User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/02/24
|Stochastic Population Dynamics|| Main project page|
Previous entry Next entry
Derin & Alan Meeting
Analytic calculations continued
Expected Variance across Ensembles
The expected variance across ensembles can be calculated from the Fokker-Planck Equation derived from the linear noise approximation. This should agree with the variance computed over a time window when the system has reached stationarity. I do these calculations and compare:
Equilibrium population size solves
Equilibrium variance should be given by
Time averaging from simulation doesn't match analytic prediction!
The analytic solution to is the general solution to a cubic so not very pretty, but easy to calculate numerically in order to give the variance estimate; using the values below I confirm that ODE solver and analytic solution above agree (), which is only about half that computed over the time window (approx 2360)! Parameter values and solutions from analytics are:
Test of higher order corrections
I try the higher order correction to the mean dynamics (accounts for inflation effect of the variance):
Though not surprisingly, the effect is negligible. In this case the average is slightly higher (572.5) and variance slightly lower (1349), so the curvature at equilibrium must be positive (in contrast to the deflation example with logistic or Levins model). Perhaps this is a consequence of the ergodicity assumption failing; I'll check against simply computing ensemble variance rather than the (mean-over-ensembles) time-averaged variance.
Ensemble averaging simulation does match theory, though time averaging doesn't
Reflections on Warning Signals literature
(Also a chance to test Mendeley library embedding)
Research collected using Mendeley
r25 | cboettig | 2010-02-25 00:12:09 -0800 (Thu, 25 Feb 2010) | 1 line
made samplefreq into a double instead of size_t, still co-opting the autocorrelations, seems to show successfully a different ensemble vs time-averaged variance
r24 | cboettig | 2010-02-24 23:46:07 -0800 (Wed, 24 Feb 2010) | 2 lines
co-opted the autoregressive data for the moment to explore direct ensemble mean and variance (without any time-averaging)
r23 | cboettig | 2010-02-24 21:53:12 -0800 (Wed, 24 Feb 2010) | 2 lines
ode integrator seems to accurately compute mean and variance dynamics, though the time-averaged variance doesn't seem to agree with the analytic ensemble variance; will have to compute that directly from simulation to compare. Runge-Kutta solver on adaptive mesh seems significantly slower than my simple Euler scheme; particularly with the (nearly irrelevant) higher order correction to the mean dynamics.