R/spde2.R
In INBO-BMK/INLA: Full Bayesian Analysis of Latent Gaussian Models using Integrated Nested Laplace Approximations
Defines functions
inla.spde2.generic
inla.internal.spde2.matern.B.tau
inla.spde2.matern.sd.basis
param2.matern.orig
inla.spde2.matern
inla.spde2.pcmatern
param2.iheat
inla.spde2.iheat
inla.spde2.theta2phi0
inla.spde2.theta2phi1
inla.spde2.theta2phi2
inla.spde2.precision
inla.spde2.result
inla.spde2.models
Documented in
inla.spde2.generic
inla.spde2.matern
inla.spde2.matern.sd.basis
inla.spde2.models
inla.spde2.pcmatern
inla.spde2.precision
inla.spde2.result
inla.spde2.theta2phi0
inla.spde2.theta2phi1
inla.spde2.theta2phi2
param2.matern.orig
## 'spde2' model functions
## Export: inla.spde.precision!inla.spde2
## Export: inla.spde.result!inla.spde2 inla.spde2.generic
## Export: inla.spde2.matern param2.matern.orig
## Export: inla.spde2.matern.sd.basis inla.spde2.models
## Export: inla.spde2.pcmatern
## Export: inla.spde2.precision inla.spde2.result
## Export: inla.spde2.theta2phi0 inla.spde2.theta2phi1 inla.spde2.theta2phi2
## Internal: inla.internal.spde2.matern.B.tau
## Internal: inla.internal.test.spde2.sd.basis
inla.spde2.generic =
function(M0, M1, M2, B0, B1, B2, theta.mu, theta.Q,
transform = c("logit","log","identity"),
theta.initial = theta.mu,
fixed = rep(FALSE, length(theta.mu)),
theta.fixed = theta.initial[fixed],
BLC = cbind(0.0, diag(nrow=length(theta.mu))),
...)
{
transform = match.arg(transform)
if (!inla.is.matrix(M0)) stop("M0 must be of a matrix type.")
if (!inla.is.matrix(M1)) stop("M1 must be of a matrix type.")
if (!inla.is.matrix(M2)) stop("M2 must be of a matrix type.")
M0 = inla.as.dgTMatrix(M0)
M1 = inla.as.dgTMatrix(M1)
M2 = inla.as.dgTMatrix(M2)
n.spde = nrow(M0)
n.theta = length(theta.mu)
spde = (list(model = "generic",
n.spde = n.spde,
n.theta = n.theta,
param.inla = list(),
f = list()
))
class(spde) = c("inla.spde2", "inla.spde", "inla.model.class")
B0 = inla.spde.homogenise_B_matrix(B0, n.spde, n.theta)
B1 = inla.spde.homogenise_B_matrix(B1, n.spde, n.theta)
B2 = inla.spde.homogenise_B_matrix(B2, n.spde, n.theta)
BLC = inla.spde.homogenise_B_matrix(BLC, nrow(BLC), n.theta)
param.inla =
list(n = n.spde,
n.theta = n.theta,
M0=M0, M1=M1, M2=M2,
B0=B0, B1=B1, B2=B2, BLC=BLC,
theta.mu=theta.mu, theta.Q=theta.Q,
transform=transform,
theta.initial=theta.initial,
fixed=fixed,
theta.fixed=theta.fixed)
## TODO: Check dimension consistency of inputs
## If any "fixed", move information from B[,1+(1:n.theta)] into
## B[,1], theta.mu, and theta.Q
## NOTE: Should this be in the f function instead?
## No, the information is also needed elsewhere.
if (any(fixed)) {
param.inla$theta.initial = theta.initial[!fixed]
param.inla$B0[,1] =
B0[,c(TRUE, fixed), drop=FALSE] %*% c(1.0, theta.fixed)
param.inla$B1[,1] =
B1[,c(TRUE, fixed), drop=FALSE] %*% c(1.0, theta.fixed)
param.inla$B2[,1] =
B2[,c(TRUE, fixed), drop=FALSE] %*% c(1.0, theta.fixed)
param.inla$BLC[,1] =
BLC[,c(TRUE, fixed), drop=FALSE] %*% c(1.0, theta.fixed)
param.inla$B0 = param.inla$B0[,c(TRUE, !fixed), drop=FALSE]
param.inla$B1 = param.inla$B1[,c(TRUE, !fixed), drop=FALSE]
param.inla$B2 = param.inla$B2[,c(TRUE, !fixed), drop=FALSE]
param.inla$BLC = param.inla$BLC[,c(TRUE, !fixed), drop=FALSE]
param.inla$theta.Q = param.inla$theta.Q[!fixed, !fixed, drop=FALSE]
if (!all(fixed)) {
param.inla$theta.mu =
theta.mu - solve(param.inla$theta.Q,
theta.Q[!fixed, fixed, drop=FALSE]
%*% (theta.fixed[fixed]-theta.mu[fixed]))
} else { ## All fixed.
param.inla$theta.mu = as.vector(matrix(0.0, 0, 1))
}
}
## Remove all BLC rows that would produce point masses, which is
## not supported by inla:
if (n.theta==0) {
param.inla$BLC = matrix(0,0,1)
} else {
param.inla$BLC =
param.inla$BLC[rowSums(abs(param.inla$BLC[,-1,drop=FALSE]))>0,,
drop=FALSE]
}
spde$param.inla = param.inla
spde$f = (list(model="spde2",
n=n.spde,
spde2.transform=transform,
hyper.default =
(list(theta1 =
list(prior="mvnorm",
param=(c(param.inla$theta.mu,
as.matrix(param.inla$theta.Q)))
)
))
))
if (n.theta>0)
for (k in 1:n.theta) {
eval(parse(text=
paste("spde$f$hyper.default$theta", k,
"$initial = param.inla$theta.mu[k]", sep="")))
}
return(spde)
}
inla.internal.spde2.matern.B.tau =
function(precision.fcn, n.theta, delta = 1e-3, ...)
{
theta.0 = rep(0, n.theta)
Q.0 = precision.fcn(theta.0, ...)
logv.0 = log(diag(inla.qinv(Q.0)))
B.tau = matrix(0, nrow(Q.0), 1L+n.theta)
for (k in seq_len(n.theta)) {
theta = theta.0
theta[k] = theta[k] + delta
Q = precision.fcn(theta, ...)
logv = log(diag(inla.qinv(Q)))
dlogv = (logv-logv.0)/delta
B.tau[, k+1] = dlogv/2
}
B.tau[,1L] = logv.0/2
return(B.tau)
}
inla.spde2.matern.sd.basis =
function(mesh, B.sd, B.range, method=1,
local.offset.compensation=FALSE,
alpha=2, ...)
{
p.sd = ncol(B.sd)-1L
p.kappa = ncol(B.range)-1L
d = inla.ifelse(inherits(mesh, "inla.mesh"), 2, 1)
nu = alpha-d/2
nu.nominal = max(0.5, nu)
alpha.nominal = max(nu.nominal+d/2, alpha)
n.spde = inla.ifelse(d==2, mesh$n, mesh$m)
## log(kappa) = 0.5*log(8*nu) - log(range)
B.kappa = (cbind(log(8*nu.nominal)/2 - B.range[,1],
-B.range[,-1,drop=FALSE] ))
if (method==1) {
spde =
inla.spde2.matern(mesh, B.tau=as.matrix(0), B.kappa=B.kappa, ...)
B.tau =
inla.internal.spde2.matern.B.tau(
precision.fcn=
function(theta, spde) {
return(inla.spde2.precision(spde, theta=theta))
},
n.theta = p.kappa,
spde = spde)
if (!local.offset.compensation) {
B.tau[,1] = rep(mean(B.tau[,1]), spde$n.spde)
}
} else {
spde =
inla.spde2.matern(mesh,
B.tau=as.matrix(0),
B.kappa=B.kappa[,1,drop=FALSE],
alpha=alpha, ...)
