splinePHLaplace.fit <-
function (x, y, id, initial.values, initial.EB = NULL, control) {
# response vectors
logT <- as.vector(y$logT)
d <- as.vector(y$d)
y <- as.vector(y$y)
# design matrices
X <- x$X
Xtime <- x$Xtime
Z <- x$Z
Ztime <- x$Ztime
W1 <- x$W
W2 <- x$W2
W2s <- x$W2s
dimnames(X) <- dimnames(Xtime) <- dimnames(Z) <- dimnames(Ztime) <- dimnames(W1) <- NULL
attr(X, "assign") <- attr(X, "contrasts") <- attr(Xtime, "assign") <- attr(Xtime, "contrasts") <- NULL
attr(Z, "assign") <- attr(Ztime, "assign") <- NULL
# sample size settings
ncx <- ncol(X)
ncz <- ncol(Z)
ncw1 <- if (is.null(W2)) 0 else ncol(W1)
nk <- ncol(W2)
n <- length(logT)
N <- length(y)
ni <- as.vector(tapply(id, id, length))
# Gauss-Kronrod rule
P <- as.vector(x$P)
id.GK <- rep(seq_along(logT), each = control$GKk)
GKk <- control$GKk
# crossproducts and others
XtX <- crossprod(X)
ZtZ <- lapply(split(Z, id), function (x) crossprod(matrix(x, ncol = ncz))); names(ZtZ) <- NULL
ZtZ <- matrix(unlist(ZtZ), n, ncz * ncz, TRUE)
outer.Ztime <- lapply(1:n, function (x) Ztime[x, ] %o% Ztime[x, ])
# initial values
betas <- as.vector(initial.values$betas)
sigma <- initial.values$sigma
gammas <- as.vector(initial.values$gammas)
gammas.bs <- as.vector(initial.values$gammas.bs)
alpha <- as.vector(initial.values$alpha)
D <- initial.values$D
diag.D <- !is.matrix(D)
if (!diag.D) dimnames(D) <- NULL else names(D) <- NULL
# initialize Laplace approximation components
b <- trc.y1 <- if (is.null(initial.EB) || !all(dim(initial.EB) == c(n, ncz))) matrix(0, n, ncz) else initial.EB
dimnames(b) <- NULL
trc.y2 <- trc.y3 <- matrix(0, n, ncz * ncz, TRUE)
trc.t1 <- trc.t2 <- numeric(n)
cons.logLik <- 0.5 * n * ncz * log(2 * pi)
# Fix environments for functions
#environment(update.logLik.Laplace) <- environment(fn) <- environment(gr) <- environment()
#environment(logsurvCH) <- environment(ScsurvCH) <- environment(SclongCH) <- environment()
#environment(LogLik.chLaplace) <- environment(Score.chLaplace) <- environment()
#old <- options(warn = (-1))
#on.exit(options(old))
# EM iterations
iter <- control$iter.EM
Y.mat <- matrix(0, iter, ncx + 1)
T.mat <- matrix(0, iter, ncw1 + nk + 1)
B.mat <- if (diag.D) matrix(0, iter, ncz) else matrix(0, iter, ncz * ncz)
lgLik <- numeric(iter)
conv <- FALSE
new.b <- update.logLik.Laplace(b, betas, sigma, gammas, gammas.bs, alpha, D)
for (it in 1:iter) {
# save parameter values in matrix
Y.mat[it, ] <- c(betas, sigma)
T.mat[it, ] <- c(gammas, gammas.bs, alpha)
B.mat[it, ] <- D
# linear predictors
b.hat <- attr(new.b, "b")
Zb <- rowSums(Z * b.hat[id, ], na.rm = TRUE)
Zsb <- rowSums(Zs * b.hat[id.GK, ], na.rm = TRUE)
eta.yx <- as.vector(X %*% betas)
eta.yxT <- as.vector(Xtime %*% betas)
Ys <- as.vector(Xs %*% betas) + Zsb
eta.s <- as.vector(alpha * Ys)
eta.ws <- as.vector(W2s %*% gammas.bs)
sc1 <- - crossprod(X, y - eta.yx - Zb) / sigma^2
