Nothing
envMU <- function(M, U, u, initial = NULL) {
dimM <- dim(M)
dimU <- dim(U)
r <- dimM[1]
if (dimM[1] != dimM[2] & dimU[1] != dimU[2]) stop("M and U should be square matrices.")
if (dimM[1] != dimU[1]) stop("M and U should have the same dimension.")
if (qr(M)$rank < r) stop("M should be positive definite.")
if (u > r & u < 0) stop("u should be between 0 and r.")
if (u == 0) {
Gammahat <- NULL
Gamma0hat <- diag(r)
MU <- M + U
tmp.MU <- eigen(MU)
objfun <- sum(log(tmp.MU$values))
} else if (u == r) {
Gammahat <- diag(r)
Gamma0hat <- NULL
tmp.M <- eigen(M)
objfun <- sum(log(tmp.M$values))
} else if (u == 1) {
maxiter = 100
ftol = 1e-3
if (!is.null(initial)) {
MU <- M + U
tmp.MU <- eigen(MU)
invMU <- sweep(tmp.MU$vectors, MARGIN = 2, 1 / tmp.MU$values, '*') %*% t(tmp.MU$vectors)
init <- initial
eig1 <- eigen(t(init) %*% M %*% init)
eig2 <- eigen(t(init) %*% invMU %*% init)
obj1 <- sum(log(eig1$values)) + sum(log(eig2$values))
} else {
MU <- M + U
tmp.MU <- eigen(MU)
invMU <- sweep(tmp.MU$vectors, MARGIN = 2, 1 / tmp.MU$values, '*') %*% t(tmp.MU$vectors)
invMU2 <- sweep(tmp.MU$vectors, MARGIN = 2, 1 / sqrt(tmp.MU$values), '*') %*% t(tmp.MU$vectors)
midmatrix <- U
startv <- function(a) t(a) %*% midmatrix %*% a
tmp2.MU <- apply(tmp.MU$vectors, 2, startv)
tmp3.MU <- sort(tmp2.MU, decreasing = TRUE, index.return = TRUE)
init <- as.matrix(tmp.MU$vectors[, tmp3.MU$ix[1]])
# if (qr(MU)$rank == r) {
eig1 <- eigen(t(init) %*% M %*% init)
eig2 <- eigen(t(init) %*% invMU %*% init)
obj1 <- sum(log(eig1$values)) + sum(log(eig2$values))
midmatrix <- invMU2 %*% tcrossprod(U, invMU2)
tmp2.MU <- apply(tmp.MU$vectors, 2, startv)
tmp3.MU <- sort(tmp2.MU, decreasing = TRUE, index.return = TRUE)
init.MU <- as.matrix(tmp.MU$vectors[, tmp3.MU$ix[1]])
e1 <- eigen(t(init.MU) %*% M %*% init.MU)
e2 <- eigen(t(init.MU) %*% invMU %*% init.MU)
obj2 <- sum(log(e1$values)) + sum(log(e2$values))
if (obj2 < obj1) {
init <- init.MU
obj1 <- obj2
}
# if (qr(M)$rank == r) {
tmp.M <- eigen(M)
midmatrix <- U
tmp2.M <- apply(tmp.M$vectors, 2, startv)
tmp3.M <- sort(tmp2.M, decreasing = TRUE, index.return = TRUE)
init.M <- as.matrix(tmp.M$vectors[, tmp3.M$ix[1]])
e1 <- eigen(t(init.M) %*% M %*% init.M)
e2 <- eigen(t(init.M) %*% invMU %*% init.M)
obj3 <- sum(log(e1$values)) + sum(log(e2$values))
if (obj3 < obj1) {
init <- init.M
obj1 <- obj3
}
invM2 <- sweep(tmp.M$vectors, MARGIN = 2, 1 / sqrt(tmp.M$values), '*') %*% t(tmp.M$vectors)
midmatrix <- invM2 %*% tcrossprod(U, invM2)
tmp2.M <- apply(tmp.M$vectors, 2, startv)
tmp3.M <- sort(tmp2.M, decreasing = TRUE, index.return = TRUE)
init.M <- as.matrix(tmp.M$vectors[, tmp3.M$ix[1]])
e1 <- eigen(t(init.M) %*% M %*% init.M)
e2 <- eigen(t(init.M) %*% invMU %*% init.M)
obj4 <- sum(log(e1$values)) + sum(log(e2$values))
if (obj4 < obj1) {
init <- init.M
obj1 <- obj4
}
}
GEidx <- GE(init)
Ginit <- init %*% solve(init[GEidx[1], ])
i <- 1
while (i < maxiter) {
fobj <- function(x) {
