com.google/broadcast?n=5&callback=GRC_p(%7Bc%3A%22blue%22%2Ct%3A%22Carl's%20shared%20items%22%2Cs%3A%22true%22%2Cn%3A%22true%22%2Cb%3A%22false%22%7D)%3Bnew%20GRC"></script> </html>

Friendfeed discussion on solving this: <html> <iframe src="http://friendfeed.com/the-life-scientists/8c7753b8/can-you-capture-snapshot-of-rss-feed-i-d-love-to-be?embed=1" frameborder="0" height="400" width="650" style="border:1px solid #aaa"></iframe> </html>

Warning Signals

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \alpha(t) } is piecewise constant.
  • I solved analytically the transition probability density in Model 2 on User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/05/06, 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