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illustration.Rmd
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---
title: Illustration of "Longitudinal dynamic functional regression" (LDFR)
author: Md Nazmul Islam, Ana-Maria Staicu, and Eric van Heugten
output: html_document
date: 2017/11
---
```{r}
source("LDFR.R")
source("data generator.R")
source("data fit.R")
x <- c("mgcv", "MASS", "ggplot2", "refund", "RColorBrewer", "reshape2", "stats", "astsa", "Matrix")
suppressMessages(unlist(lapply(x, require, character.only = TRUE)))
```
Defining inputs for a single case
```{r param}
tmin <- 0 # minimum timepoint
tmax <- 1 # maximum timepoint
rng <- tmax - tmin # range of timepoints
TT <- seq(tmin, tmax, length.out = 41) # 41 equidistant timepoints
I <- 150 # number of existing subjects
IW <- 250 # number of existing + new subjects; new subjects = IW - I
J<- length(TT) # length of TT
g <- seq(0, 100, 1) # grid points
ss <- g / 100 # vector of equidistant points
SS <- length(ss) # length of ss
minJi <- 21 # minimum number of repeated observations per subject
maxJi <- 25 # maximum number of repeated observations per subject
mm <- 5 # number of repeated observations per subject in test set
delta <- 5 # intensity at which predictor evolves over time
simul <- 1 # desired number of Monte carlo simulations
nbf <- 15 # number of knots
zetamu <- 0 # mean of the variables used in representing true covariate
zetasigma1 <- 3.5 # variance of the first gaussian variable
zetasigma2 <- 2 # variance of the second gaussian variable
zetasigma3 <- 3 # variance of the third gaussian variable
zetasigma4 <- 1.5 # variance of the fourth gaussian variable
mu.e.x1 <- 0 # mean of the first noisy term
mu.e.x2 <- 0 # mean of the second noisy term
mu.e.x3 <- 0 # mean of the third noisy term
var.e.x1 <- 0.3 # variance the first smooth noise term
var.e.x2 <- 0.7 # variance the second smooth noise term
var.e.x3 <- NULL # variance the third noise (white noise) term by SNR
errYbi0mu <- 0 # mean of random intercept
errYbijmu <- 0 # mean of random slope
errYbi1mu <- 0 # mean of white noise term
errYbi0sigma <- 1 # variance of random intercept
errYbi1sigma <- 0.5 # variance of random slope
errYbi12sigma <- 0.1 # covariance between random intercept and slope
errYbijsigma <- 0.5 # variance of white noise term
method <- "REML" # method to select smoothing parameter
Yerror <- "CS" # random-intercept used for defining covariance structure of response (compound symmetric)
full_traj <- TRUE # TRUE as full trajectory is desired
pred_interval <- TRUE # TRUE as prediction interval for response trajectory is needed
alpha <- 0.05 # nominal level of significance
Cov_Error <- "High SNR" # low noise variance and high signal
pve <- 0.95 # pre-specified percentage of variance explained to select truncation value
new_subj <- TRUE # TRUE as prediction for new subjects is desired
phi1 <- sqrt(2) * cos(2 * pi * ss)
phi2 <- sqrt(2) * sin(2 * pi * ss)
true.beta1 <- exp(- TT * delta)
true.beta2 <- delta * TT * sin (delta * TT)
phi <- cbind(phi1, phi2)
bet <- cbind(true.beta1, true.beta2)
gammaT <- phi %*% t(bet) # true functional coefficient
```
Apply function to generate a dataset and fit LDFR model.
