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--- artigos:ernesto3:sim [2008/09/26 08:40]
ernesto
+++ artigos:ernesto3:sim [2008/10/13 13:00] (atual)
ernesto
@@ Linha 1: / Linha 1: @@
 ======Simulation study ======
-===== Test beta bias =====
+===== Algorithm =====
 <code R>
-gs <- expand.grid((0:10)/10, (0:10)/10)
+#################################################################################################
-niter <- 100
+#
-arr <- array(NA, dim=c(niter, 2, 2), dimnames=list(niter=1:niter, stat=c("beta","beta.var"), dep=c(1,0)))
+#	PAPER COMPANION: SIMULATION STUDY
+#	Authors:	Ernesto Jardim <ernesto@ipimar.pt>
+#			Paulo Ribeiro Jr. <paulojus@ufpr.br>
+#	Date: 20081013
+#	Objective: 	Test paper methodology against design-based estimators regarding
+# 			association between age compositions and the age aggregated abundance.
+#			Tree options are simulated: (i) no association, (ii) week association,
+#			(iii) strong association.
+#	Note:		The association is induced by considering correlated multivariate gaussian
+#			processes as the basis for building the age aggregated abundance and age
+#			compositions.
+#
+#################################################################################################
-# gau
+# R libraries
+library(geoR)
+library(lattice)
+library(MASS)
+library(RandomFields)
+library(compositions)
+#==============================================================================================
+# SIMULATING THE PROCESSES
+#==============================================================================================
+# grid
+gs <- expand.grid((0:40)/40, (0:40)/40)
+# size of processes
+n <- nrow(gs)
+# number of replicates
+nsim <- 250
+#----------------------------------------------------------------------------------------------
+# STEP 1 : gaussian processes to build abundance and compositions
+#----------------------------------------------------------------------------------------------
+# object
+arr0 <- array(NA, dim=c(n,7,nsim), dimnames=list(loc=1:n, age=c("all","1i","2i","1w","2w","1s","2s"), nsim=1:nsim))
+# variance-covariance matrix
+s2 <- 0.5
+Sig <- diag(c(1,1,1,1,1,1,1))
+Sig[1,4] <- 0.45; Sig[4,1] <- 0.45; Sig[1,6] <- 0.9; Sig[6,1] <- 0.9; Sig[6,4] <- 0.5; Sig[4,6] <- 0.5
+Sig <- Sig*s2
+# simulation
 set.seed(111)
-gd <- grf(grid=gs, cov.pars=c(1,0.2), mean=2, nsim=niter)
+for(i in 1:nsim){
+	arr0[,,i] <- mvrnorm(nrow(gs), c(1,0,0,0,0,0,0), Sig)
+}
+#----------------------------------------------------------------------------------------------
+# STEP 2: generate 250 replicates of a log-gaussian spatial process (Diggle and Ribeiro Jr, 2007)
+#         with  mu=1, phi=0.2, sigma2=0.5, tau2=0.5
+#----------------------------------------------------------------------------------------------
+# Generate a spatial Gaussian process Z
+phi <- 0.2
+sigmasq <- 0.5
+set.seed(222)
+Z <- grf(grid=gs, cov.pars=c(sigmasq, phi), nsim=nsim)
+# build a log gausian process Y
+Ysim <- Z
+Ysim$data <- exp(Z$data+arr0[,1,])
+# Y characteristics
+Ysim.lmean <- apply(log(Ysim$data), 2, mean)
+Ysim.lvar <- apply(log(Ysim$data), 2, var)
+Ysim.lnhat <- exp(Ysim.lmean+Ysim.lvar/2)
+#----------------------------------------------------------------------------------------------
+# STEP 3: build compositions
+#----------------------------------------------------------------------------------------------
+# objects
+arr00 <- array(1, dim=c(n,3,nsim), dimnames=list(loc=1:n, age=1:3, nsim=1:nsim))
+Psim <- array(NA, dim=c(n,3,nsim,3), dimnames=list(loc=1:n, age=1:3, nsim=1:nsim, mucor=c("indep", "dep045","dep090")))
+# option 1: no association between compositions and age aggregated abundance
+arr00[,1:2,] <- exp(arr0[,2:3,])
