Ben Bolker, estimation of parameters for stochastic dynamic models

Ben Bolker, estimation of parameters for stochastic dynamic models

Stochastic simulation: discrete time and continuous time.
Simple approaches:  trajectory matching, gradient matching, comparison
Fancier method: SIMEX, Kalman filter
State space models: Markov Chain Monte Carlo.

Typical statistical model:     Typical math model:
 stochastic                                 deterministic
 static                                         dynamic
 phenomenological                    mechanistic

Standard time-series models (ARIMA, spectral/wavelet analyses) are (mostly) phenomenological. Most are assessing patterns.

For stochastic models, need to define both a process model and an observation model (measurement model).
Process model Y(t+1) = F(Y(t))
Measurement model Y_obs(t) ~ Y(t)
Might decompose process model into a deterministic model for the expectation and (additive) noises around the expectation: e.g, Y(t) = mean + noise, Y(t) ~ Poisson(exp(eta)).

My understanding is that Walker defines  process errors that induce dynamic changes in variance (cause chaotic behavior in the future?!). Observation error is the ascertainment uncertainty.
Walker joked that 'process error' are given by God. Walker considered model bias as systematic errors.

Stochastic ODEs
ODE + Wiener process (derivative of a Brownian motion)
Delicate analysis (for biologists, Turelli, 1977, Roughgarden 1995, For mathamaticians, Oksendal 2003.
Specialized integration methods
Suited for cellular, physiological, but not population model?
Gillespie algorithm, exponentially distributed time between transitions.

SIMulation-EXtrapolation method.

Kalman filter is widely used in engineering. Often works for linear (typically normal) models, but can be extended to nonlinear models. Multivaraite extensions are natural (Schnute, 1994)
Kalman filter is kind of a state-status method.

References: Random environments and stochastic calculus - ‎Turelli