lvar.0 =
log(diag(inla.qinv(inla.spde2.precision(spde, theta=rep(0,0)))))
if (!local.offset.compensation) {
lvar.0 = rep(mean(lvar.0), spde$n.spde)
}
B.tau = -B.kappa*nu
B.tau[,1] = lvar.0/2
}
log.var.scaling = lgamma(nu.nominal)-lgamma(alpha.nominal)-log(4*pi)*d/2
B.tau =
cbind(B.tau[,1]-B.sd[,1],
-B.sd[,-1,drop=FALSE],
B.tau[,-1,drop=FALSE])
B.kappa =
cbind(B.kappa[,1],
matrix(0, n.spde, p.sd),
B.kappa[,-1,drop=FALSE])
return(list(B.tau=B.tau, B.kappa=B.kappa))
}
inla.internal.test.spde2.sd.basis = function (k=1, dth=0.1, r=1000, globe=25, compensate=FALSE) {
alpha=2
mesh = inla.mesh.create(globe=globe)
B.common = inla.mesh.basis(mesh, "b.spline", n=3)
B.common = cbind(B.common, B.common[,1]*(mesh$loc[,3]>0))
B.common[,1] = B.common[,1]*(mesh$loc[,3]<=0)
B.sd = cbind(0, B.common)
B.range = cbind(log(r/6370), B.common)
sd.basis0 = inla.spde2.matern.sd.basis(mesh, B.sd=B.sd, B.range=B.range, method=0, alpha=alpha, local.offset.compensation=compensate)
sd.basis1 = inla.spde2.matern.sd.basis(mesh, B.sd=B.sd, B.range=B.range, method=1, alpha=alpha, local.offset.compensation=compensate)
spde0 =
inla.spde2.matern(mesh, alpha=alpha,
B.tau=sd.basis0$B.tau,
B.kappa=sd.basis0$B.kappa)
spde1 =
inla.spde2.matern(mesh, alpha=alpha,
B.tau=sd.basis1$B.tau,
B.kappa=sd.basis1$B.kappa)
th0 = c(0,0,0,0,0,0,0,0)
th1 = th0
th1[k] = th0[k]+dth
v0.0=diag(inla.qinv(inla.spde2.precision(spde0,theta=th0)))
v0.1=diag(inla.qinv(inla.spde2.precision(spde0,theta=th1)))
v1.0=diag(inla.qinv(inla.spde2.precision(spde1,theta=th0)))
v1.1=diag(inla.qinv(inla.spde2.precision(spde1,theta=th1)))
op=par(mfrow=c(2,2))
on.exit(par(op))
plot(asin(mesh$loc[,3]),rowSums(sd.basis0$B.kappa[,1+k,drop=FALSE]), pch=20,
ylim=(range(c(0, range(rowSums(sd.basis0$B.kappa[,1+k,drop=FALSE])),
range(-rowSums(sd.basis0$B.tau[,1+k,drop=FALSE])),
range(-rowSums(sd.basis1$B.tau[,1+k,drop=FALSE]))))),
main="Basis functions")
points(asin(mesh$loc[,3]),-rowSums(sd.basis0$B.tau[,1+k,drop=FALSE]),col=2,pch=20)
points(asin(mesh$loc[,3]),-rowSums(sd.basis1$B.tau[,1+k,drop=FALSE]),col=4,pch=20)
legend("bottomright",
legend=paste("Method", 0:1),
col=c(2,4), pch=20)
plot(asin(mesh$loc[,3]),v0.1, col=2,pch=20,
ylim=range(c(0,range(v0.0),range(v1.0),range(v0.1),range(v1.1))),
main="Absolute variances")
points(asin(mesh$loc[,3]),v1.1, col=4,pch=20)
points(asin(mesh$loc[,3]),v0.0, col=2,pch=1)
points(asin(mesh$loc[,3]),v1.0, col=4,pch=1)
legend("bottomright",
legend=paste("Method", 0:1),
col=c(2,4), pch=20)
plot(asin(mesh$loc[,3]),v0.1-v0.0, col=2,pch=20,
ylim=range(c(0,range(v0.1-v0.0),range(v1.1-v1.0))),
main="Absolute variance error")
points(asin(mesh$loc[,3]),v1.1-v1.0, col=4,pch=20)
legend("bottomright",
legend=paste("Method", 0:1),
col=c(2,4), pch=20)
plot(asin(mesh$loc[,3]),v0.1/v0.0-1, col=2,pch=20,
ylim=range(c(0,range(v0.1/v0.0-1),range(v1.1/v1.0-1))),
main="Relative variance error")
points(asin(mesh$loc[,3]),v1.1/v1.0-1, col=4,pch=20)
legend("bottomright",
legend=paste("Method", 0:1),
col=c(2,4), pch=20)
return(environment())
}
param2.matern.orig =
function(mesh,
alpha=2,
B.tau = matrix(c(0,1,0),1,3),
B.kappa = matrix(c(0,0,1),1,3),
prior.variance.nominal = 1,
prior.range.nominal = NULL,
prior.tau = NULL,
prior.kappa = NULL,
theta.prior.mean = NULL,
theta.prior.prec = 0.1)
{
## NOTE: For d==1, degree==2, the B.* must be given per basis
## function, not per knot.
inla.require.inherits(mesh, c("inla.mesh", "inla.mesh.1d"), "'mesh'")
if (is.null(B.tau))
stop("B.tau must not be NULL.")
if (is.null(B.kappa))
stop("B.kappa must not be NULL.")
is.stationary = (nrow(B.kappa)==1) && (nrow(B.tau)==1)
d = inla.ifelse(inherits(mesh, "inla.mesh"), 2, 1)
nu = alpha-d/2
nu.nominal = max(0.5, nu)
alpha.nominal = max(nu.nominal+d/2, alpha)
n.spde = inla.ifelse(d==2, mesh$n, mesh$m)
n.theta = ncol(B.kappa)-1L
B.kappa = inla.spde.homogenise_B_matrix(B.kappa, n.spde, n.theta)
B.tau = inla.spde.homogenise_B_matrix(B.tau, n.spde, n.theta)
B.prec =
cbind(+ lgamma(alpha.nominal) - lgamma(nu.nominal)
+ (d/2)*log(4*pi) + 2*nu.nominal*B.kappa[,1]
+ 2*B.tau[,1],
2*nu.nominal*B.kappa[,-1,drop=FALSE]
+ 2*B.tau[,-1,drop=FALSE] )
if (is.stationary) {
B.tau = B.tau[1,,drop=FALSE]
B.kappa = B.kappa[1,,drop=FALSE]
B.prec = B.prec[1,,drop=FALSE]
}
B.variance = -B.prec
B.range =
cbind(0.5*log(8*nu.nominal)-B.kappa[,1],
-B.kappa[,-1,drop=FALSE])
if (n.theta>0) {
B.theta = cbind(0,diag(1, n.theta))
rownames(B.theta) <- rownames(B.theta, do.NULL=FALSE, prefix="theta.")
} else {
B.theta = NULL
}
rownames(B.tau) <- rownames(B.tau, do.NULL=FALSE, prefix="tau.")
rownames(B.kappa) <- rownames(B.kappa, do.NULL=FALSE, prefix="kappa.")
rownames(B.variance) <-
rownames(B.variance, do.NULL=FALSE, prefix="variance.nominal.")
rownames(B.range) <-
rownames(B.range, do.NULL=FALSE, prefix="range.nominal.")
BLC = rbind(B.theta, B.tau, B.kappa, B.variance, B.range)
## Construct priors.
if (is.null(theta.prior.prec)) {
theta.prior.prec = diag(0.1, n.theta, n.theta)
} else {
theta.prior.prec = as.matrix(theta.prior.prec)
if (ncol(theta.prior.prec) == 1) {
theta.prior.prec =
diag(as.vector(theta.prior.prec), n.theta, n.theta)
}
if ((nrow(theta.prior.prec) != n.theta) ||
(ncol(theta.prior.prec) != n.theta)) {
stop(paste("Size of theta.prior.prec is (",
paste(dim(theta.prior.prec), collapse=",", sep=""),
") but should be (",
paste(c(n.theta, n.theta), collapse=",", sep=""),
")."))
}
}
if (is.null(theta.prior.mean)) {
if (is.null(prior.range.nominal)) {
mesh.range =
inla.ifelse(d==2,
(max(c(diff(range(mesh$loc[,1])),
diff(range(mesh$loc[,2])),
diff(range(mesh$loc[,3]))
))),
diff(mesh$interval))
prior.range.nominal = mesh.range*0.2
} else {
if (!is.numeric(prior.range.nominal) ||
(length(prior.range.nominal) != 1)) {
stop(paste0("'prior.range.nominal' must be NULL or a single scalar value.\n",
"Did you intend to supply 'prior.range' to inla.spde2.pcmatern instead?"))
}
}
if (is.null(prior.kappa)) {
prior.kappa = sqrt(8*nu.nominal)/prior.range.nominal
}
if (is.null(prior.tau)) {
prior.tau =
sqrt(gamma(nu.nominal)/gamma(alpha.nominal)/
(4*pi*prior.kappa^(2*nu.nominal)*prior.variance.nominal))
}
if (n.theta>0) {
theta.prior.mean =
qr.solve(rbind(B.tau[,-1,drop=FALSE], B.kappa[,-1,drop=FALSE]),
c(log(prior.tau) - B.tau[,1],
log(prior.kappa) - B.kappa[,1]))
} else {
theta.prior.mean = rep(0, n.theta) ## Empty vector
}
}
param =
list(is.stationary=is.stationary,
B.tau=B.tau, B.kappa=B.kappa, BLC=BLC,
theta.prior.mean=theta.prior.mean,
theta.prior.prec=theta.prior.prec)
return(param)
}
inla.spde2.matern =
function(mesh,
alpha=2,
param = NULL,
constr = FALSE,
extraconstr.int = NULL,
extraconstr = NULL,
fractional.method = c("parsimonious", "null"),
B.tau = matrix(c(0,1,0),1,3),
B.kappa = matrix(c(0,0,1),1,3),
prior.variance.nominal = 1,
prior.range.nominal = NULL,
prior.tau = NULL,
prior.kappa = NULL,
theta.prior.mean = NULL,
theta.prior.prec = 0.1,
n.iid.group = 1,
...)
