Warning Signals with SDEs

Summary

  • 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

  1. Generate a data set that does not contain a warning signal, using the OU model.
  2. 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)
[1] 3405.286
> 2*OU.lik(alpha*theta, alpha, sigma) 
[1] 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.

Misc

  • 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.