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Utils.R
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library(caret)
library(splines)
library(data.table)
library(nlme)
library(randomForest)
library(mboost)
library(expm)
library(matrixStats)
library(doMC)
library(foreach)
library(gridExtra)
library(ggplot2)
library(reshape2)
library(curatedBreastData)
library(FSelector)
library(lme4)
library(plyr)
library(tidyverse)
library(latex2exp)
### Create a vector with all 0s except 1 in the k-th position
make_basis <- function(k, p = 10) replace(numeric(p), k, 1)
### Compute the trace of a matrix
tr <- function(M){
sum(diag(M))
}
### Sample datasets from curatedOvarianData
#
# Input:
# edat_orig: list of datasets
# ndat: number of datasets to sample
# nvar: number of predictors to sample
#
# Output:
# edat: list of data sets
#
init_data <- function(edat_orig, ndat, nvar){
edat <- edat_orig
edat <- edat[sample(1:length(edat), ndat)] # Randomize dataset order
idx <- sample(1:ncol(edat[[1]]), nvar)
for(i in 1:ndat){
edat[[i]] <- edat[[i]][,idx]
edat[[i]] <- as.data.frame(edat[[i]])
colnames(edat[[i]]) <- paste0("V", 1:nvar)
}
return(edat)
}
### Calculate the boosting fit with linear learners (Algorithm 1)
#
# Input:
# edat_train: list of training studies
# edat_test: list of test studies
# lambda_opt: tuning parameter lambda
# lambdak_opt: study-specific tuning parameters
# M_merge_linear: number of boosting iterations for the merged learner
# M_ens_linear: vector of study-specific boosting iterations for the ensemble learner
# learning_rate: learning rate eta
# sigma_eps: residual error variance
#
# Output:
# R: \tilde{R}
# R_k: \tilde{R}_k
#
boost_fit <- function(edat_train, edat_test, lambda_opt, lambdak_opt, M_merge_linear, M_ens_linear, learning_rate, sigma_eps){
############################################################################
# Merged #
############################################################################
# train and test are the (merged) training and test data
train <- rbindlist(edat_train)
if(class(edat_test) == "data.frame"){
test = edat_test
}else test <- rbindlist(edat_test)
# train_X is the design matrix for the merged data set (\tilde{X})
train_X <- data.frame(train)
train_X <- train_X[, names(train_X) != "y", drop = FALSE]
train_X <- as.matrix(train_X)
# test_X is the design matrix for the test data set (\tilde{X}_0)
test_X <- data.frame(test)
test_X <- test_X[, names(test_X) != "y", drop = FALSE]
test_X <- as.matrix(test_X)
P <- ncol(train_X)
N <- nrow(train_X)
ndat <- length(edat_train)
# Calculate R (\tilde{R})
R <- vector("list", length = M_merge_linear)
for(m in 1:M_merge_linear){
R[[m]] <- ((diag(N) - learning_rate * train_X %*% solve(t(train_X) %*% train_X + lambda_opt * diag(P)) %*% t(train_X)) %^% (m - 1))
}
B <- solve(t(train_X) %*% train_X + lambda_opt * diag(P)) %*% t(train_X)
R <- learning_rate * as.matrix(B) %*% Reduce("+", R[1:length(R)])
############################################################################
# Ensemble #
############################################################################
n_k <- vector()
Z_k <- train_X_k <- R_k <- vector("list", length = length(edat_train))
# Calculate R_k (\tilde{R}_k)
for(k in 1:length(edat_train)){
# train_X_k is the design matrix for the kth dataset
train_X_k[[k]] <- data.frame(edat_train[[k]])
train_X_k[[k]] <- train_X_k[[k]][, names(train_X_k[[k]]) != "y", drop = FALSE]
train_X_k[[k]] <- as.matrix(train_X_k[[k]])
p <- ncol(train_X_k[[k]])
n_k[k] <- nrow(train_X_k[[k]])
R_k[[k]] <- vector("list", length = M_ens_linear[k])
# Calculate R
for(m in 1:M_ens_linear[k]){
R_k[[k]][[m]] <- ((diag(n_k[k]) - learning_rate * train_X_k[[k]] %*% solve(t(train_X_k[[k]]) %*% train_X_k[[k]] + lambdak_opt[k] * diag(p)) %*% t(train_X_k[[k]])) %^% (m - 1))
}
B_k <- solve(t(train_X_k[[k]]) %*% train_X_k[[k]] + lambdak_opt[k] * diag(p)) %*% t(train_X_k[[k]])
R_k[[k]] <- learning_rate * B_k %*% Reduce("+", R_k[[k]][1:length(R_k[[k]])])
}
return(list(R = R, R_k = R_k))
}
### Function to calculate the component-wise boosting fit with OLS (Algorithm 2)
#
# Input:
# mod_mboost: component-wise boosting model object
# edat_sim: data that the model was trained on
# M_m: number of boosting iterations
#
#
# Output:
# R: \tilde{R}^{\text{CW}}
#
boost_cw_ols_fit <- function(mod_mboost, edat_sim, M_m) {
