- Have a functional likelihood calculation from the full individual-based simulation, see Friday. Accuracy needs more testing, and the computation is probably too slow for optimization routines.
- Have an implementation of the linear decrease in stability model with analytic conditional probability density. Needed a couple adjustments today.
- Need to add direct simulation to the warning signals package, currently retunrs only time-averaged/ensemble averaged stats. Can be approximated by setting the window equal to the timestep and ensembles equal to one.
Revising & Testing the math
Revisions to the equations from Thursday’s entry:
- Added an alpha_0 parameter – the alpha dynamics shouldn’t start at zero.
- The variance integral had a factor of two that wasn’t carried through. Also this calculation changes as a result of the alpha_0
- The resulting analytical solution for the variance depends on a difference of error functions, which has poor numerical behavior for small beta. Implemented a flag in the R code which drops down to the analytic solution of beta=0 when it begins to run into this numerical round-off, otherwise numerics return variance of zero. Compared to analytic simulations.
Effective model choice: absence of a warning signal
- Generate a data set that does not contain a warning signal, using the OU model.
- Fit both model with changing stability and the simple OU model.
theta <- 3 alpha <- 1 sigma <- 2 X <- sde.sim(model="OU", theta= c(theta*alpha,alpha,sigma), X0=Xo, N=1000, T=1000) # (SDE package parameterizes OU differently) # These starting conditions converge to the wrong set of parameters but achieve the same likelihood Call: mle(minuslogl = warning.lik, start = list(alpha_0 = 2, theta = 1, sigma = 2, beta = 2), method = "L-BFGS-B", lower = c(0, 0, 0, 1e-09), control = list(maxit = 1000)) Coefficients: Estimate Std. Error alpha_0 0.5812878 139.62180181 theta 3.0881681 0.06202446 sigma 1.9305852 43.26824107 beta 1.0907615 279.24384653 -2 log L: 3401.722 ## These parameters converge closer to the true parameter set, and achieve much smaller Std Error Call: mle(minuslogl = warning.lik, start = list(alpha_0 = 2, theta = 1, sigma = 2, beta = 0.2), method = "L-BFGS-B", lower = c(0, 0, 0, 1e-09), control = list(maxit = 1000)) Coefficients: Estimate Std. Error alpha_0 1.10203570 NaN theta 3.08817901 0.06202212 sigma 2.09564217 0.02879976 beta 0.04950086 NaN -2 log L: 3401.722 ## Matches the parameter values from the simple OU model (beta = 0), and same likelihood mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, theta3 = 0.5), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) Coefficients: Estimate Std. Error theta1 3.479407 0.29304030 theta2 1.126721 0.09219684 theta3 2.103550 0.07886453 -2 log L: 3401.722 ## And matches (even outperforms) the likelihood of the true parameters: > 2*warning.lik(alpha_0, theta, sigma, beta)  3405.286 > 2*OU.lik(alpha*theta, alpha, sigma)  3405.837
Analysis of Results
- So bad news is fit of the richer model can depend on initial conditions, and maximizing likelihood alone doesn’t guarantee finding the right parameters.
- Luckily this alternate peak seems to have broader uncertainty
- Good news is that both approaches achieve the likelihood of the true parameter values. Any information criterion would successfully reject the change of stability model in this case.
### Code Updates
- Warning Signals project has also migrated to Github. Nicer interface, git is much faster, handles branching & merging very elegantly and this centralizes my projects.
- the optimization function in R takes control argument for maximum number of iterations as demonstrated above, though we don’t hit the default max (100) yet, which is promising for being able to optimize the individual-based model over at least a subset of parameters.
- Ironically the sde_likelihood library for this analysis has been developed in the Structured-Populations package, though it has now been integrated into the warningSignals package.
- Handy: function formals() gives the arguments/defaults of an R function.
- Should look into how mle() is calculating the standard error estimate on parameters.
- Joined Nature’s SciTable, aimed at undergraduates and professors teaching mostly. We’ll see if it’s any use.
- Statistics on Social media, youtube-style.
- Persuasive case for twitter, a social sixth sense?
- 100 twitter tips.
- added category tags to notebooks yesterday. Should help organize the subprojects in each notebook.