{
if(!is.null(list(...)[["prior.pc.rho"]]) &&
!is.null(list(...)[["prior.pc.sig"]])){
## Temporary implementation of PC prior for standard deviation and range
## - Changes parametrization to range and standard deviation
## - Sets prior according to hyperparameters for range : prior.pc.rho
## and std.dev.: prior.pc.sig
warning("You're using a deprecated experimental PC prior matern model that will be removed in a future version of the package. Use 'inla.spde2.pcmatern' instead.")
prior.pc.rho <- list(...)[["prior.pc.rho"]]
prior.pc.sig <- list(...)[["prior.pc.sig"]]
## Call inla.spde2.matern with range and standard deviation parametrization
d = inla.ifelse(inherits(mesh, "inla.mesh"), 2, 1)
nu = alpha-d/2
kappa0 = log(8*nu)/2
tau0 = 0.5*(lgamma(nu)-lgamma(nu+d/2)-d/2*log(4*pi))-nu*kappa0
spde = inla.spde2.matern(mesh = mesh,
B.tau = cbind(tau0, nu, -1),
B.kappa = cbind(kappa0, -1, 0))
## Change prior information
param = c(prior.pc.rho, prior.pc.sig)
spde$f$hyper.default$theta1$prior = "pcspdega"
spde$f$hyper.default$theta1$param = param
spde$f$hyper.default$theta1$initial = log(prior.pc.rho[1])+1
spde$f$hyper.default$theta2$initial = log(prior.pc.sig[1])-1
## End and return
return(invisible(spde))
}
## Standard code
inla.require.inherits(mesh, c("inla.mesh", "inla.mesh.1d"), "'mesh'")
fractional.method = match.arg(fractional.method)
if (is.null(param)) {
param =
param2.matern.orig(
mesh, alpha,
B.tau, B.kappa,
prior.variance.nominal,
prior.range.nominal,
prior.tau,
prior.kappa,
theta.prior.mean,
theta.prior.prec)
} else {
deprecated =
!c(missing(B.tau), missing(B.kappa),
missing(prior.variance.nominal),
missing(prior.range.nominal),
missing(prior.tau), missing(prior.kappa),
missing(theta.prior.mean), missing(theta.prior.prec))
deprecated =
c("B.tau", "B.kappa",
"prior.variance.nominal",
"prior.range.nominal",
"prior.tau", "prior.kappa",
"theta.prior.mean", "theta.prior.prec")[deprecated]
if (length(deprecated) > 0) {
warning(paste("'param' specified; ",
"Ignoring deprecated parameter(s) ",
paste(deprecated, collapse=", "), ".", sep=""))
}
}
d = inla.ifelse(inherits(mesh, "inla.mesh"), 2, 1)
nu = alpha-d/2
nu.nominal = max(0.5, nu)
alpha.nominal = max(nu.nominal+d/2, alpha)
n.spde = inla.ifelse(d==2, mesh$n, mesh$m)
n.theta = ncol(B.kappa)-1L
if (d==2) {
fem =
inla.fmesher.smorg(mesh$loc,
mesh$graph$tv,
fem=2,
output=list("c0", "c1", "g1", "g2"))
} else {
fem = inla.mesh.1d.fem(mesh)
if (mesh$degree==2) {
fem$c0 = fem$c1 ## Use higher order matrix.
}
}
if (alpha==2) {
B.phi0 = param$B.tau
B.phi1 = 2*param$B.kappa
M0 = fem$c0
M1 = fem$g1
M2 = fem$g2
} else if (alpha==1) {
B.phi0 = param$B.tau
B.phi1 = param$B.kappa
M0 = fem$c0
M1 = fem$g1*0
M2 = fem$g1
} else if (!param$is.stationary) {
stop("Non-stationary Matern with fractional alpha is not implemented.")
} else if ((alpha<2) && (alpha>1)) {
if (fractional.method == "parsimonious") {
lambda = alpha-floor(alpha)
b = matrix(c(1,0,0, 1,1,0, 1,2,1),3,3) %*%
solve(matrix(1/(c(4:2, 3:1, 2:0)+lambda), 3, 3),
1/(c(4:2)+lambda-alpha))
} else if (fractional.method == "null") {
b = c(1,alpha,alpha*(alpha-1)/2)
} else {
stop(paste("Unknown fractional.method '", fractional.method,
"'.", sep=""))
}
B.phi0 = param$B.tau + (alpha-2)*param$B.kappa
B.phi1 = 2*param$B.kappa
M0 = fem$c0*b[1]
M1 = fem$g1*b[2]/2
M2 = fem$g2*b[3]
} else if ((alpha<1) && (alpha>0)) {
if (fractional.method == "parsimonious") {
lambda = alpha-floor(alpha)
b = matrix(c(1,0,1,1),2,2) %*%
solve(matrix(1/(c(2:1, 1:0)+lambda), 2, 2),
1/(c(2:1)+lambda-alpha))
} else if (fractional.method == "null") {
b = c(1,alpha)
} else {
stop(paste("Unknown fractional.method '", fractional.method,
"'.", sep=""))
}
B.phi0 = param$B.tau + (alpha-1)*param$B.kappa
B.phi1 = param$B.kappa
M0 = fem$c0*b[1]
M1 = fem$g1*0
M2 = fem$g1*b[2]
} else {
stop(paste("Unsupported alpha value (", alpha,
"). Supported values are 0 < alpha <= 2", sep=""))
}
if (n.iid.group == 1) {
spde =
inla.spde2.generic(M0=M0, M1=M1, M2=M2,
B0=B.phi0, B1=B.phi1, B2=1,
theta.mu = param$theta.prior.mean,
theta.Q = param$theta.prior.prec,
transform = "identity",
BLC = param$BLC)
} else {
if (nrow(B.phi0) > 1) {
B.phi0 <- kronecker(matrix(1, n.iid.group, 1), B.phi0)
}
if (nrow(B.phi1) > 1) {
B.phi1 <- kronecker(matrix(1, n.iid.group, 1), B.phi1)
}
spde =
inla.spde2.generic(M0=kronecker(Diagonal(n.iid.group), M0),
M1=kronecker(Diagonal(n.iid.group), M1),
M2=kronecker(Diagonal(n.iid.group), M2),
B0=B.phi0, B1=B.phi1, B2=1,
theta.mu = param$theta.prior.mean,
theta.Q = param$theta.prior.prec,
transform = "identity",
BLC = param$BLC)
}
spde$model = "matern"
spde$BLC = param$BLC
if (constr || !is.null(extraconstr.int) || !is.null(extraconstr)) {
A.constr = matrix(numeric(0), 0, n.spde*n.iid.group)
e.constr = matrix(numeric(0), 0, 1)
if (constr) {
A.constr <- rbind(A.constr,
matrix(colSums(fem$c1)/n.iid.group,
1, n.spde*n.iid.group))
e.constr <- rbind(e.constr, 0)
}
if (!is.null(extraconstr.int)) {
if (ncol(extraconstr.int$A) == n.spde) {
A.constr <-
rbind(A.constr,
kronecker(matrix(1/n.iid.group, 1, n.iid.group),
as.matrix(extraconstr.int$A %*% fem$c1)))
} else {
A.constr <-
rbind(A.constr,
as.matrix(extraconstr.int$A %*%
kronecker(Diagonal(n.iid.group),
fem$c1)))
}
e.constr <- rbind(e.constr, as.matrix(extraconstr.int$e))
}
if (!is.null(extraconstr)) {
if (ncol(extraconstr$A) == n.spde) {
A.constr <-
rbind(A.constr,
kronecker(matrix(1/n.iid.group, 1, n.iid.group),
as.matrix(extraconstr$A)))
} else {
A.constr <- rbind(A.constr, as.matrix(extraconstr$A))
}
e.constr <- rbind(e.constr, as.matrix(extraconstr$e))
}
spde$f$constr = FALSE
spde$f$extraconstr = list(A=A.constr, e=e.constr)
}
## Attach the mesh, so downstream code can have access
spde$mesh <- mesh
return(invisible(spde))
}
inla.spde2.pcmatern =
function(mesh,
alpha = 2,
param = NULL,
constr = FALSE,
extraconstr.int = NULL,
extraconstr = NULL,
fractional.method = c("parsimonious", "null"),
n.iid.group = 1,
prior.range = NULL,
prior.sigma = NULL)
{
## Implementation of PC prior for standard deviation and range
## - Sets the parametrization to range and standard deviation
## - Sets prior according to hyperparameters for range : prior.range
## and std.dev.: prior.sigma
## Calls inla.spde2.matern to construct the object, then changes the prior
if (inherits(mesh, "inla.mesh")) {
d <- 2
} else if (inherits(mesh, "inla.mesh.1d")) {
d <- 1
} else {
stop(paste("Unknown mesh class '",
paste(class(mesh), collapse=",", sep=""),
"'.", sep=""))
}
if (missing(prior.range) || is.null(prior.range) ||
!is.vector(prior.range) || (length(prior.range) != 2)) {
stop("'prior.range' should be a length 2 vector 'c(range0,tailprob)' or a fixed range specified with 'c(range,NA)'.")
}
if (missing(prior.sigma) || is.null(prior.sigma) ||
!is.vector(prior.sigma) || (length(prior.sigma) != 2)) {
stop("'prior.sigma' should be a length 2 vector 'c(sigma0,tailprob)' or a fixed sigma specified with 'c(sigma,NA)'.")