# Obtain unique indices of selected variables
sel <- sort(unique(selected(mod_mboost)))
# Obtain response and length of data set
y <- mod_mboost$response
n <- length(y)
############################################
# Calculate the test vector v^T #
############################################
# train is the (merged) training data
if (class(edat_sim) == "list") {
train <- rbindlist(edat_sim)
} else if (class(edat_sim) == "data.frame") {
train <- edat_sim
}
# train_X is the design matrix for the merged data set
train_X <- data.frame(train)
train_X <- train_X[, names(train_X) != "y", drop = FALSE]
train_X <- as.matrix(train_X)
# Initialize objects for calculating v^T
selected_variables <- colnames(train_X)[mod_mboost$xselect()]
p <- ncol(train_X)
n <- nrow(train_X)
B <- R_t <- u <- vector("list", length = M_m)
H <- X <- vector("list", length = M_m + 1)
t_ind <- vector()
# Calculate H0
H[[1]] <- matrix(0L, nrow = n, ncol = n)
u[[1]] <- diag(n)
# Calculate Xt, Ht, and Rt
for (t in 1:M_m) {
selected_variables_t <- selected_variables[t]
t_ind[t] <- which(colnames(train) %in% selected_variables_t)
X[[t]] <- train_X[, selected_variables_t]
X[[t]] <- as.matrix(X[[t]])
B[[t]] <- solve((t(X[[t]]) %*% X[[t]])) %*% t(X[[t]])
H[[t + 1]] <- X[[t]] %*% solve(t(X[[t]]) %*% X[[t]]) %*% t(X[[t]])
I_H <- lapply(1:t, function(x)
diag(n) - learning_rate * H[[x]])
R_t[[t]] <-
learning_rate * as.matrix(make_basis(t_ind[t], p)) %*% B[[t]] %*% Reduce("%*%", rev(I_H))
u[[t]] <- Reduce("%*%", rev(I_H))
}
# Calculate R
R <- Reduce("+", R_t)
return(R)
}
### Calculate corrected AIC for Algorithm 1
#
# Input:
# edat_sim: training data
# M_m: stopping iteration
# lambda: regularization parameter
#
# Output
# AICc: corrected AICc
#
calc_AICc_Alg1 <- function(edat_sim, M_m, lambda, learning_rate){
############################################################################
# Merged #
############################################################################
# train and test are the (merged) training and test data
train <- edat_sim
Y <- train$y
# train_X is the design matrix for the merged data set (\tilde{X})
train_X <- data.frame(train)
train_X <- train_X[, names(train_X) != "y", drop = FALSE]
train_X <- as.matrix(train_X)
P <- ncol(train_X)
N <- nrow(train_X)
# Calculate \mathcal{B}_{(m)}
mathcalBm <- vector("list", length = M_m)
df <- AICc <- vector()
for(m in 1:M_m){
mathcalBm[[m]] <- diag(N) - ((diag(N) - learning_rate * train_X %*% solve(t(train_X) %*% train_X + lambda * diag(P)) %*% t(train_X)) %^% (m + 1))
df[m] <- tr(mathcalBm[[m]])
AICc[m] <- (1 + df[m]/N)/(1 - df[m] + 2)/N + log(mean((Y - mathcalBm[[m]] %*% Y)^2))
}
return(AICc[M_m])
}
# Function to calculate the transition point for boosting with linear learners (Theorems 1 & 2)
#
# Input:
# edat_train: list of training studies
# edat_test: list of test studies
# f_train: list of mean functions for training data
# f_test: list of mean functions for test data
# Z_train: list of random predictor for training data
# sigma_eps: residual error variance
# wk: study-specific weights
# lambda_opt: lambda for merged data
# lambdak_opt: vector of lambdas for study-specific data
# learning_rate: learning rate eta
# M_merge_linear: stopping iteration for merged model
# M_ens_linear: vector of stopping iterations for study-specific models
# cols_re_list: list of column indices that correspond to predictors with random effects
#
# Output:
# tau_1: lower bound of interval below which merging outperforms ensembling
# tau_2: upper bound of interval above which ensembling outperforms merging
#
tau_range <- function(edat_train, edat_test, f_train, f_test, Z_train, sigma_eps, wk, lambda_opt, lambdak_opt, learning_rate, M_merge_linear, M_ens_linear, cols_re_list){
# K = ndat
ndat = length(edat_train)
# P = nvar
nvar = P = ncol(edat_train[[1]])
# tilde(X_0), design matrix for test data
X_0 = as.matrix(rbindlist(edat_test))
f_0 = Reduce(c, f_test)
f = Reduce(c, f_train)
# tilde(X)
X = as.matrix(rbindlist(edat_train))
# N
N <- nrow(X)
# tilde(R)
R <- vector("list", length = M_merge_linear)
for(m in 1:M_merge_linear){
R[[m]] <- ((diag(N) - learning_rate * X %*% solve(t(X) %*% X + lambda_opt * diag(P)) %*% t(X)) %^% (m - 1))
}
