User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/05/04

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 * style="background-color: #EEE"|[[Image:owwnotebook_icon.png|128px]] Stochastic Population Dynamics
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 * style="background-color: #F2F2F2" align="center"|  |Main project page


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Alan Meeting

 * Goals next week: Bob & Brian beetle data: what an analysis would be with complete stochastic description.
 * Compare models with different

Full model: Parameter list from current code:


 * Add individual heterogeneity in egg and larva maturation age and in cannibalism of larva on eggs.

Outline / manuscript draft

 * Working copy

Likelihood inference for time series

 * All based on step-ahead predictions, from Markov property. Compare to deterministic skeleton's minimization of sum of squares on step-ahead predictions (i.e. assumes normal deviates).

See the monographs:
 * Iacus, S. M. (2008). Simulation and Inference for Stochastic Differential Equations With R Examples. New York: Springer.
 * Prakasa Rao, B.L.S. (1999) Statistical Inferences for Diffusion Type Processes, Oxford University Press, New York.

Exploring existing implementations of likelihood methods on SDEs through the R sde package accompanying the Iacus text.
 * Many nice methods for SDEs, general case is harder.

Conditions

 * Large sample scheme Time interval gets longer with n, while Δ is fixed time-step. Requires the additional assumptions of stationarity and/or ergodicity
 * High-frequency scheme: Δ shrinks as n increases, fixed window, need not assume ergodic.
 * Rapidly increasing design: hybrid combination with prescribed rate of mesh increase k

Underlying model
$$ dX_t = b(X_t, \theta) dt + \sigma(X_t, \theta) dW_t $$

Exact Likelihood conditions
$$ \exists \quad K \quad :: \quad \forall \quad x $$
 * Linear growth assumption:

$$ $$
 * b(x,\theta)| + |\sigma(x,\theta)| \leq K(1+|x|)

$$ $$
 * Global Lipshitz assumption:
 * b(x,\theta) -b(y,\theta) | + \sigma(x,\theta) - \sigma(y,\theta) < K|x-y|


 * Positive diffusion coefficient


 * Bounded moments


 * Smooth coefficients (will use up to 3 times differentiable)

Convergence of diffusion part estimator usually $$ \sqrt n $$, with $$ n \Delta_n^3 \to 0 $$

Numerical methods

 * Exact likelihood inference (conditional density of process must be known)
 * Euler approximation: discretization can assume linearity over small Δt
 * Elerian method (Milstein scheme)
 * Kessler (higher order Ito/Taylor expansion)
 * Simulated likelihood (approximate cdf with subdivisions in timestep over which Euler is accurate).
 * Hermite polynomial expansion of likelihood.

Steps

 * 1) Evaluate the conditional density function
 * 2) Evaluate the likelihood function (will be used as single step predictor)
 * 3) maximum likelihood estimation

Code updates

 * Swapped out my original linked list library for a more intelligent one. Not sure why pointer pointers are so useful but valgrind is happy.
 * beetle simulator now creates step-ahead realizations.
 * Considerations: replicates in C or R? would be better if R preserved the openmp code, but can always use parallel R to loop over timesteps and benefit from compiled speed on replicates.
 * Kernel density estimation for assigning probabilities? Probably reserve at R level at the moment.

Misc Reading & Notes

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