}
if (prior.range[1] <= 0){
stop("'prior.range[1]' must be a number greater than 0 specifying a spatial range")
}
if (prior.sigma[1] <= 0){
stop("'prior.sigma[1]' must be a number greater than 0 specifying a standard deviation")
}
if (!is.na(prior.range[2]) &&
((prior.range[2] <= 0) || (prior.range[2] >= 1))) {
stop("'prior.range[2]' must be a probaility strictly between 0 and 1 (or NA to specify a fixed range)")
}
if (!is.na(prior.sigma[2]) &&
((prior.sigma[2] <= 0) || (prior.sigma[2] >= 1))) {
stop("'prior.sigma[2]' must be a probaility strictly between 0 and 1 (or NA to specify a fixed sigma)")
}
nu <- alpha-d/2
if (nu <= 0) {
stop(paste("Smoothness nu = alpha-dim/2 = ", nu,
", but must be > 0.", sep=""))
}
kappa0 <- log(8*nu)/2
tau0 <- 0.5*(lgamma(nu)-lgamma(nu+d/2)-d/2*log(4*pi))-nu*kappa0
spde <- inla.spde2.matern(mesh = mesh,
B.tau = cbind(tau0, nu, -1),
B.kappa = cbind(kappa0, -1, 0),
alpha = alpha,
param = NULL,
constr = constr,
extraconstr.int = extraconstr.int,
extraconstr = extraconstr,
fractional.method = fractional.method,
n.iid.group = n.iid.group)
## Calculate hyperparameters
is.fixed.range <- is.na(prior.range[2])
if (is.fixed.range) {
lam1 <- 0
initial.range <- log(prior.range[1])
} else {
lam1 <- -log(prior.range[2])*prior.range[1]^(d/2)
initial.range <- log(prior.range[1]) + 1
}
is.fixed.sigma <- is.na(prior.sigma[2])
if (is.fixed.sigma){
lam2 <- 0
initial.sigma <- log(prior.sigma[1])
} else{
lam2 <- -log(prior.sigma[2])/prior.sigma[1]
initial.sigma <- log(prior.sigma[1]) - 1
}
pcmatern.param = c(lam1, lam2, d)
## Change prior information
spde$f$hyper.default <-
list(theta1=list(prior="pcmatern",
param=pcmatern.param,
initial=initial.range,
fixed=is.fixed.range),
theta2=list(initial=initial.sigma,
fixed=is.fixed.sigma))
## Change the model descriptor
spde$model = "pcmatern"
invisible(spde)
}
param2.iheat <- function(mesh.space,
mesh.time,
gamma.E = NULL,
alpha.E = NULL,
theta.prior.mean = NULL,
theta.prior.prec = NULL)
{
return(list(gamma.E=gamma.E,
alpha.E=alpha.E,
theta.prior.mean=theta.prior.mean,
theta.prior.prec=theta.prior.prec))
}
inla.spde2.iheat =
function(mesh.space,
mesh.time,
order=2,
param = NULL,
constr = FALSE,
extraconstr.int = NULL,
extraconstr = NULL)
{
## order=1:
## gamma.t dot(u) - Laplacian (u) = E
##
## order=2:
## gamma.t dot(u) - Laplacian (u) = v
## gamma.t dot(v) - Laplacian (v) = E
##
## (1 - gamma.E Laplacian)^(alpha.E/2) E = W/sqrt(gamma.s)
## gamma.E and alpha.E known
inla.require.inherits(mesh.space, c("inla.mesh", "inla.mesh.1d"),
"'mesh.space'")
inla.require.inherits(mesh.time, c("inla.mesh.1d"),
"'mesh.time'")
if (is.null(param)) {
stop("Use param2.iheat() to construct the prior parameter settings")
## param =
## param2.iheat(
## mesh.space, mesh.time,
## theta.prior.mean,
## theta.prior.prec)
}
d.space = inla.ifelse(inherits(mesh.space, "inla.mesh.1d"), 1, 2)
d.time = 1
n.space = inla.ifelse(d.space==2, mesh.space$n, mesh.space$m)
n.time = mesh.time$m
n.spde = n.space*n.time
n.theta = 2L ## gamma.s, gamma.t
if (d.space==2) {
fem.space = inla.mesh.fem(mesh.space$loc, 2)
} else {
fem.space = inla.mesh.1d.fem(mesh.space)
if (mesh.space$degree==2) {
fem.space$c0 = fem.space$c1 ## Use higher order matrix.
}
}
fem.time = inla.mesh.1d.fem(mesh.time)
if (mesh.time$degree==2) {
fem.time$c0 = fem.time$c1 ## Use higher order matrix.
}
## TODO: the rest
if (FALSE) {
if (alpha==2) {
B.phi0 = param$B.tau
B.phi1 = 2*param$B.kappa
M0 = fem$c0
M1 = fem$g1
M2 = fem$g2
} else if (alpha==1) {
B.phi0 = param$B.tau
B.phi1 = param$B.kappa
M0 = fem$c0
M1 = fem$g1*0
M2 = fem$g1
} else if (!param$is.stationary) {
stop("Non-stationary Matern with fractional alpha is not implemented.")
} else if ((alpha<2) && (alpha>1)) {
if (fractional.method == "parsimonious") {
lambda = alpha-floor(alpha)
b = matrix(c(1,0,0, 1,1,0, 1,2,1),3,3) %*%
solve(matrix(1/(c(4:2, 3:1, 2:0)+lambda), 3, 3),
1/(c(4:2)+lambda-alpha))
} else if (fractional.method == "null") {
b = c(1,alpha,alpha*(alpha-1)/2)
} else {
stop(paste("Unknown fractional.method '", fractional.method,
"'.", sep=""))
}
B.phi0 = param$B.tau + (alpha-2)*param$B.kappa
B.phi1 = 2*param$B.kappa
M0 = fem$c0*b[1]
M1 = fem$g1*b[2]/2
M2 = fem$g2*b[3]
} else if ((alpha<1) && (alpha>0)) {
if (fractional.method == "parsimonious") {
lambda = alpha-floor(alpha)
b = matrix(c(1,0,1,1),2,2) %*%
solve(matrix(1/(c(2:1, 1:0)+lambda), 2, 2),
1/(c(2:1)+lambda-alpha))
} else if (fractional.method == "null") {
b = c(1,alpha)
} else {
stop(paste("Unknown fractional.method '", fractional.method,
"'.", sep=""))
}
B.phi0 = param$B.tau + (alpha-1)*param$B.kappa
B.phi1 = param$B.kappa
M0 = fem$c0*b[1]
M1 = fem$g1*0
M2 = fem$g1*b[2]
} else {
stop(paste("Unsupported alpha value (", alpha,
"). Supported values are 0 < alpha <= 2", sep=""))
}
spde =
inla.spde2.generic(M0=M0, M1=M1, M2=M2,
B0=B.phi0, B1=B.phi1, B2=1,
theta.mu = param$theta.prior.mean,
theta.Q = param$theta.prior.prec,
transform = "identity",
BLC = param$BLC)
spde$model = "matern"
spde$BLC = param$BLC
if (constr || !is.null(extraconstr.int)) {
if (constr) {
A.constr = matrix(colSums(fem$c1), 1, n.spde)
e.constr = 0
} else {
A.constr = matrix(numeric(0), 0, n.spde)
e.constr = c()
}
if (!is.null(extraconstr.int)) {
A.constr =
rbind(A.constr,
matrix(extraconstr.int$A %*% fem$c1,
nrow(extraconstr.int$A), n.spde))
e.constr = c(e.constr, extraconstr.int$e)
}
if (!is.null(extraconstr)) {
A.constr = rbind(A.constr, extraconstr$A)
e.constr = c(e.constr, extraconstr$e)
}
spde$f$constr = FALSE
spde$f$extraconstr = list(A=A.constr, e=e.constr)
} else if (!is.null(extraconstr)) {
spde$f$constr = FALSE
spde$f$extraconstr = extraconstr
}
return(invisible(spde))
}
invisible(NULL)
}
inla.spde2.theta2phi0 = function(spde, theta)
{
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (spde$n.theta>0)
return(exp(spde$param.inla$B0[,1, drop=TRUE] +
spde$param.inla$B0[,-1, drop=FALSE] %*% theta))
else
return(exp(spde$param.inla$B0[,1, drop=TRUE]))
}
inla.spde2.theta2phi1 = function(spde, theta)
{
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (spde$n.theta>0)
return(exp(spde$param.inla$B1[,1, drop=TRUE] +
spde$param.inla$B1[,-1, drop=FALSE] %*% theta))
else
return(exp(spde$param.inla$B1[,1, drop=TRUE]))
}
inla.spde2.theta2phi2 = function(spde, theta)
{
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (spde$n.theta>0) {
phi = ((spde$param.inla$B2[,1, drop=TRUE] +
spde$param.inla$B2[,-1, drop=FALSE] %*% theta))
} else {
phi = ((spde$param.inla$B2[,1, drop=TRUE]))
}
if (spde$param.inla$transform == "identity") {
return(phi)
} else if (spde$param.inla$transform == "logit") {
return(cos(pi/(1+exp(-phi))))
} else if (spde$param.inla$transform == "log") {
return(2*exp(phi)-1)
} else {
warning(paste("Unknown link function '",
spde$param.inla$transform,
"' phi2. Using identity link instead.", sep=""))
return(phi)
}
}
inla.spde2.precision =
function(spde, theta=NULL,
phi0=inla.spde2.theta2phi0(spde, theta),
phi1=inla.spde2.theta2phi1(spde, theta),
phi2=inla.spde2.theta2phi2(spde, theta),
...)
{
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (spde$f$model!="spde2") {
stop("'inla.spde2.precision' only supports internal inla models 'spde2'")
}
D0 = Diagonal(spde$n.spde, phi0)
D1 = Diagonal(spde$n.spde, phi1)
D12 = Diagonal(spde$n.spde, phi1*phi2)
Q = (D0 %*% (D1 %*% spde$param.inla$M0 %*% D1 +
D12 %*% spde$param.inla$M1 +
t(spde$param.inla$M1) %*% D12 +
spde$param.inla$M2) %*% D0)
return(Q)
}
inla.spde2.result = function(inla, name, spde, do.transform=TRUE, ...)