B <- solve(t(X) %*% X + lambda_opt * diag(P)) %*% t(X)
R <- learning_rate * as.matrix(B) %*% Reduce("+", R[1:length(R)])
# tilde(R)_k
n_k <- vector()
X_k <- Z_k <- R_k <- bias_k.list <- f_k <- vector("list", length = ndat)
for(k in 1:ndat){
# X_k is the design matrix for the kth data set
X_k[[k]] = as.matrix(edat_train[[k]])
# f_k is the mean function for the kth data set
f_k[[k]] = as.matrix(f_train[[k]])
# Z_k is the subset of covariates that corresponds to the random effects
Z_k[[k]] = as.matrix(Z_train[[k]])
p = ncol(X_k[[k]])
n_k[k] = nrow(X_k[[k]])
R_k[[k]] = vector("list", length = M_ens_linear[k])
# Calculate R
for(m in 1:M_ens_linear[k]){
R_k[[k]][[m]] = ((diag(n_k[k]) - learning_rate * X_k[[k]] %*% solve(t(X_k[[k]]) %*% X_k[[k]] + lambdak_opt[k] * diag(p)) %*% t(X_k[[k]])) %^% (m - 1))
}
B_k = solve(t(X_k[[k]]) %*% X_k[[k]] + lambdak_opt[k] * diag(p)) %*% t(X_k[[k]])
R_k[[k]] = learning_rate * B_k %*% Reduce("+", R_k[[k]][1:length(R_k[[k]])])
bias_k.list[[k]] = wk[k] * (X_0 %*% R_k[[k]] %*% f_k[[k]] - f_0)
}
# Bias terms
b_ens = Reduce("+", bias_k.list)
b_merge = X_0 %*% R %*% f - f_0
# Z'
Z_prime <- bdiag(Z_k)
# Calculate the transition interval
denomk_tr <- vector()
num1k <- denom2k <- denomk <- vector("list", ndat)
num2 = tr(t(R) %*% t(X_0) %*% X_0 %*% R)
denom1 = t(Z_prime) %*% t(R) %*% t(X_0) %*% X_0 %*% R %*% Z_prime
denom1_tr <- tr(denom1)
for (k in 1:ndat) {
# wk^2 * tr(R_k^T X_0^T X_0 R_k)
num1k[[k]] = wk[k]^2 * tr(t(R_k[[k]]) %*% t(X_0) %*% X_0 %*% R_k[[k]])
# wk^2 * Z_k^T R_k^T X_0^T X_0 R_k Z_k
denom2k[[k]] = wk[k]^2 * t(Z_k[[k]]) %*% t(R_k[[k]]) %*% t(X_0) %*% X_0 %*% R_k[[k]] %*% Z_k[[k]]
# wk^2 * tr(Z_k^T R_k^T X_0^T X_0 R_k Z_k)
denomk_tr[k] = wk[k]^2 * tr(t(Z_k[[k]]) %*% t(R_k[[k]]) %*% t(X_0) %*% X_0 %*% R_k[[k]] %*% Z_k[[k]])
}
num1 = Reduce("+", num1k)
squared_bias_ens = t(b_ens) %*% b_ens
squared_bias_m = t(b_merge) %*% b_merge
if(length(cols_re_list) == 1){
# tr(wk^2 * Z_k^T R_k^T X_0^T X_0 R_k Z_k)
denom2_tr = sum(denomk_tr)
denom = denom1_tr - denom2_tr
return(c((sigma_eps^2 * (num1 - num2) + squared_bias_ens - squared_bias_m)/(nvar/ncol(Z_train[[1]]) * max(denom)),
(sigma_eps^2 * (num1 - num2) + squared_bias_ens - squared_bias_m)/(nvar/ncol(Z_train[[1]]) * min(denom))))
}else if(length(cols_re_list) > 1){
denom <- vector()
denom2 <- vector("list", length = length(cols_re_list))
for(i in 1:length(cols_re_list)){
# wk^2 (Z_k^T R_k^T X_0^T X_0 R_k Z_k)_{ii} = (wk^2 * Z_k^T R_k^T X_0^T X_0 R_k Z_k)_{ii}
denom2[[i]] = sapply(1:length(denom2k), function(x){diag(denom2k[[x]])[cols_re_list[[i]]]})
# {sum_{k=1}^K (Z'^T R^T X_0^T X_0 R_k Z')_{k x i, k x i} - wk^2 (Z_k^T R_k^T X_0^T X_0 R_k Z_k)_{ii}}/Jd
denomj <- vector()
for(j in 1:length(cols_re_list[[i]])){
if(ncol(as.matrix(denom2[[i]])) > 1){
denom2ij <- denom2[[i]][j, ]
}else denom2ij = denom2[[i]]
denomj[j] = sum(diag(denom1)[cols_re_list[[i]][j] + ncol(Z_train[[1]]) * (0:(ndat - 1))] - unlist(denom2ij))
}
denom[i] = sum(denomj)/length(cols_re_list[[i]])
}
return(c((sigma_eps^2 * (num1 - num2) + squared_bias_ens - squared_bias_m)/(nvar * max(denom)),
(sigma_eps^2 * (num1 - num2) + squared_bias_ens - squared_bias_m)/(nvar * min(denom))))
}
}
# Compute the conditional MSE for merging and ensembling in Proposition 2
#
# Input:
# mod_mboost: mboost model object
# edat_sim: training data
# M_m: number of boosting iterations
# ind: index j
# f_train: f(\tilde{X})
# cols_re: indices of random effects
# study_k: index of study k
#
# Output
# mse: conditional MSE
# bias_squared: conditional bias_squared
# var: conditional variance
#
proposition2 <- function(mod_mboost, edat_sim, ind, sigma_re, f_train, cols_re, study_k) {
# Obtain response and length of data set
y <- mod_mboost$response
n <- length(y)
M_m <- length(mod_mboost$xselect())
# Initialize mod_mboostects for Z_k, covYk, train_X_k, and n_k
if (class(edat_sim) == "list") {
Z_k <- covYk <- train_X_k <- G_k <- vector("list", length = length(edat_sim))
} else if (class(edat_sim) == "data.frame") {
Z_k <- covYk <- train_X_k <- G_k <- vector("list", length = 1)
}
n_k <- vector()
# Populate Z_k, covYk, train_X_k, and n_k
if (class(edat_sim) == "list") {
num_study = length(edat_sim)
} else if (class(edat_sim) == "data.frame") {
num_study = 1
}
for (k in 1:num_study) {
