Adding Week 5 materials

Week 5 -- 1D spatial models/Homework/Homework_Week_5_simulator.R

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+
+Sim_Fn = function( n_i=1000, Scale=2, Sigma2=1, logSD_spatial=0.1, L0=10, Linf_0=100, beta_y=0.02, growth_rate=0.1, mortality_rate=growth_rate*1.6, logSD_resid=0.05){
+  require( RandomFields )
+
+  # Sample locations
+  n_i = 1000
+  y_i = runif( n=n_i, min=32, max=49 )
+
+  # Simulate spatial process
+  RMmodel = RMgauss(var=logSD_spatial^2, scale=Scale)
+  Linf_i = Linf_0 * exp( RFsimulate(model=RMmodel, x=rep(0,n_i), y=y_i)@data[,1] - Sigma2/2 ) * exp( beta_y*(y_i-40.5) )
+  plot( y=Linf_i, x=y_i )
+
+  # Simulate ages of samples
+  Survival_a = exp( -mortality_rate * 1:100 )
+  a_i = sample( size=n_i, x=1:100, prob=Survival_a, replace=TRUE)
+  l_i = Linf_i - (Linf_i-L0) * exp( -growth_rate * a_i ) * exp( rnorm(n_i, mean=-logSD_resid^2/2, sd=logSD_resid) )
+  plot( x=a_i, y=l_i )
+
+  # Bundle and return stuff
+  DF = data.frame( "y_i"=y_i, "a_i"=a_i, "l_i"=l_i, "Linf_i"=Linf_i)
+  return(DF)
+}
+

Week 5 -- 1D spatial models/Lab/Lab 5.R

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+
+
+library(TMB)
+setwd( "C:/Users/James.Thorson/Desktop/Project_git/2018_FSH556/Week 5 -- 1D spatial models/Lab/" )
+
+###################
+# Visualize autocorrelation
+###################
+
+Rho = 0.8
+X = rep(0,10)
+for(s in 2:length(X)) X[s] = X[s-1] + rlnorm(1, meanlog=0, sdlog=1)
+Corr = Rho^X
+X_line = seq(0,max(X),length=1e4)
+Corr_line = Rho^(X_line)
+
+png( file="Autocorrelation.png", width=4, height=4, res=200, units='in')
+  plot( x=X, y=Corr, xlab="distance", ylab="Correlation", ylim=c(0,1) )
+  lines( x=X_line, y=Corr_line, xlab="distance", ylab="Correlation", ylim=c(0,1) )
+dev.off()
+
+###################
+# Unequal distance autoregressive
+###################
+
+x = 1:100
+Rho = 0.8
+Sigma2 = (0.5) ^ 2
+n_rep = 3
+beta0 = 3
+
+# Simulate locations
+loc_s = rep(NA, length(x))
+loc_s[1] = 0
+for(s in 2:length(x)) loc_s[s] = loc_s[s-1] + rlnorm(1, meanlog=0, sdlog=1)
+
+# Simulate spatial process
+epsilon_s = rep(NA, length(x))
+epsilon_s[1] = rnorm(1, mean=0, sd=sqrt(Sigma2))
+for(s in 2:length(x)) epsilon_s[s] = Rho^abs(loc_s[s]-loc_s[s-1]) * epsilon_s[s-1] + rnorm(1, mean=0, sd=sqrt(Sigma2*(1-Rho^(2*abs(loc_s[s]-loc_s[s-1])))) )
+
+# SImulate counts
+c_si = matrix( nrow=length(x), ncol=n_rep)
+for(s in 1:nrow(c_si)){
+for(i in 1:ncol(c_si)){
+  c_si[s,i] = rpois(1, exp(beta0 + epsilon_s[s]) )
+}}
+
+# Compile
+Params = list( "beta0"=0, "ln_sigma2"=0, "logit_rho"=0, "epsilon_s"=rnorm(length(x)) )
+compile( "unequal_distance_autoregressive_V1.cpp" )
+dyn.load( dynlib("unequal_distance_autoregressive_V1") )
+
+######## Version 0 -- Stochastic process with automatic sparseness detection
+# Build object
+Data = list("Options_vec"=c(0), "c_si"=c_si, "loc_s"=loc_s )
+Obj = MakeADFun( data=Data, parameters=Params, random="epsilon_s", DLL="unequal_distance_autoregressive_V1" )
+# Optimize
+Opt0 = TMBhelper::Optimize( obj=Obj, newtonsteps=1 )
+par0 = Opt0$par
+h0 = Obj$env$spHess(random=TRUE)
+
+######## Version 2 -- Covariance and built-in function
+# Build object
+Data = list("Options_vec"=c(2), "c_si"=c_si, "loc_s"=loc_s )
+Obj = MakeADFun( data=Data, parameters=Params, random="epsilon_s", DLL="unequal_distance_autoregressive_V1" )
+# Optimize
+Opt2 = TMBhelper::Optimize( obj=Obj, newtonsteps=1 )
+par2 = Opt2$par
+h2 = Obj$env$spHess(random=TRUE)
+
+# Compare timing
+Opt0$run_time
+Opt2$run_time
+
+# Compare separability
+library(INLA)
+image(h0, main="Version 0"); dev.new()
+image(h2, main="Version 2")
+