{
inla.require.inherits(inla, "inla", "'inla'")
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (!spde$f$model=="spde2") {
stop("'inla.spde2.result' only supports internal inla models 'spde2'")
}
result = list()
## Setup rownames
BLC.names = rownames(spde$param.inla$BLC)
## Values
result$summary.values = inla$summary.random[[name]]
## Marginals for values
if (!is.null(inla$marginals.random[[name]])) {
result$marginals.values = inla$marginals.random[[name]]
}
## Theta
result$summary.hyperpar =
inla.extract.el(inla$summary.hyperpar,
paste("Theta[^ ]+ for ", name, "$", sep=""))
## Marginals for theta
if (!is.null(inla$marginals$hyperpar)) {
result$marginals.hyperpar =
inla.extract.el(inla$marginals.hyperpar,
paste("Theta[^ ]+ for ", name, "$", sep=""))
}
## log-tau/kappa/variance/range
if (!is.null(inla$summary.spde2.blc[[name]])) {
rownames(inla$summary.spde2.blc[[name]]) <- BLC.names
result$summary.theta =
inla.extract.el(inla$summary.spde2.blc[[name]],
"theta\\.[^ ]+$")
result$summary.log.tau =
inla.extract.el(inla$summary.spde2.blc[[name]],
"tau\\.[^ ]+$")
result$summary.log.kappa =
inla.extract.el(inla$summary.spde2.blc[[name]],
"kappa\\.[^ ]+$")
result$summary.log.variance.nominal =
inla.extract.el(inla$summary.spde2.blc[[name]],
"variance.nominal\\.[^ ]+$")
result$summary.log.range.nominal =
inla.extract.el(inla$summary.spde2.blc[[name]],
"range.nominal\\.[^ ]+$")
}
## Marginals for log-tau/kappa/variance/range
if (!is.null(inla$marginals.spde2.blc[[name]])) {
names(inla$marginals.spde2.blc[[name]]) <- BLC.names
result$marginals.theta =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"theta\\.[^ ]+$")
result$marginals.log.tau =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"tau\\.[^ ]+$")
result$marginals.log.kappa =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"kappa\\.[^ ]+$")
result$marginals.log.variance.nominal =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"variance.nominal\\.[^ ]+$")
result$marginals.log.range.nominal =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"range.nominal\\.[^ ]+$")
if (do.transform) {
result$marginals.tau =
lapply(result$marginals.log.tau,
function(x) inla.tmarginal(function(y) exp(y), x))
result$marginals.kappa =
lapply(result$marginals.log.kappa,
function(x) inla.tmarginal(function(y) exp(y), x))
result$marginals.variance.nominal =
lapply(result$marginals.log.variance.nominal,
function(x) inla.tmarginal(function(y) exp(y), x))
result$marginals.range.nominal =
lapply(result$marginals.log.range.nominal,
function(x) inla.tmarginal(function(y) exp(y), x))
}
}
return(result)
}
inla.spde2.models = function()
{
return(c("generic", "matern", "pcmatern"))
}
## spde.common-connections:
inla.spde.precision.inla.spde2 = inla.spde2.precision
inla.spde.result.inla.spde2 = inla.spde2.result
## Deprecated:
##inla.spde.generic2 = inla.spde2.generic
INBO-BMK/INLA documentation built on Dec. 4, 2019, 11:43 p.m.
R Package Documentation
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Defines functions inla.spde2.generic inla.internal.spde2.matern.B.tau inla.spde2.matern.sd.basis param2.matern.orig inla.spde2.matern inla.spde2.pcmatern param2.iheat inla.spde2.iheat inla.spde2.theta2phi0 inla.spde2.theta2phi1 inla.spde2.theta2phi2 inla.spde2.precision inla.spde2.result inla.spde2.models
Documented in inla.spde2.generic inla.spde2.matern inla.spde2.matern.sd.basis inla.spde2.models inla.spde2.pcmatern inla.spde2.precision inla.spde2.result inla.spde2.theta2phi0 inla.spde2.theta2phi1 inla.spde2.theta2phi2 param2.matern.orig
## 'spde2' model functions
## Export: inla.spde.precision!inla.spde2
## Export: inla.spde.result!inla.spde2 inla.spde2.generic
## Export: inla.spde2.matern param2.matern.orig
## Export: inla.spde2.matern.sd.basis inla.spde2.models
## Export: inla.spde2.pcmatern
## Export: inla.spde2.precision inla.spde2.result
## Export: inla.spde2.theta2phi0 inla.spde2.theta2phi1 inla.spde2.theta2phi2
## Internal: inla.internal.spde2.matern.B.tau
## Internal: inla.internal.test.spde2.sd.basis
inla.spde2.generic =
function(M0, M1, M2, B0, B1, B2, theta.mu, theta.Q,
transform = c("logit","log","identity"),
theta.initial = theta.mu,
fixed = rep(FALSE, length(theta.mu)),
theta.fixed = theta.initial[fixed],
BLC = cbind(0.0, diag(nrow=length(theta.mu))),
...)
{
transform = match.arg(transform)
if (!inla.is.matrix(M0)) stop("M0 must be of a matrix type.")
if (!inla.is.matrix(M1)) stop("M1 must be of a matrix type.")
if (!inla.is.matrix(M2)) stop("M2 must be of a matrix type.")
M0 = inla.as.dgTMatrix(M0)
M1 = inla.as.dgTMatrix(M1)
M2 = inla.as.dgTMatrix(M2)
n.spde = nrow(M0)
n.theta = length(theta.mu)
spde = (list(model = "generic",
n.spde = n.spde,
n.theta = n.theta,
param.inla = list(),
f = list()
))
class(spde) = c("inla.spde2", "inla.spde", "inla.model.class")
B0 = inla.spde.homogenise_B_matrix(B0, n.spde, n.theta)
B1 = inla.spde.homogenise_B_matrix(B1, n.spde, n.theta)
B2 = inla.spde.homogenise_B_matrix(B2, n.spde, n.theta)
BLC = inla.spde.homogenise_B_matrix(BLC, nrow(BLC), n.theta)
param.inla =
list(n = n.spde,
n.theta = n.theta,
M0=M0, M1=M1, M2=M2,
B0=B0, B1=B1, B2=B2, BLC=BLC,
theta.mu=theta.mu, theta.Q=theta.Q,
transform=transform,
theta.initial=theta.initial,
fixed=fixed,
theta.fixed=theta.fixed)
## TODO: Check dimension consistency of inputs
## If any "fixed", move information from B[,1+(1:n.theta)] into
## B[,1], theta.mu, and theta.Q
## NOTE: Should this be in the f function instead?
## No, the information is also needed elsewhere.
if (any(fixed)) {
param.inla$theta.initial = theta.initial[!fixed]
param.inla$B0[,1] =
B0[,c(TRUE, fixed), drop=FALSE] %*% c(1.0, theta.fixed)
param.inla$B1[,1] =
B1[,c(TRUE, fixed), drop=FALSE] %*% c(1.0, theta.fixed)
param.inla$B2[,1] =
B2[,c(TRUE, fixed), drop=FALSE] %*% c(1.0, theta.fixed)
param.inla$BLC[,1] =
BLC[,c(TRUE, fixed), drop=FALSE] %*% c(1.0, theta.fixed)
param.inla$B0 = param.inla$B0[,c(TRUE, !fixed), drop=FALSE]
param.inla$B1 = param.inla$B1[,c(TRUE, !fixed), drop=FALSE]
param.inla$B2 = param.inla$B2[,c(TRUE, !fixed), drop=FALSE]
param.inla$BLC = param.inla$BLC[,c(TRUE, !fixed), drop=FALSE]
param.inla$theta.Q = param.inla$theta.Q[!fixed, !fixed, drop=FALSE]
if (!all(fixed)) {
param.inla$theta.mu =
theta.mu - solve(param.inla$theta.Q,
theta.Q[!fixed, fixed, drop=FALSE]
%*% (theta.fixed[fixed]-theta.mu[fixed]))
} else { ## All fixed.
param.inla$theta.mu = as.vector(matrix(0.0, 0, 1))
}
}
## Remove all BLC rows that would produce point masses, which is
## not supported by inla:
if (n.theta==0) {
param.inla$BLC = matrix(0,0,1)
} else {
param.inla$BLC =
param.inla$BLC[rowSums(abs(param.inla$BLC[,-1,drop=FALSE]))>0,,
drop=FALSE]
}
spde$param.inla = param.inla
spde$f = (list(model="spde2",
n=n.spde,
spde2.transform=transform,
hyper.default =
(list(theta1 =
list(prior="mvnorm",
param=(c(param.inla$theta.mu,
as.matrix(param.inla$theta.Q)))
)
))
))
if (n.theta>0)
for (k in 1:n.theta) {
eval(parse(text=
paste("spde$f$hyper.default$theta", k,
"$initial = param.inla$theta.mu[k]", sep="")))
}
return(spde)
}
inla.internal.spde2.matern.B.tau =
function(precision.fcn, n.theta, delta = 1e-3, ...)
{
theta.0 = rep(0, n.theta)
Q.0 = precision.fcn(theta.0, ...)
logv.0 = log(diag(inla.qinv(Q.0)))
B.tau = matrix(0, nrow(Q.0), 1L+n.theta)
for (k in seq_len(n.theta)) {
theta = theta.0
theta[k] = theta[k] + delta
Q = precision.fcn(theta, ...)
logv = log(diag(inla.qinv(Q)))
dlogv = (logv-logv.0)/delta
B.tau[, k+1] = dlogv/2
}
B.tau[,1L] = logv.0/2
return(B.tau)
}
inla.spde2.matern.sd.basis =
function(mesh, B.sd, B.range, method=1,
local.offset.compensation=FALSE,
alpha=2, ...)