if (class(edat_sim) == "list") {
train_X_k[[k]] <- data.frame(edat_sim[[k]])
} else if (class(edat_sim) == "data.frame") {
train_X_k[[k]] <- data.frame(edat_sim)
}
train_X_k[[k]] <-
train_X_k[[k]][, names(train_X_k[[k]]) != "y", drop = FALSE]
# Z_k is the subset of covariates that corresponds to the random effects
Z_k[[k]] <- train_X_k[[k]][, sort(cols_re)]
Z_k[[k]] <- as.matrix(Z_k[[k]])
# G_k is the covariance matrix for the k-th study
if (all(sigma_re[cols_re] == 0)) {
G_k[[k]] <-
matrix(0L, nrow = length(cols_re), ncol = length(cols_re))
} else
G_k[[k]] <- diag(sigma_re[cols_re])
n_k[k] <- nrow(train_X_k[[k]])
covYk[[k]] <-
Z_k[[k]] %*% G_k[[k]] %*% t(Z_k[[k]]) + sigma_eps ^ 2 * diag(n_k[k])
}
# Calculate covY
covY <- bdiag(covYk)
############################################
# Calculate the test vector v^T #
############################################
# train is the (merged) training data
if (class(edat_sim) == "list") {
train <- rbindlist(edat_sim)
} else if (class(edat_sim) == "data.frame") {
train <- edat_sim
}
# train_X is the design matrix for the merged data set
train_X <- data.frame(train)
train_X <- train_X[, names(train_X) != "y", drop = FALSE]
train_X <- as.matrix(train_X)
expected_val <- var <- vector()
selCourse_all <- selected(mod_mboost)
for(iter in 1:M_m){
selCourse <- selCourse_all[1:iter]
selected_variables <- colnames(train_X)[selCourse]
sel <- sort(unique(selCourse))
# Initialize objects for calculating v^T
p <- ncol(train_X)
n <- nrow(train_X)
B <- R_t <- u <- vector("list", length = iter)
H <- X <- vector("list", length = iter + 1)
t_ind <- vector()
# Calculate H0
H[[1]] <- matrix(0L, nrow = n, ncol = n)
u[[1]] <- diag(n)
# Calculate Xt, Ht, and Rt
for (t in 1:iter) {
selected_variables_t <- selected_variables[t]
t_ind[t] <- which(colnames(train) %in% selected_variables_t)
X[[t]] <- train_X[, selected_variables_t]
X[[t]] <- as.matrix(X[[t]])
B[[t]] <- solve((t(X[[t]]) %*% X[[t]])) %*% t(X[[t]])
H[[t + 1]] <- X[[t]] %*% solve(t(X[[t]]) %*% X[[t]]) %*% t(X[[t]])
I_H <- lapply(1:t, function(x)
diag(n) - learning_rate * H[[x]])
R_t[[t]] <-
learning_rate * as.matrix(make_basis(t_ind[t], p)) %*% B[[t]] %*% Reduce("%*%", rev(I_H))
u[[t]] <- Reduce("%*%", rev(I_H))
}
# Calculate R
R <- Reduce("+", R_t)
# Calculate vT (note that the coefficient estimates beta_hat = vT %*% y)
vT <- lapply(seq_len(nrow(R)), function(i) R[i,])
signCourse <-
sapply(1:iter, function(m)
sapply(mod_mboost[m]$coef(), "[[", 1))
nams <- attr(signCourse[[length(signCourse)]], "names")
# Calculate signs
signCourseS <- do.call("rbind", lapply(signCourse, function(sc) {
lenSc <- length(sc)
if (lenSc < length(nams)) {
namSc <- names(sc)
namsN <- nams[!nams %in% namSc]
sc <- c(rep(0, length(namsN)), sc)
names(sc) <- c(namsN, namSc)
sc <- sc[nams]
}
unlist(sc)
}))
signCoursePM <-
apply(rbind(rep(0, ncol(signCourseS)), signCourseS), 2, diff)
signs <- if (length(signCoursePM) == 1){
sign(signCoursePM)
}else{
rowSums(sign(signCoursePM))
}
Gamma <- unlist(lapply(1:length(selCourse), function(i) {
k = selCourse[i]
lapply(c(1:p)[-k], function(j) {
x <- rbind((signs[i] * olsFun(train_X[, k]) + olsFun(train_X[, j])) %*% u[[i]],
(signs[i] * olsFun(train_X[, k]) - olsFun(train_X[, j])) %*% u[[i]])
rownames(x) <-
c(paste(i, k, j, "+", sep = "_"), paste(i, k, j, "-", sep = "_"))
return(x)
})
}), recursive = F)
# Gamma has 2 * (p - 1) * m_stop rows and N columns
# Each row contains i_k_j_sign, where i = iteration, k = index of selected variable, and j = index of another variable neq k
# and sign is the sign of the coefficient of the selected variable
Gamma <- do.call("rbind", Gamma)
#############################################
# Calculate conditional mean and variance #
#############################################
# The linear contraints are rho %*% b <= rhs
# rho - Gamma %*% Sigma %*% V(V^T %*% Sigma V)^{-1}
# b - beta hat coefficient estimates V^TY
# rhs - - Gamma %*% Z, where Z = (I - Sigma %*% V(V^T %*% Sigma V)^{-1}V^T) %*% Y
v_mat <- data.frame(do.call(rbind, vT))
v_mat <- v_mat[apply(v_mat, 1, function(x) !all(x == 0)), ]
upper <- lower <- vector()
# loop over coefficients
for (coef in 1:nrow(v_mat)) {
C <- covY %*% t(v_mat[coef,]) %*% solve(as.matrix(v_mat[coef,]) %*% covY %*% t(as.matrix(v_mat[coef,])))
