Added Week 3 materials

Week 3 -- Temporal Models/Lab/Gompertz.R

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+
+setwd( "C:/Users/James.Thorson/Desktop/Project_git/2018_FSH556/Week 3 -- Temporal Models/Lab" )
+Use_REML = FALSE
+set.seed(2)
+
+#####################
+# Explore Gompertz model
+#####################
+
+beta= 0.2
+alpha = 1
+d_equil = exp(alpha/beta)
+
+d_1 = seq(0,d_equil*2,length=1e4)
+d_2 = d_1 * exp(alpha) * exp( - beta*log(d_1))
+
+# Dynamics
+png( file="Gompertz_dynamics.png", width=8, height=4, res=200, units="in")
+  par( mfrow=c(1,2), mar=c(3,3.5,2,0), mgp=c(1.75,0.25,0), tck=-0.02)
+  plot( x=d_1, y=d_2, type="l", lwd=3, xlab=expression(Biomass[t]), ylab=expression(Biomass[t+1]), main="Production")
+  abline( a=0, b=1, lty="dotted")
+  # Log-dynamics
+  plot( x=log(d_1[-1]), y=log(d_2[-1]/d_1[-1]), type="l", lwd=3, xlab=expression(log(Biomass[t])), ylab=expression(log(Biomass[t+1]/Biomass[t])), main="log-Biomass ratio" )
+  abline( a=1, b=0, lty="dotted")
+dev.off()
+
+######################
+# Simulate data
+######################
+
+nt = 100
+log_d0 = 3
+sigmaP = 0.5
+sigmaM = 0.5
+alpha = 1
+beta = 0.1
+
+# Simulate predictors
+log_d_t = log_b_t = rep(NA, nt)
+log_d_t[1] = log_d0
+for( t in 2:nt ){
+  log_d_t[t] = alpha + (1-beta)*log_d_t[t-1] + rnorm( 1, mean=0, sd=sigmaP )
+}
+for( t in 1:nt ){
+  log_b_t[t] = log_d_t[t] + rnorm( 1, mean=0, sd=sigmaM )
+}
+
+
+
+######################
+# Run in TMB
+######################
+
+library(TMB)
+
+# Compile model
+Version = "gompertz"
+compile( paste0(Version,".cpp") )
+
+# Build inputs
+Data = list( "nt"=nt, "log_b_t"=log_b_t )
+Parameters = list( "log_d0"=0, "log_sigmaP"=1, "log_sigmaM"=1, "alpha"=0, "rho"=0, "log_d_t"=rep(0,nt) )
+Random = c("log_d_t")
+if( Use_REML==TRUE ) Random = union( Random, c("log_d0","alpha","rho") )
+
+# Build object
+dyn.load( dynlib("gompertz") )
+Obj = MakeADFun(data=Data, parameters=Parameters, random=Random)  #
+
+# Prove that function and gradient calls work
+Obj$fn( Obj$par )
+Obj$gr( Obj$par )
+
+# Optimize
+start_time = Sys.time()
+Opt = nlminb( start=Obj$par, objective=Obj$fn, gradient=Obj$gr, control=list("trace"=1) )
+  Opt[["final_gradient"]] = Obj$gr( Opt$par )
+  Opt[["total_time"]] = Sys.time() - start_time
+
+# Get reporting and SEs
+Report = Obj$report()
+  SD = sdreport( Obj )
+
+for( z in 1:3 ){
+  png( file=paste0("gompertz_",z,".png"), width=8, height=5, res=200, units="in" )
+    par( mar=c(3,3,1,1), mgp=c(2,0.5,0), tck=-0.02 )
+    plot( x=1:nt, y=log_b_t, col="blue", cex=1.2, xlab="Time", ylab="Value", pch=20 )
+    if(z>=2){
+      points( x=1:nt, y=as.list(SD,"Estimate")$log_d_t, col="red" )
+      for( t in 1:nt) lines( x=rep(t,2), y=as.list(SD,"Estimate")$log_d_t[t]+c(-1.96,1.96)*as.list(SD,"Std. Error")$log_d_t[t], col="red" )
+    }
+    if(z>=3) lines( x=1:nt, y=log_d_t, col="black", lwd=2 )
+  dev.off()
+}
+