{
p.sd = ncol(B.sd)-1L
p.kappa = ncol(B.range)-1L
d = inla.ifelse(inherits(mesh, "inla.mesh"), 2, 1)
nu = alpha-d/2
nu.nominal = max(0.5, nu)
alpha.nominal = max(nu.nominal+d/2, alpha)
n.spde = inla.ifelse(d==2, mesh$n, mesh$m)
## log(kappa) = 0.5*log(8*nu) - log(range)
B.kappa = (cbind(log(8*nu.nominal)/2 - B.range[,1],
-B.range[,-1,drop=FALSE] ))
if (method==1) {
spde =
inla.spde2.matern(mesh, B.tau=as.matrix(0), B.kappa=B.kappa, ...)
B.tau =
inla.internal.spde2.matern.B.tau(
precision.fcn=
function(theta, spde) {
return(inla.spde2.precision(spde, theta=theta))
},
n.theta = p.kappa,
spde = spde)
if (!local.offset.compensation) {
B.tau[,1] = rep(mean(B.tau[,1]), spde$n.spde)
}
} else {
spde =
inla.spde2.matern(mesh,
B.tau=as.matrix(0),
B.kappa=B.kappa[,1,drop=FALSE],
alpha=alpha, ...)
lvar.0 =
log(diag(inla.qinv(inla.spde2.precision(spde, theta=rep(0,0)))))
if (!local.offset.compensation) {
lvar.0 = rep(mean(lvar.0), spde$n.spde)
}
B.tau = -B.kappa*nu
B.tau[,1] = lvar.0/2
}
log.var.scaling = lgamma(nu.nominal)-lgamma(alpha.nominal)-log(4*pi)*d/2
B.tau =
cbind(B.tau[,1]-B.sd[,1],
-B.sd[,-1,drop=FALSE],
B.tau[,-1,drop=FALSE])
B.kappa =
cbind(B.kappa[,1],
matrix(0, n.spde, p.sd),
B.kappa[,-1,drop=FALSE])
return(list(B.tau=B.tau, B.kappa=B.kappa))
}
inla.internal.test.spde2.sd.basis = function (k=1, dth=0.1, r=1000, globe=25, compensate=FALSE) {
alpha=2
mesh = inla.mesh.create(globe=globe)
B.common = inla.mesh.basis(mesh, "b.spline", n=3)
B.common = cbind(B.common, B.common[,1]*(mesh$loc[,3]>0))
B.common[,1] = B.common[,1]*(mesh$loc[,3]<=0)
B.sd = cbind(0, B.common)
B.range = cbind(log(r/6370), B.common)
sd.basis0 = inla.spde2.matern.sd.basis(mesh, B.sd=B.sd, B.range=B.range, method=0, alpha=alpha, local.offset.compensation=compensate)
sd.basis1 = inla.spde2.matern.sd.basis(mesh, B.sd=B.sd, B.range=B.range, method=1, alpha=alpha, local.offset.compensation=compensate)
spde0 =
inla.spde2.matern(mesh, alpha=alpha,
B.tau=sd.basis0$B.tau,
B.kappa=sd.basis0$B.kappa)
spde1 =
inla.spde2.matern(mesh, alpha=alpha,
B.tau=sd.basis1$B.tau,
B.kappa=sd.basis1$B.kappa)
th0 = c(0,0,0,0,0,0,0,0)
th1 = th0
th1[k] = th0[k]+dth
v0.0=diag(inla.qinv(inla.spde2.precision(spde0,theta=th0)))
v0.1=diag(inla.qinv(inla.spde2.precision(spde0,theta=th1)))
v1.0=diag(inla.qinv(inla.spde2.precision(spde1,theta=th0)))
v1.1=diag(inla.qinv(inla.spde2.precision(spde1,theta=th1)))
op=par(mfrow=c(2,2))
on.exit(par(op))
plot(asin(mesh$loc[,3]),rowSums(sd.basis0$B.kappa[,1+k,drop=FALSE]), pch=20,
ylim=(range(c(0, range(rowSums(sd.basis0$B.kappa[,1+k,drop=FALSE])),
range(-rowSums(sd.basis0$B.tau[,1+k,drop=FALSE])),
range(-rowSums(sd.basis1$B.tau[,1+k,drop=FALSE]))))),
main="Basis functions")
points(asin(mesh$loc[,3]),-rowSums(sd.basis0$B.tau[,1+k,drop=FALSE]),col=2,pch=20)
points(asin(mesh$loc[,3]),-rowSums(sd.basis1$B.tau[,1+k,drop=FALSE]),col=4,pch=20)
legend("bottomright",
legend=paste("Method", 0:1),
col=c(2,4), pch=20)
plot(asin(mesh$loc[,3]),v0.1, col=2,pch=20,
ylim=range(c(0,range(v0.0),range(v1.0),range(v0.1),range(v1.1))),
main="Absolute variances")
points(asin(mesh$loc[,3]),v1.1, col=4,pch=20)
points(asin(mesh$loc[,3]),v0.0, col=2,pch=1)
points(asin(mesh$loc[,3]),v1.0, col=4,pch=1)
legend("bottomright",
legend=paste("Method", 0:1),
col=c(2,4), pch=20)
plot(asin(mesh$loc[,3]),v0.1-v0.0, col=2,pch=20,
ylim=range(c(0,range(v0.1-v0.0),range(v1.1-v1.0))),
main="Absolute variance error")
points(asin(mesh$loc[,3]),v1.1-v1.0, col=4,pch=20)
legend("bottomright",
legend=paste("Method", 0:1),
col=c(2,4), pch=20)
plot(asin(mesh$loc[,3]),v0.1/v0.0-1, col=2,pch=20,
ylim=range(c(0,range(v0.1/v0.0-1),range(v1.1/v1.0-1))),
main="Relative variance error")
points(asin(mesh$loc[,3]),v1.1/v1.0-1, col=4,pch=20)
legend("bottomright",
legend=paste("Method", 0:1),
col=c(2,4), pch=20)
return(environment())
}
param2.matern.orig =
function(mesh,
alpha=2,
B.tau = matrix(c(0,1,0),1,3),
B.kappa = matrix(c(0,0,1),1,3),
prior.variance.nominal = 1,
prior.range.nominal = NULL,
prior.tau = NULL,
prior.kappa = NULL,
theta.prior.mean = NULL,
theta.prior.prec = 0.1)
{
## NOTE: For d==1, degree==2, the B.* must be given per basis
## function, not per knot.
inla.require.inherits(mesh, c("inla.mesh", "inla.mesh.1d"), "'mesh'")
if (is.null(B.tau))
stop("B.tau must not be NULL.")
if (is.null(B.kappa))
stop("B.kappa must not be NULL.")
is.stationary = (nrow(B.kappa)==1) && (nrow(B.tau)==1)
d = inla.ifelse(inherits(mesh, "inla.mesh"), 2, 1)
nu = alpha-d/2
nu.nominal = max(0.5, nu)
alpha.nominal = max(nu.nominal+d/2, alpha)
n.spde = inla.ifelse(d==2, mesh$n, mesh$m)
n.theta = ncol(B.kappa)-1L
B.kappa = inla.spde.homogenise_B_matrix(B.kappa, n.spde, n.theta)
B.tau = inla.spde.homogenise_B_matrix(B.tau, n.spde, n.theta)
B.prec =
cbind(+ lgamma(alpha.nominal) - lgamma(nu.nominal)
+ (d/2)*log(4*pi) + 2*nu.nominal*B.kappa[,1]
+ 2*B.tau[,1],
2*nu.nominal*B.kappa[,-1,drop=FALSE]
+ 2*B.tau[,-1,drop=FALSE] )
if (is.stationary) {
B.tau = B.tau[1,,drop=FALSE]
B.kappa = B.kappa[1,,drop=FALSE]
B.prec = B.prec[1,,drop=FALSE]
}
B.variance = -B.prec
B.range =
cbind(0.5*log(8*nu.nominal)-B.kappa[,1],
-B.kappa[,-1,drop=FALSE])
if (n.theta>0) {
B.theta = cbind(0,diag(1, n.theta))
rownames(B.theta) <- rownames(B.theta, do.NULL=FALSE, prefix="theta.")
} else {
B.theta = NULL
}
rownames(B.tau) <- rownames(B.tau, do.NULL=FALSE, prefix="tau.")
rownames(B.kappa) <- rownames(B.kappa, do.NULL=FALSE, prefix="kappa.")
rownames(B.variance) <-
rownames(B.variance, do.NULL=FALSE, prefix="variance.nominal.")
rownames(B.range) <-
rownames(B.range, do.NULL=FALSE, prefix="range.nominal.")
BLC = rbind(B.theta, B.tau, B.kappa, B.variance, B.range)
## Construct priors.
if (is.null(theta.prior.prec)) {
theta.prior.prec = diag(0.1, n.theta, n.theta)
} else {
theta.prior.prec = as.matrix(theta.prior.prec)
if (ncol(theta.prior.prec) == 1) {
theta.prior.prec =
diag(as.vector(theta.prior.prec), n.theta, n.theta)
}
if ((nrow(theta.prior.prec) != n.theta) ||
(ncol(theta.prior.prec) != n.theta)) {
stop(paste("Size of theta.prior.prec is (",
paste(dim(theta.prior.prec), collapse=",", sep=""),
") but should be (",
paste(c(n.theta, n.theta), collapse=",", sep=""),
")."))
}
}
if (is.null(theta.prior.mean)) {
if (is.null(prior.range.nominal)) {
mesh.range =
inla.ifelse(d==2,
(max(c(diff(range(mesh$loc[,1])),
diff(range(mesh$loc[,2])),
diff(range(mesh$loc[,3]))
))),
diff(mesh$interval))
prior.range.nominal = mesh.range*0.2
} else {
if (!is.numeric(prior.range.nominal) ||
(length(prior.range.nominal) != 1)) {
stop(paste0("'prior.range.nominal' must be NULL or a single scalar value.\n",
"Did you intend to supply 'prior.range' to inla.spde2.pcmatern instead?"))