Z_star <- (diag(n) - as.matrix(C) %*% as.matrix(v_mat[coef, ])) %*% as.matrix(y)
Gamma_C <- as.numeric(Gamma %*% C)
neg_Gamma_Z_star <- -Gamma %*% Z_star
plus_index <- which(Gamma_C > 0)
neg_index <- which(Gamma_C < 0)
if (length(plus_index) == 0) {
lower[coef] = -Inf
} else{
lower[coef] <- max(neg_Gamma_Z_star[plus_index] / Gamma_C[plus_index])
}
if (length(neg_index) == 0) {
upper[coef] = Inf
} else{
upper[coef] <- min(neg_Gamma_Z_star[neg_index] / Gamma_C[neg_index])
}
}
if(n == length(f_train[[1]])){
mu <- f_train[[study_k]]
}else if(n == length(Reduce("c", f_train))){
mu <- Reduce("c", f_train)
}
v_jT <- vT[[ind]]
mu_bar_j <- as.numeric(v_jT %*% mu)
vartheta_j <- as.numeric(t(v_jT) %*% covY %*% v_jT)
sqrt_vartheta_j <- sqrt(vartheta_j)
lower <- "[<-"(numeric(p), sel, lower)
upper <- "[<-"(numeric(p), sel, upper)
alpha_j <- as.numeric((lower[ind] - mu_bar_j)/sqrt_vartheta_j)
xi_j <- as.numeric((upper[ind] - mu_bar_j)/sqrt_vartheta_j)
expected_val[iter] <- mu_bar_j - sqrt_vartheta_j * (dnorm(xi_j)- dnorm(alpha_j))/(pnorm(xi_j) - pnorm(alpha_j))
var[iter] <- vartheta_j * (1 - (xi_j * dnorm(xi_j) - alpha_j * dnorm(alpha_j))/(pnorm(xi_j) - pnorm(alpha_j)) - ((dnorm(xi_j)- dnorm(alpha_j))/(pnorm(xi_j) - pnorm(alpha_j)))^2)
}
return(list(expected_val = expected_val, var = var))
}
sim_prop2 <- function(edat_train, edat_test, f_train, f_test, Z_train, Z_test, sigma_re, sigma_eps, wk, learning_rate, M_upp, ind, cols_re, true_coefs){
nvar = ncol(edat_train[[1]])
# Generate the random effects and outcomes for the training data
edat_sim = edat_train
for (k in 1:length(edat_train)) {
dataset = edat_train[[k]]
f_k = as.matrix(f_train[[k]])
Z_k = as.matrix(Z_train[[k]])
gamma = rnorm(ncol(Z_k), 0, sigma_re)
eps = rnorm(nrow(dataset), 0, sigma_eps)
dataset$y = f_k + Z_k %*% gamma + eps
dataset$y = scale(dataset$y, center = TRUE, scale = FALSE)
edat_sim[[k]] = dataset
}
# Generate random effects and outcomes for test data
edat_sim_test = edat_test
for (k in 1:length(edat_test)) {
dataset = edat_test[[k]]
f_0 = as.matrix(f_test[[k]])
Z_0 = as.matrix(Z_test[[k]])
gamma2 = rnorm(ncol(Z_k), 0, sigma_re)
eps2 = rnorm(nrow(dataset), 0, sigma_eps)
dataset$y = f_0 + Z_0 %*% gamma2 + eps2
edat_sim_test[[k]] = dataset
}
train = as.data.frame(rbindlist(edat_sim))
if(class(edat_sim_test) == "data.frame"){
edat_sim_test <- list(edat_sim_test)
}
if(class(edat_sim_test) == "list"){
test <- as.data.frame(rbindlist(edat_sim_test))
}else test <- edat_sim_test
test_X <- data.frame(test)
test_X <- test_X[, names(test_X) != "y", drop = FALSE]
test_X <- as.matrix(test_X)
all_data <- c(edat_sim, edat_sim_test)
ndat_total <- length(all_data)
######################################################
# Algorithm 2 (OLS) #
######################################################
fmla <- as.formula(paste0('y ~ ', paste0('bols(', setdiff(names(train),
'y'), ', intercept = FALSE)', collapse= " + ")))
M_merge <- mstop(aic <- AIC(glmboost(y ~., data = train, offset = 0, control = boost_control(mstop = M_upp, nu = learning_rate)), "corrected"))
prop2_merge <- proposition2(mod_mboost = mboost(fmla, data = train, offset = 0, control= boost_control(mstop = M_upp, nu = learning_rate)), edat_sim = train, ind = ind, sigma_re = sigma_re, f_train = f_train, cols_re = cols_re)
#############################################
# #
# Ens #
# #
#############################################
#################################################
# Algorithm 2 #
#################################################
obj_ens <- coefficients_ens <- prop2_ens <- vector("list", length(edat_sim))
M_ens <- vector()
for(k in 1:length(edat_sim)){
# Boosting with OLS (Algorithm 2)
fmla <- as.formula(paste0('y ~ ', paste0('bols(', setdiff(names(edat_sim[[k]]),
'y'), ', intercept = FALSE)', collapse= " + ")))
edat_sim[[k]] <- do.call(data.frame, edat_sim[[k]])
M_ens[k] <- mstop(aic <- AIC(glmboost(y ~., data = edat_sim[[k]], offset = 0, control = boost_control(mstop = M_upp, nu = learning_rate)), "corrected"))
obj_ens[[k]] <- mboost(fmla, data = edat_sim[[k]], offset = 0, control= boost_control(mstop = M_upp, nu = learning_rate))
prop2_ens[[k]] <- proposition2(mod_mboost = obj_ens[[k]], edat_sim = edat_sim[[k]], ind = ind, f_train = f_train, sigma_re = sigma_re, cols_re = cols_re, study_k = k)
}