Week 3 -- Temporal Models/Lab/gompertz.cpp

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+#include <TMB.hpp>
+template<class Type>
+Type objective_function<Type>::operator() ()
+{
+  // Data
+  DATA_INTEGER( nt );
+  DATA_VECTOR( log_b_t );
+
+  // Parameters
+  PARAMETER( log_d0 );
+  PARAMETER( log_sigmaP );
+  PARAMETER( log_sigmaM );
+  PARAMETER( alpha );
+  PARAMETER( rho );
+  PARAMETER_VECTOR( log_d_t );
+
+  // Objective funcction
+  Type jnll = 0;
+
+  // Probability of random coefficients
+  jnll -= dnorm( log_d_t(0), log_d0, exp(log_sigmaP), true );
+  for( int t=1; t<nt; t++){
+    jnll -= dnorm( log_d_t(t), alpha + rho*log_d_t(t-1), exp(log_sigmaP), true );
+  }
+
+  // Probability of data conditional on fixed and random effect values
+  for( int t=0; t<nt; t++){
+    jnll -= dnorm( log_b_t(t), log_d_t(t), exp(log_sigmaM), true );
+  }
+
+  // Reporting
+  Type sigmaP = exp(log_sigmaP);
+  Type sigmaM = exp(log_sigmaM);
+
+  REPORT( sigmaP );
+  REPORT( sigmaM );
+  REPORT( log_d_t );
+
+  ADREPORT( sigmaP );
+  ADREPORT( sigmaM );
+
+  return jnll;
+}