T1 <- crossprod(x, x)
T2 <- crossprod(x, M) %*% x
T3 <- crossprod(x, invMU) %*% x
-2 * log(T1) + log(T2) + log(T3)
}
gobj <- function(x) {
W1 <- crossprod(x, x)
T1 <- x / as.vector(W1)
W2 <- crossprod(x, M) %*% x
T2 <- M %*% x / as.vector(W2)
W3 <- crossprod(x, invMU) %*% x
T3 <- invMU %*% x / as.vector(W3)
-2 * T1 + T2 + T3
}
res <- stats::optim(Ginit, fobj, gobj, method = "BFGS")
g <- as.matrix(res$par)
a <- qr(g)
Gammahat <- qr.Q(a)
e1 <- eigen(t(Gammahat) %*% M %*% Gammahat)
e2 <- eigen(t(Gammahat) %*% invMU %*% Gammahat)
obj5 <- sum(log(e1$values)) + sum(log(e2$values))
if (abs(obj1 - obj5) < ftol * abs(obj1)) {
break
} else {
obj1 <- obj5
i <- i + 1
}
}
Gamma0hat <- qr.Q(a, complete = TRUE)[, (u+1):r]
objfun <- obj5 + sum(log(tmp.MU$values))
Gammahat <- as.matrix(Gammahat)
Gamma0hat <- as.matrix(Gamma0hat)
} else if (u == r - 1 & u != 1) {
maxiter = 100
ftol = 1e-3
if (!is.null(initial)) {
MU <- M + U
tmp.MU <- eigen(MU)
invMU <- sweep(tmp.MU$vectors, MARGIN = 2, 1 / tmp.MU$values, '*') %*% t(tmp.MU$vectors)
init <- initial
eig1 <- eigen(t(init) %*% M %*% init)
eig2 <- eigen(t(init) %*% invMU %*% init)
obj1 <- sum(log(eig1$values)) + sum(log(eig2$values))
} else {
MU <- M + U
tmp.MU <- eigen(MU)
invMU <- sweep(tmp.MU$vectors, MARGIN = 2, 1 / tmp.MU$values, '*') %*% t(tmp.MU$vectors)
invMU2 <- sweep(tmp.MU$vectors, MARGIN = 2, 1 / sqrt(tmp.MU$values), '*') %*% t(tmp.MU$vectors)
midmatrix <- U
startv <- function(a) t(a) %*% midmatrix %*% a
tmp2.MU <- apply(tmp.MU$vectors, 2, startv)
tmp3.MU <- sort(tmp2.MU, decreasing = TRUE, index.return = TRUE)
init <- as.matrix(tmp.MU$vectors[, tmp3.MU$ix[1:u]])
# if (qr(MU)$rank == r) {
eig1 <- eigen(t(init) %*% M %*% init)
eig2 <- eigen(t(init) %*% invMU %*% init)
obj1 <- sum(log(eig1$values)) + sum(log(eig2$values))
midmatrix <- invMU2 %*% tcrossprod(U, invMU2)
tmp2.MU <- apply(tmp.MU$vectors, 2, startv)
tmp3.MU <- sort(tmp2.MU, decreasing = TRUE, index.return = TRUE)
init.MU <- as.matrix(tmp.MU$vectors[, tmp3.MU$ix[1:u]])
e1 <- eigen(t(init.MU) %*% M %*% init.MU)
e2 <- eigen(t(init.MU) %*% invMU %*% init.MU)
obj2 <- sum(log(e1$values)) + sum(log(e2$values))
if (obj2 < obj1) {
init <- init.MU
obj1 <- obj2
}
# if (qr(M)$rank == r) {
tmp.M <- eigen(M)
midmatrix <- U
tmp2.M <- apply(tmp.M$vectors, 2, startv)
tmp3.M <- sort(tmp2.M, decreasing = TRUE, index.return = TRUE)
init.M <- as.matrix(tmp.M$vectors[, tmp3.M$ix[1:u]])
e1 <- eigen(t(init.M) %*% M %*% init.M)
e2 <- eigen(t(init.M) %*% invMU %*% init.M)
obj3 <- sum(log(e1$values)) + sum(log(e2$values))
if (obj3 < obj1) {
init <- init.M
obj1 <- obj3
}
invM2 <- sweep(tmp.M$vectors, MARGIN = 2, 1 / sqrt(tmp.M$values), '*') %*% t(tmp.M$vectors)
midmatrix <- invM2 %*% tcrossprod(U, invM2)
tmp2.M <- apply(tmp.M$vectors, 2, startv)
tmp3.M <- sort(tmp2.M, decreasing = TRUE, index.return = TRUE)
init.M <- as.matrix(tmp.M$vectors[, tmp3.M$ix[1:u]])
e1 <- eigen(t(init.M) %*% M %*% init.M)