sc2 <- - colSums(alpha * d * Xtime - exp(c(W1 %*% gammas)) * P * rowsum(exp(eta.ws + eta.s) * alpha * Xs, id.GK, reorder = FALSE))
c(sc1 + sc2)
# E-step -- compute EB estimates and traces
for (i in 1:n) {
# individual values
eta.twi <- eta.tw[i]
Ztime.i <- Ztime[i, ]
# EB estimates, i.e., posterior modes
bb <- attr(new.b, "b")[i, ]
hes.b <- attr(new.b, "hes.b")[i, ]
var.b <- attr(new.b, "vb")[i, ]
dim(var.b) <- dim(hes.b) <- c(ncz, ncz)
# traces
Ztime.b <- sum(Ztime.i * bb)
eta.ti <- eta.twi + alpha * (eta.yxT[i] + Ztime.b)
exp.eta.ti <- exp(eta.ti)
trc1 <- - (alpha^3 * exp.eta.ti) * outer.Ztime[[i]]
trc2 <- colSums(Ztime[i, ] * var.b)
K <- var.b %*% trc1
tr.var.b.trc1 <- - 0.5 * sum(diag(K))
trc.y1[i, ] <- tr.var.b.trc1 * trc2
trc.y2[i, ] <- - 0.5 * sum(- K * t(K)) * c(trc2 %o% trc2)
L <- alpha * c(trc2 %o% trc2)
M <- c(trc2 %o% colSums(Ztime.i * (K %*% var.b)))
trc.y3[i, ] <- tr.var.b.trc1 * (L + M)
ZtSZ <- c(crossprod(Ztime.i, solve(hes.b, Ztime.i)))
P <- alpha * exp.eta.ti * ZtSZ - 1 / alpha
trc.t1[i] <- tr.var.b.trc1 * P
Q <- exp.eta.ti * ZtSZ * (alpha * Ztime.b + 1) - (alpha * Ztime.b + 2) / alpha^2
trc.t2[i] <- tr.var.b.trc1 * Q
}
b <- attr(new.b, "b")
hes.b <- attr(new.b, "hes.b")
b.hat <- b + trc.y1
vb.hat <- attr(new.b, "vb") + trc.y2 + trc.y3
Zb <- rowSums(Z * b.hat[id, ])
btZtZb <- drop(crossprod(Zb))
outer.b.hat <- if (ncz == 1) apply(b.hat, 1, function (x) x %o% x) else t(apply(b.hat, 1, function (x) x %o% x))
tr.tZZvarb <- sum(ZtZ * vb.hat)
# compute log-likelihood and check convergence
lgLik[it] <- as.vector(new.b)
if (it > 2) {
if (lgLik[it] < lgLik[it - 1]) {
betas <- Y.mat[it - 1, 1:ncx]; sigma <- Y.mat[it - 1, ncx + 1]
gammas <- T.mat[it - 1, 1:ncww]; alpha <- T.mat[it - 1, ncww + 1]
D <- B.mat[it - 1, ]
if (!diag.D) dim(D) <- c(ncz, ncz)
break
} else {
thets1 <- c(Y.mat[it - 1, ], T.mat[it - 1, ], B.mat[it - 1, ])
thets2 <- c(Y.mat[it, ], T.mat[it, ], B.mat[it, ])
check1 <- max(abs(thets2 - thets1) / (abs(thets1) + control$tol1)) < control$tol2
check2 <- (lgLik[it] - lgLik[it - 1]) < control$tol3 * (abs(lgLik[it - 1]) + control$tol3)
if (any(check1, check2)) {
conv <- TRUE
if (control$verbose) cat("\n\nconverged!\n")
break
}
}
}
# print results if verbose
if (control$verbose) {
cat("\n\niter:", it, "\n")
cat("log-likelihood:", lgLik[it], "\n")
cat("betas:", round(betas, 4), "\n")
cat("sigma:", round(sigma, 4), "\n")
cat("gammas:", round(gammas, 4), "\n")
cat("alpha:", round(alpha, 4), "\n")
cat("D:", if (!diag.D) round(D[lower.tri(D, TRUE)], 4) else round(D, 4), "\n")
}
# M-step
mu <- y - eta.yx
sigman <- sqrt(drop(crossprod(mu, mu - 2 * Zb) + btZtZb + tr.tZZvarb) / N)
Dn <- matrix(colMeans(vb.hat + outer.b.hat), ncz, ncz)
Dn <- if (diag.D) diag(Dn) else 0.5 * (Dn + t(Dn))
Hbetas <- nearPD(fd.vec(betas, SclongCH))
scbetas <- SclongCH(betas)
betasn <- betas - c(solve(Hbetas, scbetas))
thetas <- c(gammas, alpha)
thetas[2:nk] <- log(diff(thetas[1:nk]))