Week 3 -- Temporal Models/Lecture/kalman_filter.R

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+
+setwd( "C:/Users/James.Thorson/Desktop/Project_git/2018_FSH556/Week 3 -- Temporal Models/Lecture" )
+Use_REML = FALSE
+set.seed(2)
+
+######################
+# Simulate data
+######################
+
+nt = 100
+x0 = 3
+sigmaP = 0.2
+sigmaM = 0.2
+alpha = 0.04
+
+# Simulate predictors
+x_t = y_t = rep(NA, nt)
+x_t[1] = x0
+for( t in 2:nt ){
+  x_t[t] = x_t[t-1] + rnorm( 1, mean=alpha, sd=sigmaP )
+}
+for( t in 1:nt ){
+  y_t[t] = x_t[t] + rnorm( 1, mean=0, sd=sigmaM )
+}
+
+######################
+# Linear model in R
+######################
+
+Lm = lm( y_t ~ I(1:nt) )
+ypred_t = predict(Lm, se=TRUE)
+
+for( z in 1:3 ){
+  png( file=paste0("lm_",z,".png"), width=8, height=5, res=200, units="in" )
+    par( mar=c(3,3,1,1), mgp=c(2,0.5,0), tck=-0.02 )
+    plot( x=1:nt, y=y_t, col="blue", cex=1.2, xlab="Time", ylab="Value", pch=20 )
+    if(z>=2){
+      points( x=1:nt, y=ypred_t$fit, col="red" )
+      for( t in 1:nt) lines( x=rep(t,2), y=ypred_t$fit[t]+c(-1.96,1.96)*ypred_t$se.fit[t], col="red" )
+    }
+    if(z>=3) lines( x=1:nt, y=x_t, col="black", lwd=2 )
+  dev.off()
+}
+
+
+######################
+# Loess smoother in R
+######################
+
+Loess = loess( y_t ~ I(1:nt) )
+ypred_t = predict(Loess, se=TRUE)
+
+for( z in 1:3 ){
+  png( file=paste0("loess_",z,".png"), width=8, height=5, res=200, units="in" )
+    par( mar=c(3,3,1,1), mgp=c(2,0.5,0), tck=-0.02 )
+    plot( x=1:nt, y=y_t, col="blue", cex=1.2, xlab="Time", ylab="Value", pch=20 )
+    if(z>=2){
+      points( x=1:nt, y=ypred_t$fit, col="red" )
+      for( t in 1:nt) lines( x=rep(t,2), y=ypred_t$fit[t]+c(-1.96,1.96)*ypred_t$se.fit[t], col="red" )
+    }
+    if(z>=3) lines( x=1:nt, y=x_t, col="black", lwd=2 )
+  dev.off()
+}
+
+
+######################
+# Smoother in R
+######################
+
+library(mgcv)
+
+Gam = gam( y_t ~ s(I(1:nt)) )
+ypred_t = predict(Gam, se=TRUE)
+
+for( z in 1:3 ){
+  png( file=paste0("gam_",z,".png"), width=8, height=5, res=200, units="in" )
+    par( mar=c(3,3,1,1), mgp=c(2,0.5,0), tck=-0.02 )
+    plot( x=1:nt, y=y_t, col="blue", cex=1.2, xlab="Time", ylab="Value", pch=20 )
+    if(z>=2){
+      points( x=1:nt, y=ypred_t$fit, col="red" )
+      for( t in 1:nt) lines( x=rep(t,2), y=ypred_t$fit[t]+c(-1.96,1.96)*ypred_t$se.fit[t], col="red" )
+    }
+    if(z>=3) lines( x=1:nt, y=x_t, col="black", lwd=2 )
+  dev.off()
+}
+
+######################
+# Run in TMB
+######################
+
+library(TMB)
+
+# Compile model
+Version = "kalman_filter"
+compile( paste0(Version,".cpp") )
+
+# Build inputs
+Data = list( "nt"=nt, "y_t"=y_t )
+Parameters = list( "x0"=0, "log_sigmaP"=1, "log_sigmaM"=1, "alpha"=0, "x_t"=rep(0,nt) )
+Random = c("x_t")
+if( Use_REML==TRUE ) Random = union( Random, c("x0","alpha") )
+
+# Build object
+dyn.load( dynlib("kalman_filter") )
+Obj = MakeADFun(data=Data, parameters=Parameters, random=Random)  #
+
+# Prove that function and gradient calls work
+Obj$fn( Obj$par )
+Obj$gr( Obj$par )
+
+# Optimize
+start_time = Sys.time()
+Opt = nlminb( start=Obj$par, objective=Obj$fn, gradient=Obj$gr, control=list("trace"=1) )
+  Opt[["final_gradient"]] = Obj$gr( Opt$par )
+  Opt[["total_time"]] = Sys.time() - start_time
+
+# Get reporting and SEs
+Report = Obj$report()
+  SD = sdreport( Obj )
+
+for( z in 1:3 ){
+  png( file=paste0("kalman_",z,".png"), width=8, height=5, res=200, units="in" )
+    par( mar=c(3,3,1,1), mgp=c(2,0.5,0), tck=-0.02 )
+    plot( x=1:nt, y=y_t, col="blue", cex=1.2, xlab="Time", ylab="Value", pch=20 )
+    if(z>=2){
+      points( x=1:nt, y=as.list(SD,"Estimate")$x_t, col="red" )
+      for( t in 1:nt) lines( x=rep(t,2), y=as.list(SD,"Estimate")$x_t[t]+c(-1.96,1.96)*as.list(SD,"Std. Error")$x_t[t], col="red" )
+    }
+    if(z>=3) lines( x=1:nt, y=x_t, col="black", lwd=2 )
+  dev.off()
+}
+

Week 3 -- Temporal Models/Lecture/kalman_filter.cpp

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+#include <TMB.hpp>
+template<class Type>
+Type objective_function<Type>::operator() ()
+{
+  // Data
+  DATA_INTEGER( nt );
+  DATA_VECTOR( y_t );
+
+  // Parameters
+  PARAMETER( x0 );
+  PARAMETER( log_sigmaP );
+  PARAMETER( log_sigmaM );
+  PARAMETER( alpha );
+  PARAMETER_VECTOR( x_t );
+
+  // Objective funcction
+  Type jnll = 0;
+
+  // Probability of random coefficients
+  jnll -= dnorm( x_t(0), x0, exp(log_sigmaP), true );
+  for( int t=1; t<nt; t++){
+    jnll -= dnorm( x_t(t), x_t(t-1) + alpha, exp(log_sigmaP), true );
+  }
+
+  // Probability of data conditional on fixed and random effect values
+  for( int t=0; t<nt; t++){
+    jnll -= dnorm( y_t(t), x_t(t), exp(log_sigmaM), true );
+  }
+
+  // Reporting
+  Type sigmaP = exp(log_sigmaP);
+  Type sigmaM = exp(log_sigmaM);
+
+  REPORT( sigmaP );
+  REPORT( sigmaM );
+  REPORT( x_t );
+
+  ADREPORT( sigmaP );
+  ADREPORT( sigmaM );
+  ADREPORT( x_t );
+
+  return jnll;
+}