+Psim[,,,1] <- aperm(apply(arr00,c(1,3),function(x) x/sum(x)),c(2,1,3))
+# option 2: week association between compositions and age aggregated abundance
+arr00[,1:2,] <- exp(arr0[,4:5,])
+Psim[,,,2] <- aperm(apply(arr00,c(1,3),function(x) x/sum(x)),c(2,1,3))
+# option 3: strong association between compositions and age aggregated abundance
+arr00[,1:2,] <- exp(arr0[,6:7,])
+Psim[,,,3] <- aperm(apply(arr00,c(1,3),function(x) x/sum(x)),c(2,1,3))
+# P characteristics
+#----------------------------------------------------------------------------------------------
+# STEP 4: build abundance at age
+#----------------------------------------------------------------------------------------------
-for(i in 1:niter){
+# objects
-cat(i)
+Isim.ln <- array(NA, dim=c(n,3,nsim,3), dimnames=list(loc=1:n, age=1:3, nsim=1:nsim, mucor=c("indep", "dep045","dep090")))
-	ggdd <- gd
+# sim
-	ggdd$data <- gd$data[,i]
+for(i in 1:3){
-	lf <- likfit(ggdd, ini.cov.pars=c(1,0.2), messages=FALSE)
+	for(j in 1:3){
-	arr[i,,1] <- c(lf$beta, lf$beta.var)
+		Isim.ln[,i,,j] <- Psim[,i,,j]*Ysim$data
+	}
 }
+# I characteristics
+Isim.lnmean <- apply(log(Isim.ln), c(2,3,4), mean)
+Isim.lnvar <- apply(log(Isim.ln), c(2,3,4), var)
+Isim.lnhat <- exp(Isim.lnmean+Isim.lnvar/2)
-# gau independente
+#==============================================================================================
-set.seed(111)
+# ESTIMATION
-gd <- grf(grid=gs, cov.pars=c(0,0), mean=2, nsim=niter, nugget=1)
+#==============================================================================================
+# number of samples to be drawn from each of the 250 replicate
+ns <- 2
-for(i in 1:niter){
+#----------------------------------------------------------------------------------------------
-cat(i)
+# STEP 5: estimation with methodology proposed by the paper
-	ggdd <- gd
+#----------------------------------------------------------------------------------------------
-	ggdd$data <- gd$data[,i]
-	lf <- likfit(ggdd, ini.cov.pars=c(0.5,0.5), messages=F)
+# objects
-	arr[i,,2] <- c(lf$beta, lf$beta.var)
+Yres <- array(NA, dim=c(ns,4,nsim), dimnames=list(samp=1:ns, stat=c("Ybar", "Ybarvar", "Yhat", "Yhatvar"), nsim=1:nsim))
+Ihat <- array(NA, dim=c(ns,3,nsim,3), dimnames=list(samp=1:ns, age=1:3, nsim=1:nsim, mucor=c("indep", "dep045","dep090")))
+xd <- expand.grid(dimnames(Ihat)[-c(1,2)])
+# estimation
+set.seed(222)
+for(j in 1:ns){
+cat("\n",j ,":", date(),"-", sep="")
+	samp <- sample(1:n, 100)
+	for(i in 1:nrow(xd)){
+		x <- xd[i,]
+		cat(".", sep="")
+		# building the sample
+		Isamp <- Isim.ln[samp,,x[[1]],x[[2]]]
+		locsamp <- gs[samp,]
+		Ysamp <- apply(Isamp,1,sum)
+		# estimation of age aggregated abundance
+		lf <- likfit(as.geodata(cbind(locsamp, Ysamp)), lambda=0, ini.cov.pars=c(1,0.2), messages=FALSE)
+		Yhat <- exp(lf$beta+lf$tausq/2+lf$sigmasq/2)
+		Yhatvar <- exp(2*lf$beta+lf$tausq+lf$sigmasq)*(exp(lf$tausq+lf$sigmasq)-1)
+		Yres[j,c("Yhat", "Yhatvar"),x[[1]]] <- c(Yhat, Yhatvar)
+		# estimation of abundance-at-age
+		prop <- acomp(Isamp)
+		Ihat[j,,x[[1]],x[[2]]] <- Yhat*c(mean(prop))
+	}
 }
-pdf("betabias.pdf")
+#----------------------------------------------------------------------------------------------
-par(mfrow=c(2,2))
+# STEP 6: estimation with design-based estimators
-hist(arr[,1,1], main="beta for mu=2 s2=1 phi=0.2 t2=0")
+#----------------------------------------------------------------------------------------------
-hist(arr[,1,2], main="beta for mu=2 s2=0 phi=0 t2=1")
-hist(arr[,2,1], main="beta.var")
-hist(arr[,2,2], main="beta.var")
-dev.off()
-</code>
-<note>
+# objects
-O problema está na variância do beta que não é um estimador da variância do processo. Logo quando fazemos a backtrans há uma subestimação da média do processo.