sc.thets <- ScsurvCH(thetas)
# apply positive-definite modifications to the Hessian, if required
H <- nearPD(cd.vec(thetas, ScsurvCH))
step.thetas <- c(solve(H, sc.thets))
# check if the step is too big, and scale if required
sum.step <- sqrt(sum(step.thetas * step.thetas))
sum.thet <- sqrt(sum(thetas * thetas))
if (sum.step > (step.max <- control$step.max * max(sum.thet, ncww + 1)))
step.thetas <- step.thetas * step.max / sum.step
thetasn <- thetas - step.thetas
# use backtracking if the log-likelihood has not increased
new.b <- update.bCH(b, hes.b, betasn, sigman, thetasn, Dn, TRUE)
dg0 <- - c(crossprod(sc.thets, step.thetas))
g1 <- new.b
g2 <- 0
backtrack.step <- 0
backtrack.fail <- FALSE
while (!is.finite(g1) || g1 < lgLik[it] - 1e-04 * dg0) {
if (!g2) {
g2 <- g1
lambda2 <- 1
lambda1 <- 0.5 * dg0 / as.vector(g1 - lgLik[it] - dg0)
if (is.na(lambda1) || (lambda1 > 1 | lambda1 < 0.1))
lambda1 <- 0.1
g1 <- update.bCH(b, hes.b, betasn, sigman, thetas - lambda1 * step.thetas, Dn, TRUE)
} else {
L1 <- cbind(c(1 / lambda1^2, -lambda2 / lambda1^2), c(-1 / lambda2^2, lambda1 / lambda2^2))
L2 <- c(g1 - dg0 * lambda1 - lgLik[it], g2 - dg0 * lambda2 - lgLik[it])
k <- c(L1 %*% L2) / (lambda1 - lambda2)
g2 <- g1
lambda2 <- lambda1
lambda1 <- (- k[2] + sqrt(k[2]^2 - 3 * k[1] * dg0)) / (3 * k[1])
if (is.na(lambda1) || (lambda1 > 1 | lambda1 < 0.1))
lambda1 <- 0.1
g1 <- update.bCH(b, hes.b, betasn, sigman, thetas - lambda1 * step.thetas, Dn, TRUE)
}
new.b <- g1
thetasn <- thetas - lambda1 * step.thetas
backtrack.step <- backtrack.step + 1
if (backtrack.fail <- backtrack.step > control$backtrackSteps)
break
}
if (control$verbose && backtrack.step > 0)
cat("backtrack:", backtrack.step, "\tlambda:", lambda1, "\n")
if (backtrack.fail) {
if ((new.b <- update.bCH(b, hes.b, betas, sigma, thetasn, Dn, TRUE)) > lgLik[it] - 1e-04 * dg0) {
betasn <- betas
sigman <- sigma
} else if ((new.b <- update.bCH(b, hes.b, betasn, sigman, thetasn, D, TRUE)) > lgLik[it] - 1e-04 * dg0) {
Dn <- D
} else {
break
}
}
# update parameter values
betas <- betasn
sigma <- sigman
D <- Dn
gammas <- thetasn[1:ncww]
gammas[1:nk] <- cumsum(c(gammas[1], exp(gammas[2:nk])))
alpha <- thetasn[ncww + 1]
}
thetsT <- c(gammas, alpha)
thetsT[2:nk] <- log(diff(thetsT[1:nk]))
thetas <- c(betas, log(sigma), thetsT, if (diag.D) log(D) else chol.transf(D))
lgLik <- - LogLik.chLaplace(thetas, b = b)
# if not converged, start quasi-Newton iterations
if (!conv && !control$only.EM) {
if (is.null(control$parscale))
control$parscale <- rep(0.01, length(thetas))
if (control$verbose)
cat("\n\nquasi-Newton iterations start.\n\n")
out <- if (control$optimizer == "optim") {
optim(thetas, LogLik.chLaplace, Score.chLaplace, method = "BFGS",
control = list(maxit = control$iter.qN, parscale = control$parscale,
trace = 10 * control$verbose), b = b)
} else {