Week 5 -- 1D spatial models/Lab/unequal_distance_autoregressive_V1.cpp

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+
+#include <TMB.hpp>
+
+// Space time
+template<class Type>
+Type objective_function<Type>::operator() ()
+{
+  // Settings
+  DATA_IVECTOR( Options_vec );
+  // Slot 0: method for calculating probability of random effects
+
+  // Data
+  DATA_MATRIX( c_si );
+  DATA_VECTOR( loc_s );
+
+  // Parameters
+  PARAMETER( beta0 );
+  PARAMETER( ln_sigma2 );
+  PARAMETER( logit_rho );
+
+  // Random effects
+  PARAMETER_VECTOR( epsilon_s );
+
+  // Objective funcction
+  int n_s = c_si.col(0).size();
+  int n_i = c_si.row(0).size();
+  vector<Type> jnll_comp(2);
+  jnll_comp.setZero();
+  Type sigma2 = exp(ln_sigma2);
+  Type rho = 1 / (1 + exp(-logit_rho));
+
+  // Probability of random effects
+  using namespace density;
+  vector<Type> dist_s(n_s);
+  if( Options_vec(0)==0 ){
+    jnll_comp(1) -= dnorm( epsilon_s(0), Type(0.0), pow(sigma2,0.5), true );
+    for(int s=1; s<n_s; s++){
+      dist_s(s) = loc_s(s)-loc_s(s-1);
+      jnll_comp(1) -= dnorm( epsilon_s(s), pow(rho,dist_s(s))*epsilon_s(s-1), pow(sigma2*(1-pow(rho,2*dist_s(s))),0.5), true );
+    }
+  }
+  if( Options_vec(0)==1 ){
+    // NOT IMPLEMENTED //
+  }
+  if( Options_vec(0)==2 ){
+    matrix<Type> Cov_ss(n_s,n_s);
+    for(int s1=0; s1<n_s; s1++){
+    for(int s2=s1; s2<n_s; s2++){
+      Cov_ss(s1,s2) = sigma2 * pow( rho, loc_s(s2)-loc_s(s1) );
+      if(s1!=s2) Cov_ss(s2,s1) = Cov_ss(s1,s2);
+    }}
+    REPORT( Cov_ss );
+    jnll_comp(1) += MVNORM(Cov_ss)( epsilon_s );
+  }
+  if( Options_vec(0)==3 ){
+    // NOT IMPLEMENTED //
+  }
+
+  // Probability of data conditional on random effects
+  for( int s=0; s<n_s; s++){
+  for( int i=0; i<n_i; i++){
+    jnll_comp(0) -= dpois( c_si(s,i), exp(beta0 + epsilon_s(s)), true );
+  }}
+
+  // Reporting
+  Type jnll = jnll_comp.sum();
+  REPORT( jnll_comp );
+  REPORT( jnll );
+  REPORT( sigma2 );
+  REPORT( rho );
+
+  return jnll;
+}