e2 <- eigen(t(init.M) %*% invMU %*% init.M)
obj4 <- sum(log(e1$values)) + sum(log(e2$values))
if (obj4 < obj1) {
init <- init.M
obj1 <- obj4
}
}
# }
# }
GEidx <- GE(init)
Ginit <- init %*% solve(init[GEidx[1:u], ])
j <- GEidx[r]
g <- as.matrix(Ginit[j, ])
t2 <- crossprod(Ginit[-j, ], as.matrix(M[-j, j])) / M[j, j]
t3 <- crossprod(Ginit[-j, ], as.matrix(invMU[-j, j])) / invMU[j, j]
GUGt2 <- g + t2
GUG <- crossprod(Ginit, (M %*% Ginit)) - tcrossprod(GUGt2, GUGt2) * M[j, j]
GVGt2 <- g + t3
GVG <- crossprod(Ginit, (invMU %*% Ginit)) - tcrossprod(GVGt2, GVGt2) * invMU[j, j]
invC1 <- chol2inv(chol(GUG))
invC2 <- chol2inv(chol(GVG))
fobj <- function(x) {
tmp2 <- x + t2
tmp3 <- x + t3
T2 <- invC1 %*% tmp2
T3 <- invC2 %*% tmp3
-2 * log(1 + sum(x^2)) + log(1 + M[j, j] * crossprod(tmp2, T2)) + log(1 + invMU[j, j] * crossprod(tmp3, T3))
}
gobj <- function(x) {
tmp2 <- x + t2
tmp3 <- x + t3
T2 <- invC1 %*% tmp2
T3 <- invC2 %*% tmp3
-4 * x %*% solve(1 + sum(x^2)) + 2 * T2 / as.numeric(1 / M[j, j] + crossprod(tmp2, T2)) + 2 * T3 / as.numeric(1 / invMU[j, j] + crossprod(tmp3, T3))
}
i <- 1
while (i < maxiter) {
res <- stats::optim(Ginit[j,], fobj, gobj, method = "BFGS")
Ginit[j, ] <- res$par
a <- qr(Ginit)
Gammahat <- qr.Q(a)
e1 <- eigen(t(Gammahat) %*% M %*% Gammahat)
e2 <- eigen(t(Gammahat) %*% invMU %*% Gammahat)
obj5 <- sum(log(e1$values)) + sum(log(e2$values))
if (abs(obj1 - obj5) < ftol * abs(obj1)) {
break
} else {
obj1 <- obj5
i <- i + 1
}
}
Gamma0hat <- qr.Q(a, complete = TRUE)[, (u+1):r, drop = FALSE]
objfun <- obj5 + sum(log(tmp.MU$values))
Gammahat <- as.matrix(Gammahat)
Gamma0hat <- as.matrix(Gamma0hat)
} else {
maxiter = 100
ftol = 1e-3
if (!is.null(initial)) {
MU <- M + U
tmp.MU <- eigen(MU)
invMU <- sweep(tmp.MU$vectors, MARGIN = 2, 1 / tmp.MU$values, '*') %*% t(tmp.MU$vectors)
init <- initial
eig1 <- eigen(t(init) %*% M %*% init)
eig2 <- eigen(t(init) %*% invMU %*% init)
obj1 <- sum(log(eig1$values)) + sum(log(eig2$values))
} else {
MU <- M + U
tmp.MU <- eigen(MU)
invMU <- sweep(tmp.MU$vectors, MARGIN = 2, 1 / tmp.MU$values, '*') %*% t(tmp.MU$vectors)
invMU2 <- sweep(tmp.MU$vectors, MARGIN = 2, 1 / sqrt(tmp.MU$values), '*') %*% t(tmp.MU$vectors)
midmatrix <- U
startv <- function(a) t(a) %*% midmatrix %*% a
tmp2.MU <- apply(tmp.MU$vectors, 2, startv)
tmp3.MU <- sort(tmp2.MU, decreasing = TRUE, index.return = TRUE)
init <- as.matrix(tmp.MU$vectors[, tmp3.MU$ix[1:u]])
# if (qr(MU)$rank == r) {
eig1 <- eigen(t(init) %*% M %*% init)
eig2 <- eigen(t(init) %*% invMU %*% init)
obj1 <- sum(log(eig1$values)) + sum(log(eig2$values))
midmatrix <- invMU2 %*% tcrossprod(U, invMU2)
tmp2.MU <- apply(tmp.MU$vectors, 2, startv)
tmp3.MU <- sort(tmp2.MU, decreasing = TRUE, index.return = TRUE)
init.MU <- as.matrix(tmp.MU$vectors[, tmp3.MU$ix[1:u]])
e1 <- eigen(t(init.MU) %*% M %*% init.MU)
e2 <- eigen(t(init.MU) %*% invMU %*% init.MU)
obj2 <- sum(log(e1$values)) + sum(log(e2$values))