```{r apply}
hh0 <- seq (1,9999999 , I + 13)
set.seed(1331)
seednum0 <- sample(hh0, simul, replace = F)
gf <- lapply(seednum0 , function(dd) try(Yestim(A = dd, I, IW, TT, ss, zetamu, zetasigma1,
zetasigma2, zetasigma3, zetasigma4, delta, minJi, maxJi, Cov_Error, mu.e.x1, mu.e.x2,
mu.e.x3, var.e.x1, var.e.x2, var.e.x3, errYbi0mu, errYbijmu, errYbi0sigma, errYbijsigma,
errYbi1mu, errYbi1sigma, errYbi12sigma, mm, J, Yerror, nbf, method, pve, full_traj,
pred_interval, alpha, new_subj)))
```
## Observed data
```{r}
par(mfrow = c(1, 1))
set.seed(1177)
n <- sample(seq_len(I), 1)
Y <- gf[[1]]$fit$Y
fpr <- gf[[1]]$fit$fpr
ID <- gf[[1]]$fit$ID_obs
lty <- rep(1, length(which(ID == n)))
plot(Y[which(ID == n)], type = "b", col = "navyblue", ylim = c(-15, 25),ylab = "Yield",
xlab = expression(n[i]), lwd = 2, axes = TRUE, cex.lab = 1, cex.axis = 1, cex.main = 1,
main = paste("Response for subject : ", ID[which(ID==n)][1]))
matplot(ss, t(fpr[which(ID == n),]), type ="l", lty = 1, col = "grey",main =
paste("Observed profiles for subject : ", ID[which(ID == n)][1], "", sep = ""),
cex.main = 1, cex.lab = 1, cex.axis = 1, lwd = 2, ylab = "", xlab = "Grid (s)", ylim = c(-6, 20))
matlines(ss, t(fpr[c(8,17,2),]), type ="l", lwd = 2, lty = 1, col = c("black", "green", "magenta"))
```
## Smooth functional predictors
```{r smooth}
zetaw.hat <- gf[[1]]$fit$zetaw_hat
K.hat <- gf[[1]]$fit$Fnp
phi.hat <- gf[[1]]$fit$phi
Mean <- gf[[1]]$fit$Mean
AS <- list(); BS <- list()
set.seed(1)
k <- sample(seq_len(I), 3)
for (sb in 1:length(k)){
for (ui in 1 : length(which(ID == k[sb]))) {
AS[[ui]] <- zetaw.hat[which(ID == k[sb])[ui], 1] * phi.hat[,1] +
zetaw.hat[which(ID == k[sb])[ui], 2] * phi.hat[,2] +
zetaw.hat[which(ID == k[sb])[ui], 3] * phi.hat[,3]
BS[[ui]] <- AS[[ui]] + Mean[which(ID == k[sb])[ui],]
}
dd2 <- t(as.matrix(do.call(cbind, lapply(BS, function(a) a))))
par(mfrow = c(1, 1))
matplot(ss, t(fpr[which(ID == k[sb]),]), type = "l",
main = paste("Observed profiles for subject : ", ID[which(ID == k[sb])][1], "", sep = ""),
cex.main = 1, cex.lab = 1, cex.axis = 1, lwd = 1.5, lty = 1, ylab = "", xlab = "Grid (s)",
col = "grey", ylim = c(-6, 20))
matlines(ss, t(fpr[which(ID == k[sb])[c(3, 9, 5)],]),
lty = c(2, 2, 2), col = c("black", "red", "purple"), lwd = 2)
matlines(ss, t(dd2[c(3, 9, 5),]), lty =c(1, 1, 1), col = c("black", "red","purple"), lwd = 2)
legend(-0, 18, inset = 1, legend = c("observed", "smooth"), lty = c(2, 1),
cex = 1, horiz = FALSE, bty = "n")
}
```
# Estimates of the covariates related components
## Empirical basis functions
```{r plotbasis}
freq <- gf[[1]]$fit$freq
par(mfrow = c(1, 1))
plot(ss, phi.hat[, 1], type ="l", col = 1, main = expression(hat(phi)[1](s)),
xlab = "Grid (s)", lwd = 2, ylim = c(-2,2), ylab = "")
plot(ss, phi.hat[, 2], type ="l", col = 2, main = expression(hat(phi)[2](s)),
xlab = "Grid (s)", lwd = 2, ylim = c(-2,2), ylab = "")
plot(ss, phi.hat[, 3], type ="l", col = 3, main = expression(hat(phi)[3](s)),
xlab = "Grid (s)", lwd = 2, ylim = c(-2,2), ylab = "")
```
## Estimated basis-coefficients
```{r plotxi}
xiF <- gf[[1]]$fit$xiF
fID <- gf[[1]]$fit$ID_full
par(mfrow = c(1,1))