# Calculate the MSE for merged and ensemble estimators
merge_bias_sq <- (prop2_merge$expected_val - true_coefs[ind])^2
merge_variance <- prop2_merge$var
merge_mse <- merge_bias_sq + merge_variance
prop2_ens_bias <- lapply(1:length(f_train), function(x){
prop2_ens[[x]]$expected_val
})
prop2_ens_var <- lapply(1:length(f_train), function(x){
prop2_ens[[x]]$var
})
ens_bias_sq <- (colWeightedMeans(do.call("rbind", prop2_ens_bias), wk) - true_coefs[ind])^2
ens_variance <- colWeightedMeans(do.call("rbind", prop2_ens_var), wk^2)
ens_mse <- ens_bias_sq + ens_variance
return(list(merge_mse = merge_mse, merge_bias_sq = merge_bias_sq, merge_variance = merge_variance,
ens_mse = ens_mse, ens_bias_sq = ens_bias_sq, ens_variance = ens_variance, M_merge = M_merge, M_ens = mean(M_ens)))
}
sim_prop2_multiple <- function(nreps, edat_train, edat_test, f_train, f_test, Z_train, Z_test, sigma_re, sigma_eps, wk, learning_rate, M_upp, ind, cols_re, true_coefs){
registerDoMC(cores = 48)
results = foreach (j = 1:nreps, .combine = rbind) %dopar% {
print(paste("Iteration =", j))
sim_prop2(edat_train = edat_train, edat_test = edat_test, f_train = f_train, f_test = f_test, Z_train = Z_train, Z_test = Z_test,
sigma_re = sigma_re, sigma_eps = sigma_eps, wk = wk, learning_rate = learning_rate, M_upp = M_upp, ind = ind, cols_re = cols_re, true_coefs = true_coefs)
}
}
# Calculate performance ratio asymptote
#
# Input:
# edat_train: list of training studies
# edat_test: list of test studies
# f_train: list of mean functions for training data
# f_test: list of mean functions for test data
# Z_train: list of random predictor for training data
# sigma_eps: residual error variance
# wk: study-specific weights
# lambda_opt: lambda for merged data
# lambdak_opt: vector of lambdas for study-specific data
# learning_rate: learning rate eta
# M_merge_linear: stopping iteration for merged model
# M_ens_linear: vector of stopping iterations for study-specific models
# cols_re_list: list of column indices that correspond to predictors with random effects
#
# Output:
# asymptote: asymptote from corollary 1
#
cor1 <- function(edat_train, edat_test, f_train, f_test, Z_train, sigma_eps, wk, lambda_opt, lambdak_opt, learning_rate, M_merge_linear, M_ens_linear, cols_re_list){
# K = ndat
ndat = length(edat_train)
# P = nvar
nvar = P = ncol(edat_train[[1]])
# tilde(X_0), design matrix for test data
X_0 = as.matrix(rbindlist(edat_test))
f_0 = Reduce(c, f_test)
f = Reduce(c, f_train)
# tilde(X)
X = as.matrix(rbindlist(edat_train))
# N
N <- nrow(X)
# tilde(R)
R <- vector("list", length = M_merge_linear)
for(m in 1:M_merge_linear){
R[[m]] <- ((diag(N) - learning_rate * X %*% solve(t(X) %*% X + lambda_opt * diag(P)) %*% t(X)) %^% (m - 1))
}
B <- solve(t(X) %*% X + lambda_opt * diag(P)) %*% t(X)
R <- learning_rate * as.matrix(B) %*% Reduce("+", R[1:length(R)])
# tilde(R)_k
n_k <- vector()
X_k <- Z_k <- R_k <- bias_k.list <- f_k <- vector("list", length = ndat)
for(k in 1:ndat){
# X_k is the design matrix for the kth data set
X_k[[k]] = as.matrix(edat_train[[k]])
# f_k is the mean function for the kth data set
f_k[[k]] = as.matrix(f_train[[k]])
# Z_k is the subset of covariates that corresponds to the random effects
Z_k[[k]] = as.matrix(Z_train[[k]])
p = ncol(X_k[[k]])
n_k[k] = nrow(X_k[[k]])
R_k[[k]] = vector("list", length = M_ens_linear[k])
# Calculate R
for(m in 1:M_ens_linear[k]){
R_k[[k]][[m]] = ((diag(n_k[k]) - learning_rate * X_k[[k]] %*% solve(t(X_k[[k]]) %*% X_k[[k]] + lambdak_opt[k] * diag(p)) %*% t(X_k[[k]])) %^% (m - 1))
}
B_k = solve(t(X_k[[k]]) %*% X_k[[k]] + lambdak_opt[k] * diag(p)) %*% t(X_k[[k]])
R_k[[k]] = learning_rate * B_k %*% Reduce("+", R_k[[k]][1:length(R_k[[k]])])
bias_k.list[[k]] = wk[k] * (X_0 %*% R_k[[k]] %*% f_k[[k]] - f_0)
}
# Bias terms
b_ens = Reduce("+", bias_k.list)
b_merge = X_0 %*% R %*% f - f_0
# Z'
Z_prime <- bdiag(Z_k)
# Calculate the transition interval
denomk_tr <- vector()
num1k <- denom2k <- denomk <- vector("list", ndat)
num2 = tr(t(R) %*% t(X_0) %*% X_0 %*% R)
denom1 = t(Z_prime) %*% t(R) %*% t(X_0) %*% X_0 %*% R %*% Z_prime
denom1_tr <- tr(denom1)
for (k in 1:ndat) {
# wk^2 * tr(R_k^T X_0^T X_0 R_k)
num1k[[k]] = wk[k]^2 * tr(t(R_k[[k]]) %*% t(X_0) %*% X_0 %*% R_k[[k]])
# wk^2 * Z_k^T R_k^T X_0^T X_0 R_k Z_k
denom2k[[k]] = wk[k]^2 * t(Z_k[[k]]) %*% t(R_k[[k]]) %*% t(X_0) %*% X_0 %*% R_k[[k]] %*% Z_k[[k]]
