[Pomp-commits] r1024 - pkg/pompExamples/man

[noreply at r-forge.r-project.org]

Author: kingaa
Date: 2014-12-17 20:12:26 +0100 (Wed, 17 Dec 2014)
New Revision: 1024

Modified:
   pkg/pompExamples/man/budmoth.Rd
Log:
- update budmoth model documentation

Modified: pkg/pompExamples/man/budmoth.Rd
===================================================================
--- pkg/pompExamples/man/budmoth.Rd	2014-12-17 19:12:21 UTC (rev 1023)
+++ pkg/pompExamples/man/budmoth.Rd	2014-12-17 19:12:26 UTC (rev 1024)
@@ -4,8 +4,43 @@
 \title{Larch budmoth model POMPs with real and simulated data.}
 \description{
   \code{pompExample(budmoth.sim)} constructs a \code{pomp} object containing the larch budmoth model and simulated budmoth density, parasitism rate, and food quality (needle-length) data.
-  Four datasets, representing four distinct parameter regimes, are avaiable.
+  Four datasets, representing four distinct parameter regimes, are available.
+
+  The model has three state variables: 
+  \enumerate{
+    \item{\eqn{Q_t} (measure of food quality on \eqn{[0,1]})} 
+    \item{\eqn{N_t} (budmoth density)}
+    \item{\eqn{S_t} (fraction of budmoth larvae infected with parasitoids)}
+  }
+
+  There are three observables:
+  \enumerate{
+    \item{\eqn{\hat Q_t} (needle length)}
+    \item{\eqn{\hat N_t}}
+    \item{\eqn{\hat S_t}}
+  }
 }
+\section{State process}{
+  Uncorrelated random effects, for \eqn{t=1,\dots,T}:
+  \deqn{\alpha_t \sim \mathrm{LogitNormal}(\mathrm{logit}(\alpha),\sigma_{\alpha}^2)}
+  \deqn{\lambda_t \sim \mathrm{Gamma}(\lambda,\sigma_{\lambda}^2)}
+  \deqn{a_t \sim \mathrm{LogNormal}(\log(a),\sigma_{a}^2)}
+  Note: \eqn{X} is \eqn{\mathrm{LogitNormal}(\mu,\sigma)} if \eqn{\mathrm{logit}(X)} is \eqn{\mathrm{Normal}(\mu,\sigma)}. 
+  
+  The inverse of \eqn{\mathrm{logit}} is \eqn{\mathrm{expit}}.
+  \pkg{R} functions \code{logit}, \code{expit}, \code{rlogitnorm}, \code{dlogitnorm} are defined in \pkg{pompExamples}.
+  
+  The state process, for \eqn{t=1,\dots,T}:
+  \deqn{Q_{t} = (1-\alpha_{t})\frac{\gamma}{\gamma+N_{t-1}} +\alpha_{t}Q_{t-1}}
+  \deqn{N_{t} = \lambda_t N_{t-1} (1-S_{t-1})\exp\big\{-gN_{t-1}-\delta(1-Q_{t-1})\big\}}
+  \deqn{S_{t} =  1-\exp\left(\frac{-a_tS_{t-1}N_{t-1}}{1+a_twS_{t-1}N_{t-1}} \right)}
+}  
+\section{Measurement process}{
+  For \eqn{t=1,\dots,T}:
+  \deqn{\hat Q_t \sim \mathrm{LogNormal}(\log(\beta_0+\beta_1Q_t),\sigma_Q)}
+  \deqn{\hat N_t \sim \mathrm{LogNormal}(\log(N_t),\sigma_N)}
+  \deqn{\hat S_t \sim \mathrm{LogitNormal}(\mathrm{logit}(uS_t),\sigma_S)}
+}
 \examples{
 ## three regimes, high and low noise regimes for parasitism and tritrophic
 bm <- pompExample(budmoth,envir=NULL)
@@ -14,4 +49,4 @@
 plot(bm$para2)
 plot(bm$tri)
 }
-\keyword{datasets}
+\keyword{models}