for ( i in 1 : K.hat){
if(i == 1) Main = expression(hat(xi)[1](t))
if(i == 2) Main = expression(hat(xi)[2](t))
if(i == 3) Main = expression(hat(xi)[3](t))
plot(x = TT, xiF[which(fID == 1), i], type = "l", lwd = 2,
ylab = "", xlab = "Longitudinal time (t)", main = Main,
col = "grey", ylim = c(-6, 6))
for (ii in 2 : I){
lines(x = TT, xiF[which(fID == ii),i], type ="l", lwd = 2,
ylab = "", xlab = "", col = "grey", ylim = c(-6,6))
}
lines(x = TT, xiF[which(fID == 11), i], type ="l", lwd = 2, col = "blue")
lines(x = TT, xiF[which(fID == 35), i], type ="l", lwd = 2, col = "magenta")
lines(x = TT, xiF[which(fID == 46), i], type ="l", lwd = 2, col = "red")
legend(0.6, -3, inset = 1, legend = c("subject 11", "subject 35", "subject 46"),
fill = c("blue", "magenta", "red"), cex = 1, horiz = FALSE, bty = "n")
}
```
# Fit assessment
## In-sample and out-of-sample prediction error
```{r plotIN}
a <- c(gf[[1]]$IN.MPE.y, gf[[1]]$OUT.MPE.y)
barplot(a, names.arg=c("in-sample", "out-of-sample"), xlab = "", ylim = c(0, 2),
col = c("violetred3", "chocolate4"), main = "Root Mean Prediction Errors",
cex.names = 1.4, cex.axis = 1.5, space = 1.5)
print(a)
```
## Prediction of full trajectory for randomly selected 10 subjects
```{r plotF}
set.seed(7777)
idN <- sample(seq_len(I), 10);
V <- gf[[1]]$fit$V
Y <- gf[[1]]$fit$Y
pred <- gf[[1]]$fit$pred
f_pred <- gf[[1]]$fit$full_pred_obs
boundYF <- gf[[1]]$fit$boundYF
boundXF <- gf[[1]]$fit$boundXF
IDF <- unlist(lapply(seq_len(I), function(gh) rep(gh, J)))
TijF <- rep(TT, I)
par(mfrow = c(1, 1))
for(f in 1 : length(idN)) {
dayT <- seq_len(J)
nw <- V[which(ID == idN[f])]
dayT[-nw] <- 1000
plot(x = TT, y = f_pred[which(IDF == idN[f])], type = "b", ylab = "Yield", xlab =
"Longitudinal time (t)", lwd = 2, cex.lab = 1, pch = ifelse (dayT == 1000, 21, 1),
cex = ifelse(dayT == 1000, 1.4, 1), bg = "orange", ylim = c(-20, 20), main =
paste("Subject:", idN[f], ""))
polygon(c(TT, rev(TT)), c(boundYF[which(IDF == idN[f]), 1],
rev(boundYF[which(IDF == idN[f]), 2])), col = 'grey80', border = NA)
lines(x = TT, y = f_pred[which(IDF == idN[f])], type = "b", lwd = 2, cex.lab = 1,
pch = ifelse (dayT == 1000, 21, 1), cex = ifelse(dayT == 1000, 1.4, 1),
bg = "orange")
A1 <- rep(NA, J)
A1[V[which(ID == idN[f])]] <- Y[which(ID == idN[f])]
lines(x = TT, A1, type = "b", lwd = 2, col = "red")
legend(0, -8, inset = 1, legend = c("predicted", "observed", "recovered", "95% prediction band"),
fill = c("white", "red", "orange", "grey"), border = c("black", "red", "black", "grey"),
cex = 1, horiz = FALSE, bty = "n")
}
```
## Prediction coverage and length of prediction band
```{r}
cvs <- cbind(boundYF[,1], gf[[1]]$YijF, boundYF[,2])
cvsx <- cbind(boundXF[,1], gf[[1]]$Yij.XF, boundXF[,2])
ali <- alix <- list()
find_cov <- function(XX){
if(XX[2] <= XX[3] & XX[2] >= XX[1]) {
ind = 1
} else {
ind = 0
}
ind
}
for(i in 1 : I){
inx <- which(IDF == i)
ali[[i]] <- unlist(lapply(seq_len(length(inx)), function(jj) find_cov(cvs[inx,][jj,])))
}
IDXF <- unlist(lapply(seq_len(IW), function(gh) rep(gh, J)))
IDXF <- IDXF[which(IDXF > I)]
for(i in 1 : (IW - I)){
inx <- which(IDXF == unique(IDXF)[i])