if (obj2 < obj1) {
init <- init.MU
obj1 <- obj2
}
# if (qr(M)$rank == r) {
tmp.M <- eigen(M)
midmatrix <- U
tmp2.M <- apply(tmp.M$vectors, 2, startv)
tmp3.M <- sort(tmp2.M, decreasing = TRUE, index.return = TRUE)
init.M <- as.matrix(tmp.M$vectors[, tmp3.M$ix[1:u]])
e1 <- eigen(t(init.M) %*% M %*% init.M)
e2 <- eigen(t(init.M) %*% invMU %*% init.M)
obj3 <- sum(log(e1$values)) + sum(log(e2$values))
if (obj3 < obj1) {
init <- init.M
obj1 <- obj3
}
invM2 <- sweep(tmp.M$vectors, MARGIN = 2, 1 / sqrt(tmp.M$values), '*') %*% t(tmp.M$vectors)
midmatrix <- invM2 %*% tcrossprod(U, invM2)
tmp2.M <- apply(tmp.M$vectors, 2, startv)
tmp3.M <- sort(tmp2.M, decreasing = TRUE, index.return = TRUE)
init.M <- as.matrix(tmp.M$vectors[, tmp3.M$ix[1:u]])
e1 <- eigen(t(init.M) %*% M %*% init.M)
e2 <- eigen(t(init.M) %*% invMU %*% init.M)
obj4 <- sum(log(e1$values)) + sum(log(e2$values))
if (obj4 < obj1) {
init <- init.M
obj1 <- obj4
}
}
# }
# }
GEidx <- GE(init)
Ginit <- init %*% solve(init[GEidx[1:u], ])
GUG <- crossprod(Ginit, (M %*% Ginit))
GVG <- crossprod(Ginit, (invMU %*% Ginit))
t4 <- crossprod(Ginit[GEidx[(u+1):r],], Ginit[GEidx[(u+1):r], ]) + diag(u)
i <- 1
while (i < maxiter) {
for (j in GEidx[(u+1):r]) {
g <- as.matrix(Ginit[j, ])
t2 <- crossprod(Ginit[-j, ], as.matrix(M[-j, j])) / M[j, j]
t3 <- crossprod(Ginit[-j, ], as.matrix(invMU[-j, j])) / invMU[j, j]
GUGt2 <- g + t2
GUG <- GUG - tcrossprod(GUGt2, GUGt2) * M[j, j]
GVGt2 <- g + t3
GVG <- GVG - tcrossprod(GVGt2, GVGt2) * invMU[j, j]
t4 <- t4 - tcrossprod(g, g)
invC1 <- chol2inv(chol(GUG))
invC2 <- chol2inv(chol(GVG))
invt4 <- chol2inv(chol(t4))
fobj <- function(x) {
tmp2 <- x + t2
tmp3 <- x + t3
T1 <- invt4 %*% x
T2 <- invC1 %*% tmp2
T3 <- invC2 %*% tmp3
-2 * log(1 + x %*% T1) + log(1 + M[j, j] * crossprod(tmp2, T2)) + log(1 + invMU[j, j] * crossprod(tmp3, T3))
}
gobj <- function(x) {
tmp2 <- x + t2
tmp3 <- x + t3
T1 <- invt4 %*% x
T2 <- invC1 %*% tmp2
T3 <- invC2 %*% tmp3
-4 * T1 / as.numeric(1 + x %*% T1) + 2 * T2 / as.numeric(1 / M[j, j] + crossprod(tmp2, T2)) + 2 * T3 / as.numeric(1 / invMU[j, j] + crossprod(tmp3, T3))
}
res <- stats::optim(Ginit[j,], fobj, gobj, method = "BFGS")
Ginit[j, ] <- res$par
g <- as.matrix(Ginit[j, ])
t4 <- t4 + tcrossprod(g, g)
GUGt2 <- g + t2
GUG <- GUG + tcrossprod(GUGt2, GUGt2) * M[j, j]
GVGt2 <- g + t3
GVG <- GVG + tcrossprod(GVGt2, GVGt2) * invMU[j, j]
}
a <- qr(Ginit)
Gammahat <- qr.Q(a)
e1 <- eigen(t(Gammahat) %*% M %*% Gammahat)
e2 <- eigen(t(Gammahat) %*% invMU %*% Gammahat)
obj5 <- sum(log(e1$values)) + sum(log(e2$values))
if (abs(obj1 - obj5) < ftol * abs(obj1)) {
break
} else {
obj1 <- obj5
i <- i + 1
}
}
Gamma0hat <- qr.Q(a, complete = TRUE)[, (u+1):r]
objfun <- obj5 + sum(log(tmp.MU$values))
Gammahat <- as.matrix(Gammahat)
Gamma0hat <- as.matrix(Gamma0hat)
}
return(list(Gammahat = Gammahat, Gamma0hat = Gamma0hat, objfun = objfun))
}
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