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data generator.R
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data generator.R
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# --- READ ME
# title: R code for generating data that have the structure described in:
# "Longitudinal dynamic functional regression" (LDFR)
# by Md Nazmul Islam, Ana-Maria Staicu, and Eric van Heugten
# created by Md Nazmul Islam
# date: 2017/11
# ---
###########################################################################################################
### ------------------------- defining inputs to generate datasets (Gaussian) ------------------------- ###
###########################################################################################################
# tmin : minimum timepoint
# tmax : maximum timepoint
# rng : range of timepoints
# TT : vector of 41 equidistant timepoints
# I : Number of subjects
# J : length of the vector "TT"
# g : grid points over which functional predictor is observed at each time point
# ss : vector of equidistant points
# SS : length of the vector "ss"
# minJi : minimum number of repeated observations per subject
# maxJi : maximum number of repeated observations per subject
# mm : number of repeated observations per subject in the test set
# delta : the intensity at which functional predictor evolves over time
# A : seed number
# simul : desired number of Monte carlo simulations
# nbf : number of knots
# zetamu : mean of the variables that are used in representing the true functional predictor
# zetasigma1 : variance of the first gaussian variable used in representing the true functional predictor
# zetasigma2 : variance of the second gaussian variable used in representing the true functional predictor
# zetasigma3 : variance of the third gaussian variable used in representing the true functional predictor
# zetasigma4 : variance of the fourth gaussian variable used in representing the true functional predictor
# mu.e.x1 : mean of the first noisy term of the observed functional predictor
# mu.e.x2 : mean of the second noisy term of the observed functional predictor
# mu.e.x3 : mean of the third noisy term of the observed functional predictor
# var.e.x1 : variance the first noisy term of the observed functional predictor
# var.e.x2 : variance the second noisy term of the observed functional predictor
# var.e.x3 : variance the third noisy term of the observed functional predictor; determined by signal-to-noise ratio (SNR)
# errYbi0mu : mean of the random intercept
# errYbi1mu : mean of the random slope
# errYbijmu : mean of the white noise term
# errYbi0sigma : variance of the random intercept
# errYbi1sigma : variance of the random slope
# errYbi12sigma : covariance between random intercept and slope
# errYbijsigma : variance of the white noise term
# method : selection of smoothing parameter; "GCV.Cp", "REML", "ML" or others
# Yerror : type of dependence structure for responses; "CS" for compound symmetric type
# or "REM" for both subject-specific random intercept and slope. We illustarte
# the method using former here.
# full_traj : logical argument; TRUE if full trajectory is desired; default is TRUE
# Cov_Error : "High SNR" or "Low SNR"
# pve : percentage of variance explained; defualt is 0.95
###################################################################################################
### ---------------------- defining functions to generate datasets (Gaussian) ----------------- ###
###################################################################################################
### -------------------------------------------------------------------------------- ###
### ------------------------- mean of functional predictor ------------------------- ###
### -------------------------------------------------------------------------------- ###
mufun <- function (ss, Tij)
{
mu <- 1 + 2 * ss + 3 * Tij + 4 * ss * Tij
mu
}