alix[[i]] <- unlist(lapply(seq_len(length(inx)), function(jj) find_cov(cvsx[inx,][jj,])))
}
cov95 <- sum(unlist(ali)) / length(IDF)
L95 <- mean(boundYF[, 2] - boundYF[, 1])
cov95X <- sum(unlist(alix)) / length(IDXF)
L95X <- mean(boundXF[, 2] - boundXF[, 1])
b <- c(cov95, cov95X)
barplot(b, names.arg = c("Existing subject", "New subject"), xlab = "", ylim = c(0, 1),
col = c("black", "grey"), cex.names = 1.4, cex.axis = 1.5, space = 1.5,
main = paste0("Coverage probability at nominal level 0.05"))
abline(h = 0.95, col = "Red", lwd = 2)
print(b)
e <- c(L95, L95X)
barplot(e, names.arg = c("Existing subject", "New subject"), xlab = "", ylim = c(0, 7),
col = c("black", "grey"), main = "Average length of prediction band",
cex.names = 1.4, cex.axis = 1.5, space = 1.5)
print(e)
```
## Prediction error of full response trajectory
```{r plotRMPE}
b <- c(gf[[1]]$RMPE_trj, gf[[1]]$RMPE_trjX)
barplot(b, names.arg=c("Existing subject", "New subject"), xlab = "", ylim = c(0, 2),
col = c("violetred3", "chocolate4"), main = "RMPE_traj",
cex.names = 1.4, cex.axis = 1.5, space = 1.5)
print(b)
```
## Model selection criteria
```{r model}
gf[[1]]$fit$model_adeq
```
## Checking the validity of Gaussian assumption
```{r resd}
resd <- Y - pred
par(mfrow = c(1, 1))
hist(resd, prob = TRUE, main = "Histogram", xlab = "Residuals",
ylab = "Probability", ylim = c(0, 0.7), cex.lab = 2)
qqnorm(resd, main = "Normal Q-Q plot for residuals")
qqline(resd, lwd = 2)
plot(pred, resd, xlab = "Predicted", ylab = "Marginal residuals", pch = 1,
cex.lab = 1, cex.axis = 1, ylim = c(-7, 7), xlim = c(-15, 30))
abline(h = 0,lwd = 2)
lines(smooth.spline(pred, resd), col = "red", lwd = 2)
```
# Estimates of response related components
## Estiamtes of dynamic parameters $\widehat\beta_{k}(t)$
```{r}
beta <- gf[[1]]$fit$beta
bounds <- gf[[1]]$fit$bounds
for (bb in 1 : K.hat) {
if(bb == 1) Main = expression(hat(beta)[1](t))
if(bb == 2) Main = expression(hat(beta)[2](t))
if(bb == 3) Main = expression(hat(beta)[3](t))
lb <- min(beta[[bb]]) - 1; ub <- max(beta[[bb]]) + 1
plot(x = TT, y = beta[[bb]], type = "b", ylab = "", xlab = "Longitudinal time (t)",
col = "black", main = Main, lwd = 2, pch = 1, axes = TRUE, cex.lab = 1.3,
cex.axis = 1.3, cex.main = 1.4, ylim = c(lb, ub) )
polygon(c(TT, rev(TT)), c(bounds[[bb]][, 1], rev(bounds[[bb]][, 2])),
col = 'grey80', border = NA)
lines(x = TT, y = beta[[bb]], type = "b", col = "black", lwd = 2, pch = 1)
}
```
## Predicted random effect $\widehat b_{i}$
```{r random}
if(Yerror == "CS"){
rand_inter <- gf[[1]]$fit$SS_int
Estimates <- as.vector( c(rand_inter) )
Terms <- as.character(c(rep("intercept", I)))
mrg <- data.frame(as.matrix(as.data.frame(cbind(Estimates, Terms))))
mrg$Estimates <- as.numeric(mrg$Estimates)
mrg$Terms <- as.factor(mrg$Terms)
mrg$Estimates <- Estimates
tb0 <- expression(""~hat(b)[i0])
ggplot(mrg, aes(x = Terms, y = Estimates)) + theme_bw() + theme(text = element_text(size = 17)) +
geom_boxplot(outlier.shape=NA) + geom_jitter(position=position_jitter(width=.1, height = 0)) +
annotate("text", x = 1, y = 3.5, parse = T, label = as.character(tb0), size = 6)
}
if(Yerror == "IS"){