# wk^2 * tr(Z_k^T R_k^T X_0^T X_0 R_k Z_k)
denomk_tr[k] = wk[k]^2 * tr(t(Z_k[[k]]) %*% t(R_k[[k]]) %*% t(X_0) %*% X_0 %*% R_k[[k]] %*% Z_k[[k]])
}
num1 = Reduce("+", num1k)
squared_bias_ens = t(b_ens) %*% b_ens
squared_bias_m = t(b_merge) %*% b_merge
# tr(wk^2 * Z_k^T R_k^T X_0^T X_0 R_k Z_k)
denom2_tr = sum(denomk_tr)
denom = denom1_tr - denom2_tr
return(denom2_tr/denom1_tr)
}
# Simulation
#
# Input:
# edat_train: list of training data
# edat_test: list of test data
# f_train: list of mean function for fixed effects (training data)
# f_test: list of mean function for fixed effects (test data)
# Z_train: list of covariates with random effects (training data)
# Z_test: list of covariates with random effects (test data)
# sigma_re: variance for random effects
# wk: study-specific weights
# lambda_opt: optimal lambda for merged model
# lambdak_opt: optimal lambads for study-specific models
# learning_rate: learning rate
# M_merge_linear: stopping iteration for Alg 1 (Merged)
# M_merge_cw: stopping iteration for Alg 2 (Merged)
#
sim_each <- function(edat_train, edat_test, f_train, f_test, Z_train, Z_test, sigma_re, sigma_eps, wk, lambda_opt, lambdak_opt, learning_rate,
M_merge_linear, M_merge_cw, M_ens_linear, M_ens_cw, M_merge_cw_bsplines, M_ens_cw_bsplines, M_merge_tree, M_ens_tree){
nvar = ncol(edat_train[[1]])
# Generate the random effects and outcomes for the training data
edat_sim = edat_train
for (k in 1:length(edat_train)) {
dataset = edat_train[[k]]
f_k = as.matrix(f_train[[k]])
Z_k = as.matrix(Z_train[[k]])
gamma = rnorm(ncol(Z_k), 0, sigma_re)
eps = rnorm(nrow(dataset), 0, sigma_eps)
dataset$y = f_k + Z_k %*% gamma + eps
dataset$y = scale(dataset$y, center = TRUE, scale = FALSE)
edat_sim[[k]] = dataset
}
# Generate random effects and outcomes for test data
edat_sim_test = edat_test
for (k in 1:length(edat_test)) {
dataset = edat_test[[k]]
f_0 = as.matrix(f_test[[k]])
Z_0 = as.matrix(Z_test[[k]])
gamma2 = rnorm(ncol(Z_k), 0, sigma_re)
eps2 = rnorm(nrow(dataset), 0, sigma_eps)
dataset$y = f_0 + Z_0 %*% gamma2 + eps2
edat_sim_test[[k]] = dataset
}
train = as.data.frame(rbindlist(edat_sim))
if(class(edat_sim_test) == "data.frame"){
edat_sim_test <- list(edat_sim_test)
}
if(class(edat_sim_test) == "list"){
test <- as.data.frame(rbindlist(edat_sim_test))
}else test <- edat_sim_test
test_X <- data.frame(test)
test_X <- test_X[, names(test_X) != "y", drop = FALSE]
test_X <- as.matrix(test_X)
all_data <- c(edat_sim, edat_sim_test)
ndat_total <- length(all_data)
# Formulas for RE models
# ind.features = 1:nvar
# lm.formula <- as.formula(paste("y ~ ", paste(c(names(train)[ind.features], "0"), collapse = "+")))
# reStruct.formula = as.formula(paste(paste("~", paste(c(names(Z_train[[1]]), "0"), collapse = "+")), "|study"))
#############################################
# #
# Merging #
# #
#############################################
#################################################
# Algorithm 1 #
#################################################
out <- boost_fit(edat_train = edat_sim, edat_test = test, lambda_opt = lambda_opt, lambdak_opt = lambdak_opt,
M_merge_linear = M_merge_linear, M_ens_linear = M_ens_linear, learning_rate = learning_rate)
R <- out$R
R_k <- out$R_k
train_y <- as.matrix(train$y)
R <- as.matrix(R)
Rboost_merge <- mean((test$y - test_X %*% R %*% train_y)^2)
######################################################
# Algorithm 2 (OLS) #
######################################################
fmla <- as.formula(paste0('y ~ ', paste0('bols(', setdiff(names(train),
'y'), ', intercept = FALSE)', collapse= " + ")))
mod_mboost <- mboost(fmla, data = train, offset = 0, control= boost_control(mstop = M_merge_cw, nu = learning_rate))
Mboost_merge <- mean((test$y - predict(mod_mboost, newdata = as.data.frame(test_X)))^2)
###########################################################
# Algorithm 2 (Bsplines) #
###########################################################
fmla_bbs <- as.formula(paste0('y ~ ', paste0('bbs(', setdiff(names(train),
'y'), ', knots = 3, df = 3)', collapse= " + ")))
mod_gamboost <- gamboost(fmla_bbs, data = train, offset = 0, control= boost_control(mstop = M_merge_cw_bsplines, nu = learning_rate))