}
}
if (is.null(prior.kappa)) {
prior.kappa = sqrt(8*nu.nominal)/prior.range.nominal
}
if (is.null(prior.tau)) {
prior.tau =
sqrt(gamma(nu.nominal)/gamma(alpha.nominal)/
(4*pi*prior.kappa^(2*nu.nominal)*prior.variance.nominal))
}
if (n.theta>0) {
theta.prior.mean =
qr.solve(rbind(B.tau[,-1,drop=FALSE], B.kappa[,-1,drop=FALSE]),
c(log(prior.tau) - B.tau[,1],
log(prior.kappa) - B.kappa[,1]))
} else {
theta.prior.mean = rep(0, n.theta) ## Empty vector
}
}
param =
list(is.stationary=is.stationary,
B.tau=B.tau, B.kappa=B.kappa, BLC=BLC,
theta.prior.mean=theta.prior.mean,
theta.prior.prec=theta.prior.prec)
return(param)
}
inla.spde2.matern =
function(mesh,
alpha=2,
param = NULL,
constr = FALSE,
extraconstr.int = NULL,
extraconstr = NULL,
fractional.method = c("parsimonious", "null"),
B.tau = matrix(c(0,1,0),1,3),
B.kappa = matrix(c(0,0,1),1,3),
prior.variance.nominal = 1,
prior.range.nominal = NULL,
prior.tau = NULL,
prior.kappa = NULL,
theta.prior.mean = NULL,
theta.prior.prec = 0.1,
n.iid.group = 1,
...)
{
if(!is.null(list(...)[["prior.pc.rho"]]) &&
!is.null(list(...)[["prior.pc.sig"]])){
## Temporary implementation of PC prior for standard deviation and range
## - Changes parametrization to range and standard deviation
## - Sets prior according to hyperparameters for range : prior.pc.rho
## and std.dev.: prior.pc.sig
warning("You're using a deprecated experimental PC prior matern model that will be removed in a future version of the package. Use 'inla.spde2.pcmatern' instead.")
prior.pc.rho <- list(...)[["prior.pc.rho"]]
prior.pc.sig <- list(...)[["prior.pc.sig"]]
## Call inla.spde2.matern with range and standard deviation parametrization
d = inla.ifelse(inherits(mesh, "inla.mesh"), 2, 1)
nu = alpha-d/2
kappa0 = log(8*nu)/2
tau0 = 0.5*(lgamma(nu)-lgamma(nu+d/2)-d/2*log(4*pi))-nu*kappa0
spde = inla.spde2.matern(mesh = mesh,
B.tau = cbind(tau0, nu, -1),
B.kappa = cbind(kappa0, -1, 0))
## Change prior information
param = c(prior.pc.rho, prior.pc.sig)
spde$f$hyper.default$theta1$prior = "pcspdega"
spde$f$hyper.default$theta1$param = param
spde$f$hyper.default$theta1$initial = log(prior.pc.rho[1])+1
spde$f$hyper.default$theta2$initial = log(prior.pc.sig[1])-1
## End and return
return(invisible(spde))
}
## Standard code
inla.require.inherits(mesh, c("inla.mesh", "inla.mesh.1d"), "'mesh'")
fractional.method = match.arg(fractional.method)
if (is.null(param)) {
param =
param2.matern.orig(
mesh, alpha,
B.tau, B.kappa,
prior.variance.nominal,
prior.range.nominal,
prior.tau,
prior.kappa,
theta.prior.mean,
theta.prior.prec)
} else {
deprecated =
!c(missing(B.tau), missing(B.kappa),
missing(prior.variance.nominal),
missing(prior.range.nominal),
missing(prior.tau), missing(prior.kappa),
missing(theta.prior.mean), missing(theta.prior.prec))
deprecated =
c("B.tau", "B.kappa",
"prior.variance.nominal",
"prior.range.nominal",
"prior.tau", "prior.kappa",
"theta.prior.mean", "theta.prior.prec")[deprecated]
if (length(deprecated) > 0) {
warning(paste("'param' specified; ",
"Ignoring deprecated parameter(s) ",
paste(deprecated, collapse=", "), ".", sep=""))
}
}
d = inla.ifelse(inherits(mesh, "inla.mesh"), 2, 1)
nu = alpha-d/2
nu.nominal = max(0.5, nu)
alpha.nominal = max(nu.nominal+d/2, alpha)
n.spde = inla.ifelse(d==2, mesh$n, mesh$m)
n.theta = ncol(B.kappa)-1L
if (d==2) {
fem =
inla.fmesher.smorg(mesh$loc,
mesh$graph$tv,
fem=2,
output=list("c0", "c1", "g1", "g2"))
} else {
fem = inla.mesh.1d.fem(mesh)
if (mesh$degree==2) {
fem$c0 = fem$c1 ## Use higher order matrix.
}
}
if (alpha==2) {
B.phi0 = param$B.tau
B.phi1 = 2*param$B.kappa
M0 = fem$c0
M1 = fem$g1
M2 = fem$g2
} else if (alpha==1) {
B.phi0 = param$B.tau
B.phi1 = param$B.kappa
M0 = fem$c0
M1 = fem$g1*0
M2 = fem$g1
} else if (!param$is.stationary) {
stop("Non-stationary Matern with fractional alpha is not implemented.")
} else if ((alpha<2) && (alpha>1)) {
if (fractional.method == "parsimonious") {
lambda = alpha-floor(alpha)
b = matrix(c(1,0,0, 1,1,0, 1,2,1),3,3) %*%
solve(matrix(1/(c(4:2, 3:1, 2:0)+lambda), 3, 3),
1/(c(4:2)+lambda-alpha))
} else if (fractional.method == "null") {
b = c(1,alpha,alpha*(alpha-1)/2)
} else {
stop(paste("Unknown fractional.method '", fractional.method,
"'.", sep=""))
}
B.phi0 = param$B.tau + (alpha-2)*param$B.kappa
B.phi1 = 2*param$B.kappa
M0 = fem$c0*b[1]
M1 = fem$g1*b[2]/2
M2 = fem$g2*b[3]
} else if ((alpha<1) && (alpha>0)) {
if (fractional.method == "parsimonious") {
lambda = alpha-floor(alpha)
b = matrix(c(1,0,1,1),2,2) %*%
solve(matrix(1/(c(2:1, 1:0)+lambda), 2, 2),
1/(c(2:1)+lambda-alpha))
} else if (fractional.method == "null") {
b = c(1,alpha)
} else {
stop(paste("Unknown fractional.method '", fractional.method,
"'.", sep=""))
}
B.phi0 = param$B.tau + (alpha-1)*param$B.kappa
B.phi1 = param$B.kappa
M0 = fem$c0*b[1]
M1 = fem$g1*0
M2 = fem$g1*b[2]
} else {
stop(paste("Unsupported alpha value (", alpha,
"). Supported values are 0 < alpha <= 2", sep=""))
}
if (n.iid.group == 1) {
spde =
inla.spde2.generic(M0=M0, M1=M1, M2=M2,
B0=B.phi0, B1=B.phi1, B2=1,
theta.mu = param$theta.prior.mean,
theta.Q = param$theta.prior.prec,
transform = "identity",
BLC = param$BLC)
} else {
if (nrow(B.phi0) > 1) {
B.phi0 <- kronecker(matrix(1, n.iid.group, 1), B.phi0)
}
if (nrow(B.phi1) > 1) {
B.phi1 <- kronecker(matrix(1, n.iid.group, 1), B.phi1)
}
spde =
inla.spde2.generic(M0=kronecker(Diagonal(n.iid.group), M0),
M1=kronecker(Diagonal(n.iid.group), M1),
M2=kronecker(Diagonal(n.iid.group), M2),
B0=B.phi0, B1=B.phi1, B2=1,
theta.mu = param$theta.prior.mean,
theta.Q = param$theta.prior.prec,
transform = "identity",
BLC = param$BLC)
}
spde$model = "matern"
spde$BLC = param$BLC
if (constr || !is.null(extraconstr.int) || !is.null(extraconstr)) {
A.constr = matrix(numeric(0), 0, n.spde*n.iid.group)
e.constr = matrix(numeric(0), 0, 1)
if (constr) {
A.constr <- rbind(A.constr,
matrix(colSums(fem$c1)/n.iid.group,
1, n.spde*n.iid.group))
e.constr <- rbind(e.constr, 0)
}
if (!is.null(extraconstr.int)) {
if (ncol(extraconstr.int$A) == n.spde) {
A.constr <-
rbind(A.constr,
kronecker(matrix(1/n.iid.group, 1, n.iid.group),
as.matrix(extraconstr.int$A %*% fem$c1)))
} else {
A.constr <-
rbind(A.constr,
as.matrix(extraconstr.int$A %*%
kronecker(Diagonal(n.iid.group),
fem$c1)))
}
e.constr <- rbind(e.constr, as.matrix(extraconstr.int$e))
}
if (!is.null(extraconstr)) {
if (ncol(extraconstr$A) == n.spde) {
A.constr <-
rbind(A.constr,
kronecker(matrix(1/n.iid.group, 1, n.iid.group),
as.matrix(extraconstr$A)))
} else {
A.constr <- rbind(A.constr, as.matrix(extraconstr$A))
}
e.constr <- rbind(e.constr, as.matrix(extraconstr$e))
}
spde$f$constr = FALSE
spde$f$extraconstr = list(A=A.constr, e=e.constr)
}
## Attach the mesh, so downstream code can have access
spde$mesh <- mesh
return(invisible(spde))
}
inla.spde2.pcmatern =
function(mesh,
alpha = 2,
param = NULL,
constr = FALSE,
extraconstr.int = NULL,
extraconstr = NULL,
fractional.method = c("parsimonious", "null"),
n.iid.group = 1,
prior.range = NULL,
prior.sigma = NULL)
{
## Implementation of PC prior for standard deviation and range
## - Sets the parametrization to range and standard deviation
## - Sets prior according to hyperparameters for range : prior.range
## and std.dev.: prior.sigma
## Calls inla.spde2.matern to construct the object, then changes the prior
if (inherits(mesh, "inla.mesh")) {
d <- 2
} else if (inherits(mesh, "inla.mesh.1d")) {
d <- 1
} else {
stop(paste("Unknown mesh class '",
paste(class(mesh), collapse=",", sep=""),
"'.", sep=""))
}
if (missing(prior.range) || is.null(prior.range) ||
!is.vector(prior.range) || (length(prior.range) != 2)) {
stop("'prior.range' should be a length 2 vector 'c(range0,tailprob)' or a fixed range specified with 'c(range,NA)'.")