rand_inter <- gf[[1]]$fit$SS_int
rand_slp <- gf[[1]]$fit$SS_slp
Estimates <- as.vector( c(rand_inter, rand_slp) )
Terms <- as.character(c(rep("intercept", I), rep("slope", I)))
mrg <- data.frame(as.matrix(as.data.frame(cbind(Estimates, Terms))))
mrg$Estimates <- as.numeric(mrg$Estimates)
mrg$Terms <- as.factor(mrg$Terms)
mrg$Estimates <- Estimates
tb0 <- expression(""~hat(b)[i0])
tb1 <- expression(""~hat(b)[i1])
ggplot(mrg, aes(x = Terms, y = Estimates)) + theme_bw() + theme(text = element_text(size = 17)) +
geom_boxplot(outlier.shape=NA) + geom_jitter(position=position_jitter(width=.1, height = 0)) +
annotate("text", x = 1, y = 3.5, parse = T, label = as.character(tb0), size = 6) +
annotate("text", x = 2, y = 3.5, parse = T, label = as.character(tb1), size = 6)
}
```
## True functional coefficient $\gamma(s,t)$
```{r gammaT}
gamma <- gammaT
Functional_hours <- paste0("s_", 1 : SS)
Longitudinal_time <- paste0("t_", 1 : J)
sq <- Functional_hours[seq(1,length(Functional_hours), 10)]
sq2 <- Longitudinal_time[seq(1,length(Longitudinal_time), 5)]
Longitudinal_time <- paste0("t_", 1 : J)
rownames(gamma) <- Functional_hours[1 : dim(gamma)[1]]
colnames(gamma) <- Longitudinal_time[1 : dim(gamma)[2]]
longData <- melt(gamma)
names(longData) <- c("Functional_hours", "Longitudinal_time", "value")
longData$Longitudinal_time <- factor(longData$Longitudinal_time)
longData$Functional_hours <- factor(longData$Functional_hours)
myPalette <- colorRampPalette(brewer.pal(11, "Spectral"), space="rgb", bias = 1)
zp1 <- ggplot(longData, aes(y = Functional_hours, x = Longitudinal_time, fill = value))
zp1 <- zp1 + theme(text = element_text(size = 15), axis.text.x =
element_text(size = 15, angle=0), axis.text.y = element_text(size = 15, angle=0))
zp1 <- zp1 + geom_tile()
zp1 <- zp1 + scale_fill_gradientn(colours = myPalette(200), limits=c(-9, 7))
zp1 <- zp1 + scale_y_discrete(breaks = sq)
zp1 <- zp1 + scale_x_discrete(breaks = sq2)
print(zp1)
```
## Estimate of functional coefficient $\widehat\gamma(s,t)$
```{r gamma}
gamma <- gf[[1]]$fit$gamma
Functional_hours <- paste0("s_", 1 : SS)
Longitudinal_time <- paste0("t_", 1 : J)
sq <- Functional_hours[seq(1,length(Functional_hours), 10)]
sq2 <- Longitudinal_time[seq(1,length(Longitudinal_time), 5)]
Longitudinal_time <- paste0("t_", 1:J)
rownames(gamma) <- Functional_hours[1 : dim(gamma)[1]]
colnames(gamma) <- Longitudinal_time[1 : dim(gamma)[2]]
longData <- melt(gamma)
names(longData) <- c("Functional_hours", "Longitudinal_time", "value")
longData$Longitudinal_time <- factor(longData$Longitudinal_time)
longData$Functional_hours <- factor(longData$Functional_hours)
myPalette <- colorRampPalette(brewer.pal(11, "Spectral"), space="rgb", bias = 1)
zp1 <- ggplot(longData, aes(y = Functional_hours, x = Longitudinal_time, fill = value))
zp1 <- zp1 + theme(text = element_text(size = 15), axis.text.x =
element_text(size = 15, angle = 0), axis.text.y = element_text(size = 15, angle = 0))
zp1 <- zp1 + geom_tile()
zp1 <- zp1 + scale_fill_gradientn(colours = myPalette(200), limits=c(-9, 7))
zp1 <- zp1 + scale_y_discrete(breaks = sq)
zp1 <- zp1 + scale_x_discrete(breaks = sq2)
print(zp1)
```