### -------------------------------------------------------------------------------------------------------------------- ###
### ------------------------- true basis functions used in functional predictor representation ------------------------- ###
### -------------------------------------------------------------------------------------------------------------------- ###
phi_fun <- function (ss)
{
phi1 <- sqrt(2) * cos (2 * pi * ss)
phi2 <- sqrt(2) * sin (2 * pi * ss)
return = list (phi1 = phi1, phi2 = phi2)
}
### ---------------------------------------------------------------------------------------------------- ###
### ------------------------- complete data (without sparsity) for one subject ------------------------- ###
### ---------------------------------------------------------------------------------------------------- ###
onesubj <- function (A, i, TT, ss, zetamu, zetasigma1, zetasigma2, zetasigma3,
zetasigma4, delta, 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, J, Yerror)
{
set.seed (A)
J <- length(TT)
SS <- length(ss)
Si.i11 <- rnorm (1, mean = zetamu, sd = sqrt(zetasigma1))
Si.i12 <- rnorm (1, mean = zetamu, sd = sqrt(zetasigma2))
Si.i21 <- rnorm(1, mean = zetamu, sd = sqrt(zetasigma3))
Si.i22 <- rnorm(1, mean = zetamu, sd = sqrt(zetasigma4))
zetai1 <- Si.i11 * cos(2 * pi * TT) + Si.i12 * sin(2 * pi * TT)
zetai2 <- Si.i21 * cos(4 * pi * TT) + Si.i22 * sin(4 * pi * TT)
MU.Xi <- do.call ( rbind, lapply(TT, function(cc) as.vector(mufun(ss, Tij = cc ))))
pf <- phi_fun(ss)
true.alpha <- 7 * sin(3 * pi * TT)
true.beta1 <- exp (- TT * delta)
true.beta2 <- delta * TT * sin ( delta * TT)
### --------------------------------------------------------------------------------------- ###
### ------------------------- defining true functional predictors ------------------------- ###
### --------------------------------------------------------------------------------------- ###
true.func.cov <- MU.Xi + do.call(rbind, lapply(zetai2, function (feb)
feb * pf$phi2)) + do.call(rbind, lapply(zetai1, function (jan) jan * pf$phi1))
### ----------------------------------------------------------------------------------------- ###
### ------------------------- defining error structure for response ------------------------- ###
### ----------------------------------------------------------------------------------------- ###
if (Yerror == "IS") {
meanYij3 <- errYbi0mu + errYbi1mu * TT + errYbijmu
Z <- cbind(1, TT)
D <- matrix( c(errYbi0sigma, errYbi12sigma, errYbi12sigma, errYbi1sigma ),2, 2)
varYij3 <- (Z %*% D) %*% t(Z) + diag(errYbijsigma, J)
Randi <- mvrnorm(1, meanYij3, varYij3)
}
else if (Yerror == "CS") {
meanYij3 <- rep(errYbi0mu + errYbijmu, J)
varYij3 <- diag(errYbijsigma, J) + matrix(errYbi0sigma, nrow = J, ncol = J)
Randi <- mvrnorm(1, meanYij3, varYij3)
}
rng <- max(TT) - min(TT)
### ---------------------------------------------------------------------------------------------------- ###
### ------------------------------------------- Generating responses ----------------------------------- ###
### ---------------------------------------------------------------------------------------------------- ###
Yi.r <- apply(true.func.cov, 1, function(v){mean(v * pf$phi2 ) * rng }) * (true.beta2) +
apply(true.func.cov, 1, function(v){mean(v * pf$phi1 ) * rng }) * (true.beta1) + Randi + true.alpha
### ---------------------------------------------------------------------------------------------------- ###
### ------------------------- defining error structure for observed predictors ------------------------- ###
### ---------------------------------------------------------------------------------------------------- ###
if ( Cov_Error == "High SNR" ) {
SNR <- 2.5
var.e.x3 <- (0.5 * (zetasigma1 + zetasigma2 + zetasigma3 + zetasigma4) - SNR * (var.e.x1 + var.e.x2)) / SNR
e1 <- rnorm(J, mean = mu.e.x1, sd = sqrt(var.e.x1))
e2 <- rnorm(J, mean = mu.e.x2, sd = sqrt(var.e.x2))
e3 <- matrix(rnorm(J * SS , mean = mu.e.x3, sd = sqrt(var.e.x3)), nrow = J, byrow = T)
error1 <- do.call(rbind, lapply(e1, function (d) d * pf$phi1)) + do.call(rbind, lapply(e2, function (d) d * pf$phi2))
