# 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.

• 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?