nlminb(thetas, LogLik.chLaplace, Score.chLaplace, scale = control$parscale,
control = list(iter.max = control$iter.qN, trace = 1 * control$verbose), b = b)
}
if ((conv <- out$convergence) == 0 || - out[[2]] > lgLik) {
lgLik <- - out[[2]]
thetas <- out$par
betas <- thetas[1:ncx]
sigma <- exp(thetas[ncx + 1])
gammas <- thetas[seq(ncx + 2, ncx + 1 + ncww)]
gammas[1:nk] <- cumsum(c(gammas[1], exp(gammas[2:nk])))
alpha <- thetas[ncx + ncww + 2]
D <- thetas[seq(ncx + ncww + 3, length(thetas))]
D <- if (diag.D) exp(D) else chol.transf(D)
it <- it + if (control$optimizer == "optim") out$counts[1] else out$iterations
# compute posterior moments for thetas after quasi-Newton
new.b <- update.bCH(b, hes.b, betas, sigma, c(gammas, alpha), D)
for (i in 1:n) {
# individual values
eta.twi <- eta.tw[i]
Ztime.i <- Ztime[i, ]
# EB estimates, i.e., posterior modes
bb <- attr(new.b, "b")[i, ]
hes.b <- attr(new.b, "hes.b")[i, ]
var.b <- attr(new.b, "vb")[i, ]
dim(var.b) <- dim(hes.b) <- c(ncz, ncz)
# traces
Ztime.b <- sum(Ztime.i * bb)
eta.ti <- eta.twi + alpha * (eta.yxT[i] + Ztime.b)
exp.eta.ti <- exp(eta.ti)
trc1 <- - (alpha^3 * exp.eta.ti) * outer.Ztime[[i]]
trc2 <- colSums(Ztime[i, ] * var.b)
K <- var.b %*% trc1
tr.var.b.trc1 <- - 0.5 * sum(diag(K))
trc.y1[i, ] <- tr.var.b.trc1 * trc2
trc.y2[i, ] <- - 0.5 * sum(- K * t(K)) * c(trc2 %o% trc2)
L <- alpha * c(trc2 %o% trc2)
M <- c(trc2 %o% colSums(Ztime.i * (K %*% var.b)))
trc.y3[i, ] <- tr.var.b.trc1 * (L + M)
ZtSZ <- c(crossprod(Ztime.i, solve(hes.b, Ztime.i)))
P <- alpha * exp.eta.ti * ZtSZ - 1 / alpha
trc.t1[i] <- tr.var.b.trc1 * P
Q <- exp.eta.ti * ZtSZ * (alpha * Ztime.b + 1) - (alpha * Ztime.b + 2) / alpha^2
trc.t2[i] <- tr.var.b.trc1 * Q
}
b <- attr(new.b, "b")
hes.b <- attr(new.b, "hes.b")
b.hat <- b + trc.y1
vb.hat <- attr(new.b, "vb") + trc.y2 + trc.y3
Zb <- rowSums(Z * b.hat[id, ])
}
}
# calculate Hessian matrix
Hessian <- if (control$numeriDeriv == "fd") {
fd.vec(thetas, Score.chLaplace, b = b, eps = control$eps.Hes)
} else {
cd.vec(thetas, Score.chLaplace, b = b, eps = control$eps.Hes)
}
names(betas) <- names(initial.values$betas)
if (!diag.D) dimnames(D) <- dimnames(initial.values$D) else names(D) <- names(initial.values$D)
names(gammas) <- c(paste("bs.", 1:nk, sep = ""), colnames(W1))
nams <- c(paste("Y.", c(names(betas), "sigma"), sep = ""), paste("T.", c(names(gammas), "alpha"), sep = ""),
paste("B.", if (!diag.D) paste("D", seq(1, ncz * (ncz + 1) / 2), sep = "") else names(D), sep = "")
)
dimnames(Hessian) <- list(nams, nams)
colnames(b.hat) <- colnames(x$Z)
list(coefficients = list(betas = betas, sigma = sigma, gammas = gammas, alpha = alpha, D = as.matrix(D)),
Hessian = Hessian, logLik = lgLik, EB = list(post.b = b.hat, post.vb = vb.hat, Zb = Zb,
Ztimeb = rowSums(Ztime * b.hat)), knots = kn, iters = it, convergence = conv, n = n, N = N, ni = ni, d = d,
id = id)
}
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