+Ibar <- array(NA, dim=c(ns,3,nsim,3), dimnames=list(samp=1:ns, page=1:3, nsim=1:nsim, mucor=c("indep", "dep045","dep090")))
-</note>
+# estimation
+set.seed(222)
+for(j in 1:ns){
+cat("\n",j ,":", date(),"-", sep="")
+	samp <- sample(1:n, 100)
+	for(i in 1:nrow(xd)){
+		x <- xd[i,]
+		cat(".", sep="")
+		# building the sample
+		Isamp <- Isim.ln[samp,,x[[1]],x[[2]]]
+		Ysamp <- apply(Isamp,1,sum)
+		# store means
+		Yres[j,c("Ybar", "Ybarvar"),x[[1]]] <- c(mean(Ysamp), var(Ysamp))
+		# estimation of abundance-at-age
+		Ibar[j,,x[[1]],x[[2]]] <- apply(Isamp,2,mean)
+	}
+}
-===== Algorítmo =====
+#==============================================================================================
-<code R>
+# RESULTS
-## Modelo: variáveis com mu dependente e correlação componente espacial parcialmente comum
+#==============================================================================================
-##	Y1 = mu1 + S00 + S11 + e1 = mu1 + sig00 R + sig11 R11 + e1
-##	Y2 = mu2 + S00 + S22 + e2 = mu1 + sig00 R + sig22 R22 + e2
-##	Y3 = mu3 + S00 + S33 + e3 = mu1 + sig00 R + sig33 R33 + e3
-##	Constraint by:
-##		sig00 + sig## = 1; e# = 1
-##		mu ~ MVG(c(mu1, mu2, mu3), Sigma)
-##		mu1 = mu2 = mu3 = 1
-##		diag(Sigma) = e#
-</code>
-**i indexa localizações
+#----------------------------------------------------------------------------------------------
-j indexa idades
+# STEP 7: summary statistics
-**
+#----------------------------------------------------------------------------------------------
-  * Definir grid
+# objects
-  * Definir sig00 (usado para gerir associação espacial entre idades)
+sim.res <- array(NA, dim=c(nsim,3,3,2,3), dimnames=list(iter=1:nsim, age=1:3, mucor=c("indep", "dep045","dep090"), meth=c("geo","samp"), stats=c("bias", "mse", "ICcov")))
-  * Simular Ij gaussiana
+# computation
-     * set sig00 e phi00 = sig00/5
+for(i in 1:nsim){
-     * set sig## = 1-sig00 e phi## = sig##/5
+	lmusim <- Isim.lnhat[,i,]
-     * set mu## = 0.4
+	# geo
-     * sim S ~ MVN(0,Sig) Sig=f(sig, phi)
+	m <- Ihat[,,i,]
-     * sim mu ~ MVN({mu1,mu2,m3}, Sigmu) Sigmu = {indep, dep}
+	q025 <- apply(m,c(2,3), quantile, probs=0.025, na.rm=T)
-     * set Ij = muj + S00 + Sjj
+	q975 <- apply(m,c(2,3), quantile, probs=0.975, na.rm=T)
-  * Construir Y = prod(exp{Ij})
+	sim.res[i,,,"geo","ICcov"] <- q025<=lmusim & q975>=lmusim
-  * Construir Pj = exp{Ij}/sum_j(exp{Ij})
+	for(j in 1:ns){
-  * Construir Ij = Y*Pj para garantir que Y = sum_j(Ij)
+		m[j,,] <- m[j,,]-lmusim
-  * Aleatorizar vector de localizações (100)
+	}
-  * Construir amostras de Y, P e I
+	sim.res[i,,,"geo","bias"] <- apply(m, c(2,3), mean)
-  * Ajustar modelo geo
+	sim.res[i,,,"geo","mse"] <- apply(m, c(2,3), var)
-  * Estimar Y
-    * Yhat = exp{beta+beta.var/2}
-    * Ybar = sum_i(Yi)/n
-  * Estimar Ij
-    * Ihatj = Yhat * 1/n sum_i(Pij) sendo Pij = Iij/sum_j(Iij)
-    * Ibarj = sum_i(Iij)/n
-<code R>
+	# samp
+	m <- Ibar[,,i,]
+	q025 <- apply(m,c(2,3), quantile, probs=0.025, na.rm=T)
+	q975 <- apply(m,c(2,3), quantile, probs=0.975, na.rm=T)
+	sim.res[i,,,"samp","ICcov"] <- q025<=lmusim & q975>=lmusim
+	for(j in 1:ns){
+		m[j,,] <- m[j,,]-lmusim
+	}
+	sim.res[i,,,"samp","bias"] <- apply(m, c(2,3), mean)
+	sim.res[i,,,"samp","mse"] <- apply(m, c(2,3), var)
+}
+# table
+tab.res <-  apply(sim.res, c(2,3,4,5), mean)
+#################################################################################################
+# THE END (or the beginning ...)
+#################################################################################################
 </code>
+===== Results =====

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