errorX <- error1 + e3
obs.func.cov <- true.func.cov + errorX
zetaiw1 <- zetai1 + apply(error1, 1, function(v){mean(v * pf$phi1) * rng })
zetaiw2 <- zetai2 + apply(error1, 1, function(v){mean(v * pf$phi2) * rng })
}
else if ( Cov_Error == "Low SNR" ) {
SNR <- 0.5
var.e.x3 <- (0.5 * (zetasigma1 + zetasigma2 + zetasigma3 + zetasigma4) - SNR * (var.e.x1 + var.e.x2)) / SNR
e1 <- rnorm(J, mean = mu.e.x1, sd = sqrt(var.e.x1))
e2 <- rnorm(J, mean = mu.e.x2, sd = sqrt(var.e.x2))
e3 <- matrix(rnorm(J * SS , mean = mu.e.x3, sd = sqrt(var.e.x3)), nrow = J, byrow = T)
error1 <- do.call(rbind, lapply(e1, function (d) d * pf$phi1)) + do.call(rbind, lapply(e2, function (d) d * pf$phi2))
errorX <- error1 + e3
obs.func.cov <- true.func.cov + errorX
zetaiw1 <- zetai1 + apply(error1, 1, function(v){mean(v * pf$phi1) * rng })
zetaiw2 <- zetai2 + apply(error1, 1, function(v){mean(v * pf$phi2) * rng })
}
index <- c((1 + (i - 1) * J) : (i * J))
data <- cbind(i = rep(i, J), j = seq_len( J ), v=seq_len( J ), Tij = TT,
Randi = Randi, zetai1 = zetai1, zetai2 = zetai2, true.alpha = true.alpha,
true.beta1 = true.beta1, true.beta2 = true.beta2, obs.func.cov = obs.func.cov,
index = index, Yij = Yi.r, zetaiw1 = zetaiw1, zetaiw2 = zetaiw2, MU.Xi = MU.Xi,
true.func.cov = true.func.cov)
}
### ------------------------------------------------------------------------------------- ###
### ------------------------- functions creating sparse dataset ------------------------- ###
### ------------------------------------------------------------------------------------- ###
sparsefun <- function ( A, i, minJi, maxJi, TT, data )
{
set.seed (A )
J <- length(TT)
mi <- sample ( minJi : maxJi, size = 1 )
ith <- sort ( sample ( 1 : J, size = mi), decreasing = FALSE )
ith.data <- data [ which ( data$i == i ), ]
k1 <- ith.data [ ith, ]
k1
}
### ------------------------------------------------------------------------- ###
### -------------------------- data generation ------------------------------ ###
### ------------------------------------------------------------------------- ###
data_func <- function (A , I, 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)
{
set.seed (A)
n <- I
### ------------------------------------------------------------------------------------- ###
### ---------------------------- Full data set for all subjects ------------------------- ###
### ------------------------------------------------------------------------------------- ###
dat.full <- data.frame(do.call(rbind, lapply(seq_len(I), function(i)
onesubj (A = i + A , i, TT, ss, zetamu, zetasigma1, zetasigma2, zetasigma3,
zetasigma4, delta, 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, J, Yerror))))
### ------------------------------------------------------------------------------------- ###
### ------------------ Sparse data set generated from full data set --------------------- ###
### ------------------------------------------------------------------------------------- ###
dat.sparse <- do.call(rbind, lapply(seq_len(I), function(i)
sparsefun(A = i + A , i, minJi, maxJi, TT, data = dat.full)))
### -------------------------------------------------------------------------------------- ###
### ---- creating training and test set used for investigation of prediction accuracy ---- ###
### -------------------------------------------------------------------------------------- ###
m1 <- sample(seq_len(I), n, replace = F );
m2 <- as.matrix(aggregate(dat.sparse[, 2], by = list(dat.sparse[, 1]), function(a)
{
sample(a, mm)
} )[m1, ])
row <- matrix(0, nrow = n, ncol = (dim(m2)[2]-1))
for (k in 1 : (dim(m2)[2]-1)) {
row[, k] <- as.vector (do.call(rbind, lapply(1 : dim(m2)[1],
function (a) which ( ( dat.sparse[,1] == m2[ a, 1]) *
( dat.sparse[,2] == m2[ a, (k + 1)]) == 1))))
}
row <- as.vector(row)
dat.test <- dat.sparse[row,]; dat.train <- dat.sparse [- row, ]
output = list ( dat.full = dat.full, dat.sparse = dat.sparse,
dat.train = dat.train, row = row,
dat.test = dat.test , m2 = m2)
}