Gamboost_merge <- mean((test$y - predict(mod_gamboost, newdata = as.data.frame(test_X)))^2)
# #################################################
# # LME #
# #################################################
# # equal variances
# edat_sim_lme <- edat_sim
# for(i in 1:length(edat_sim_lme)){
# edat_sim_lme[[i]]$study <- i
# }
# train_lme <- do.call(rbind, edat_sim_lme[1:ndat])
# fit.lme2 = tryCatch(do.call(lme, list(lm.formula, data = train_lme,
# random = reStruct(reStruct.formula, pdClass = "pdIdent"))),
# error = function(e) NA)
# pred.lme2 = tryCatch(predict(fit.lme2, newdata = data.frame(test_X), level = 0), error = function(e) rep(NA, nrow(test)))
# LME_merge = mean((test$y - pred.lme2)^2)
#
###########################################################
# Boosting with trees #
###########################################################
Tboost_merge_mod <- grad_boost(data = train, learning_rate = learning_rate, M = M_merge_tree, grad.fun = grad.fun, loss.fun = loss.fun, max_depth = max_depth)
Tboost_merge_pred <- grad_boost_pred(mod = Tboost_merge_mod[[1]], initial = mean(train$y), newdata = data.frame(test_X), learning_rate = learning_rate)
Tboost_merge <- mean((test$y - Tboost_merge_pred)^2)
#############################################
# #
# Ens #
# #
#############################################
#################################################
# Algorithm 2 #
#################################################
obj_ens <- coefficients_ens <- Rboost_pred <- Tboost_pred <- Tboost_mod <- Gamboost_mod <- Gamboost_pred <- vector("list", length(edat_sim))
for(k in 1:length(edat_sim)){
# Boosting with OLS (Algorithm 2)
fmla <- as.formula(paste0('y ~ ', paste0('bols(', setdiff(names(edat_sim[[k]]),
'y'), ', intercept = FALSE)', collapse= " + ")))
edat_sim[[k]] <- do.call(data.frame, edat_sim[[k]])
obj_ens[[k]] <- mboost(fmla, data = edat_sim[[k]], offset = 0, control= boost_control(mstop = M_ens_cw[k], nu = learning_rate))
sel <- sort(unique(selected(obj_ens[[k]])))
coefficients_ens[[k]] <- as.numeric(unlist(coef(obj_ens[[k]])))
coefficients_ens[[k]] <- "[<-"(numeric(nvar), sel, coefficients_ens[[k]])
# Boosting with linear learners (Algorithm 1)
Rboost_pred[[k]] <- t(test_X %*% R_k[[k]] %*% edat_sim[[k]]$y)
# Boosting with Bsplines (Algorithm 2)
Gamboost_mod[[k]] <- gamboost(fmla_bbs, data = edat_sim[[k]], offset = 0, control= boost_control(mstop = M_ens_cw_bsplines[k], nu = learning_rate))
Gamboost_pred[[k]] <- predict(Gamboost_mod[[k]], newdata = as.data.frame(test_X))[, 1]
# Boosting with trees
Tboost_mod[[k]] <- grad_boost(data = edat_sim[[k]], learning_rate = learning_rate, M = M_ens_tree[k], grad.fun = grad.fun, loss.fun = loss.fun, max_depth = max_depth)
Tboost_pred[[k]] <- grad_boost_pred(mod = Tboost_mod[[k]][[1]], initial = mean(edat_sim[[k]]$y), newdata = data.frame(test_X), learning_rate = learning_rate)
}
#################################################
# Algorithm 1 #
#################################################
Rboost_ens <- mean((test$y - colWeightedMeans(do.call(rbind, Rboost_pred), w = wk))^2)
######################################################
# Algorithm 2 (OLS) #
######################################################
coefficients_ens <- colWeightedMeans(do.call(rbind, coefficients_ens), w = wk)
Mboost_ens <- mean((test$y - test_X %*% coefficients_ens)^2)
######################################################
# Algorithm 2 (OLS) #
######################################################
Gamboost_ens <- mean((test$y - colWeightedMeans(do.call("rbind", Gamboost_pred), w = wk))^2)
# ######################################################
# # Boosting with Trees #
# ######################################################
Tboost_ens <- mean((test$y - colWeightedMeans(do.call(rbind, Tboost_pred), w = wk))^2)
return(c(Mboost_merge = Mboost_merge, Mboost_ens = Mboost_ens,
Rboost_merge = Rboost_merge, Rboost_ens = Rboost_ens,
Tboost_merge = Tboost_merge, Tboost_ens = Tboost_ens,
Gamboost_merge = Gamboost_merge, Gamboost_ens = Gamboost_ens))
}
sim_multiple <- function(nreps, edat_train, edat_test, f_train, f_test, Z_train, Z_test, sigma_re, sigma_eps, wk, lambda_opt, lambdak_opt, learning_rate,
M_merge_linear, M_merge_cw, M_ens_linear, M_ens_cw, M_merge_cw_bsplines, M_ens_cw_bsplines, M_merge_tree, M_ens_tree){