Week 5 -- 1D spatial models/Lecture/Lecture_5.R

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+
+
+library(TMB)
+setwd( "C:/Users/James.Thorson/Desktop/Project_git/2018_FSH556/Week 5 -- 1D spatial models/Lecture/" )
+
+############################
+# Visualize Gaussian process
+############################
+
+library(RandomFields)
+
+png( "Gaussian_process.png", width=8, height=5, res=200, units="in")
+  par( mfrow=c(2,2), mar=c(2,2,2,0), mgp=c(2,0.5,0) )
+  for(i in 1:4){
+    Scale = c(10, 10, 25, 25)[i]
+    SD = c(0.5, 1, 0.5, 1)[i]
+    x_i = seq(0,100,length=100)
+    RMmodel = RMgauss(var=SD^2, scale=Scale)
+    plot( 1, type="n", xlim=range(x_i), ylim=c(-3,3), main=paste0("SD=",SD," Scale=",Scale), xlab="", ylab="")
+    for(j in 1:20){
+      gp_i = RFsimulate(model=RMmodel, x=x_i, y=rep(0,length(x_i)))@data[,1]
+      lines( x=x_i, y=gp_i )
+    }
+  }
+dev.off()
+
+###################
+# Visualize autocorrelation
+###################
+
+Rho = 0.8
+X = 0:10
+Corr = Rho^X
+
+png( file="Autocorrelation.png", width=4, height=4, res=200, units='in')
+  plot( x=X, y=Corr, xlab="distance", ylab="Correlation", ylim=c(0,1) )
+dev.off()
+
+###################
+# Visualize autocorrelation vs. random-walk
+###################
+
+Rho = c(-0.5,0,0.5,0.99)
+Marginal_Sigma = 1
+Conditional_Sigma = sqrt( 1-Rho^2 ) * Marginal_Sigma
+
+Y = X = 1:1e2
+png( file="Autocorrelated_series.png", width=4, height=6, res=200, units='in')
+  par( mfrow=c(4,1), mgp=c(2,0.5,0), mar=c(2,2,1,0), xaxs="i")
+  for(i in 1:4){
+    Y[1] = 0
+    for(s in 2:length(X)) Y[s] = Y[s-1]*Rho[i] + rnorm(1, mean=0, sd=Conditional_Sigma[i])
+    plot( x=X, y=Y, xlab="location", ylab="value", ylim=c(-3,3), type="l", main=paste0("Rho = ",Rho[i]) )
+  }
+dev.off()
+
+###################
+# Math check on inverse-covariance matrix
+###################
+Rho = 0.5
+Sigma2 = 2 ^ 2
+x = 1:10
+Dist = outer(x, x, FUN=function(a,b){abs(a-b)})
+
+# Calculate Q directly
+Cov = Sigma2 / (1-Rho^2) * Rho^Dist
+Q = solve(Cov)
+
+# Calculate Q analytically
+M = diag(length(x)) * (1+Rho^2)
+M[ cbind(1:9,2:10) ] = -Rho
+M[ cbind(2:10,1:9) ] = -Rho
+Q2 = 1/Sigma2 * M
+
+###################
+# Equal distance autoregressive
+###################
+
+x = 1:100
+Rho = 0.8
+Sigma2 = (0.5) ^ 2
+n_rep = 3
+beta0 = 3
+
+# Simulate spatial process
+epsilon_s = rep(NA, length(x))
+epsilon_s[1] = rnorm(1, mean=0, sd=sqrt(Sigma2))
+for(s in 2:length(x)) epsilon_s[s] = Rho*epsilon_s[s-1] + rnorm(1, mean=0, sd=sqrt(Sigma2))
+
+# SImulate counts
+c_si = matrix( nrow=length(x), ncol=n_rep)
+for(s in 1:nrow(c_si)){
+for(i in 1:ncol(c_si)){
+  c_si[s,i] = rpois(1, exp(beta0 + epsilon_s[s]) )
+}}
+
+# Compile
+Params = list( "beta0"=0, "ln_sigma2"=0, "logit_rho"=0, "epsilon_s"=rnorm(length(x)) )
+compile( "autoregressive_V1.cpp" )
+dyn.load( dynlib("autoregressive_V1") )
+
+######## Version 0 -- Stochastic process with automatic sparseness detection
+# Build object
+Data = list("Options_vec"=c(0), "c_si"=c_si )
+Obj = MakeADFun( data=Data, parameters=Params, random="epsilon_s", DLL="autoregressive_V1" )
+# Optimize
+Opt0 = TMBhelper::Optimize( obj=Obj, newtonsteps=1 )
+par0 = Opt0$par
+h0 = Obj$env$spHess( random=TRUE )
+
+######## Version 1 -- Analytic precision matrix
+# Build object
+Data = list("Options_vec"=c(1), "c_si"=c_si )
+Obj = MakeADFun( data=Data, parameters=Params, random="epsilon_s", DLL="autoregressive_V1" )
+# Optimize
+Opt1 = TMBhelper::Optimize( obj=Obj, newtonsteps=1 )
+par1 = Opt1$par
+h1 = Obj$env$spHess(random=TRUE)
+
+######## Version 2 -- Covariance and built-in function
+# Build object
+Data = list("Options_vec"=c(2), "c_si"=c_si )
+Obj = MakeADFun( data=Data, parameters=Params, random="epsilon_s", DLL="autoregressive_V1" )
+# Optimize
+Opt2 = TMBhelper::Optimize( obj=Obj, newtonsteps=1 )
+par2 = Opt2$par
+h2 = Obj$env$spHess(random=TRUE)
+
+######## Version 3 -- Built-in function for AR process
+# Build object
+Data = list("Options_vec"=c(3), "c_si"=c_si )
+Obj = MakeADFun( data=Data, parameters=Params, random="epsilon_s", DLL="autoregressive_V1" )
+# Optimize
+Opt3 = TMBhelper::Optimize( obj=Obj, newtonsteps=1 )
+par3 = Opt3$par
+h3 = Obj$env$spHess(random=TRUE)
+
+# Compare timing
+Opt0$run_time
+Opt1$run_time
+Opt2$run_time
+Opt3$run_time
+
+# Compare separability
+library(INLA)
+image(h0, main="Version 0"); dev.new()
+image(h1, main="Version 1"); dev.new()
+image(h2, main="Version 2"); dev.new()
+image(h3, main="Version 3")
+