}
if (missing(prior.sigma) || is.null(prior.sigma) ||
!is.vector(prior.sigma) || (length(prior.sigma) != 2)) {
stop("'prior.sigma' should be a length 2 vector 'c(sigma0,tailprob)' or a fixed sigma specified with 'c(sigma,NA)'.")
}
if (prior.range[1] <= 0){
stop("'prior.range[1]' must be a number greater than 0 specifying a spatial range")
}
if (prior.sigma[1] <= 0){
stop("'prior.sigma[1]' must be a number greater than 0 specifying a standard deviation")
}
if (!is.na(prior.range[2]) &&
((prior.range[2] <= 0) || (prior.range[2] >= 1))) {
stop("'prior.range[2]' must be a probaility strictly between 0 and 1 (or NA to specify a fixed range)")
}
if (!is.na(prior.sigma[2]) &&
((prior.sigma[2] <= 0) || (prior.sigma[2] >= 1))) {
stop("'prior.sigma[2]' must be a probaility strictly between 0 and 1 (or NA to specify a fixed sigma)")
}
nu <- alpha-d/2
if (nu <= 0) {
stop(paste("Smoothness nu = alpha-dim/2 = ", nu,
", but must be > 0.", sep=""))
}
kappa0 <- log(8*nu)/2
tau0 <- 0.5*(lgamma(nu)-lgamma(nu+d/2)-d/2*log(4*pi))-nu*kappa0
spde <- inla.spde2.matern(mesh = mesh,
B.tau = cbind(tau0, nu, -1),
B.kappa = cbind(kappa0, -1, 0),
alpha = alpha,
param = NULL,
constr = constr,
extraconstr.int = extraconstr.int,
extraconstr = extraconstr,
fractional.method = fractional.method,
n.iid.group = n.iid.group)
## Calculate hyperparameters
is.fixed.range <- is.na(prior.range[2])
if (is.fixed.range) {
lam1 <- 0
initial.range <- log(prior.range[1])
} else {
lam1 <- -log(prior.range[2])*prior.range[1]^(d/2)
initial.range <- log(prior.range[1]) + 1
}
is.fixed.sigma <- is.na(prior.sigma[2])
if (is.fixed.sigma){
lam2 <- 0
initial.sigma <- log(prior.sigma[1])
} else{
lam2 <- -log(prior.sigma[2])/prior.sigma[1]
initial.sigma <- log(prior.sigma[1]) - 1
}
pcmatern.param = c(lam1, lam2, d)
## Change prior information
spde$f$hyper.default <-
list(theta1=list(prior="pcmatern",
param=pcmatern.param,
initial=initial.range,
fixed=is.fixed.range),
theta2=list(initial=initial.sigma,
fixed=is.fixed.sigma))
## Change the model descriptor
spde$model = "pcmatern"
invisible(spde)
}
param2.iheat <- function(mesh.space,
mesh.time,
gamma.E = NULL,
alpha.E = NULL,
theta.prior.mean = NULL,
theta.prior.prec = NULL)
{
return(list(gamma.E=gamma.E,
alpha.E=alpha.E,
theta.prior.mean=theta.prior.mean,
theta.prior.prec=theta.prior.prec))
}
inla.spde2.iheat =
function(mesh.space,
mesh.time,
order=2,
param = NULL,
constr = FALSE,
extraconstr.int = NULL,
extraconstr = NULL)
{
## order=1:
## gamma.t dot(u) - Laplacian (u) = E
##
## order=2:
## gamma.t dot(u) - Laplacian (u) = v
## gamma.t dot(v) - Laplacian (v) = E
##
## (1 - gamma.E Laplacian)^(alpha.E/2) E = W/sqrt(gamma.s)
## gamma.E and alpha.E known
inla.require.inherits(mesh.space, c("inla.mesh", "inla.mesh.1d"),
"'mesh.space'")
inla.require.inherits(mesh.time, c("inla.mesh.1d"),
"'mesh.time'")
if (is.null(param)) {
stop("Use param2.iheat() to construct the prior parameter settings")
## param =
## param2.iheat(
## mesh.space, mesh.time,
## theta.prior.mean,
## theta.prior.prec)
}
d.space = inla.ifelse(inherits(mesh.space, "inla.mesh.1d"), 1, 2)
d.time = 1
n.space = inla.ifelse(d.space==2, mesh.space$n, mesh.space$m)
n.time = mesh.time$m
n.spde = n.space*n.time
n.theta = 2L ## gamma.s, gamma.t
if (d.space==2) {
fem.space = inla.mesh.fem(mesh.space$loc, 2)
} else {
fem.space = inla.mesh.1d.fem(mesh.space)
if (mesh.space$degree==2) {
fem.space$c0 = fem.space$c1 ## Use higher order matrix.
}
}
fem.time = inla.mesh.1d.fem(mesh.time)
if (mesh.time$degree==2) {
fem.time$c0 = fem.time$c1 ## Use higher order matrix.
}
## TODO: the rest
if (FALSE) {
if (alpha==2) {
B.phi0 = param$B.tau
B.phi1 = 2*param$B.kappa
M0 = fem$c0
M1 = fem$g1
M2 = fem$g2
} else if (alpha==1) {
B.phi0 = param$B.tau
B.phi1 = param$B.kappa
M0 = fem$c0
M1 = fem$g1*0
M2 = fem$g1
} else if (!param$is.stationary) {
stop("Non-stationary Matern with fractional alpha is not implemented.")
} else if ((alpha<2) && (alpha>1)) {
if (fractional.method == "parsimonious") {
lambda = alpha-floor(alpha)
b = matrix(c(1,0,0, 1,1,0, 1,2,1),3,3) %*%
solve(matrix(1/(c(4:2, 3:1, 2:0)+lambda), 3, 3),
1/(c(4:2)+lambda-alpha))
} else if (fractional.method == "null") {
b = c(1,alpha,alpha*(alpha-1)/2)
} else {
stop(paste("Unknown fractional.method '", fractional.method,
"'.", sep=""))
}
B.phi0 = param$B.tau + (alpha-2)*param$B.kappa
B.phi1 = 2*param$B.kappa
M0 = fem$c0*b[1]
M1 = fem$g1*b[2]/2
M2 = fem$g2*b[3]
} else if ((alpha<1) && (alpha>0)) {
if (fractional.method == "parsimonious") {
lambda = alpha-floor(alpha)
b = matrix(c(1,0,1,1),2,2) %*%
solve(matrix(1/(c(2:1, 1:0)+lambda), 2, 2),
1/(c(2:1)+lambda-alpha))
} else if (fractional.method == "null") {
b = c(1,alpha)
} else {
stop(paste("Unknown fractional.method '", fractional.method,
"'.", sep=""))
}
B.phi0 = param$B.tau + (alpha-1)*param$B.kappa
B.phi1 = param$B.kappa
M0 = fem$c0*b[1]
M1 = fem$g1*0
M2 = fem$g1*b[2]
} else {
stop(paste("Unsupported alpha value (", alpha,
"). Supported values are 0 < alpha <= 2", sep=""))
}
spde =
inla.spde2.generic(M0=M0, M1=M1, M2=M2,
B0=B.phi0, B1=B.phi1, B2=1,
theta.mu = param$theta.prior.mean,
theta.Q = param$theta.prior.prec,
transform = "identity",
BLC = param$BLC)
spde$model = "matern"
spde$BLC = param$BLC
if (constr || !is.null(extraconstr.int)) {
if (constr) {
A.constr = matrix(colSums(fem$c1), 1, n.spde)
e.constr = 0
} else {
A.constr = matrix(numeric(0), 0, n.spde)
e.constr = c()
}
if (!is.null(extraconstr.int)) {
A.constr =
rbind(A.constr,
matrix(extraconstr.int$A %*% fem$c1,
nrow(extraconstr.int$A), n.spde))
e.constr = c(e.constr, extraconstr.int$e)
}
if (!is.null(extraconstr)) {
A.constr = rbind(A.constr, extraconstr$A)
e.constr = c(e.constr, extraconstr$e)
}
spde$f$constr = FALSE
spde$f$extraconstr = list(A=A.constr, e=e.constr)
} else if (!is.null(extraconstr)) {
spde$f$constr = FALSE
spde$f$extraconstr = extraconstr
}
return(invisible(spde))
}
invisible(NULL)
}
inla.spde2.theta2phi0 = function(spde, theta)
{
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (spde$n.theta>0)
return(exp(spde$param.inla$B0[,1, drop=TRUE] +
spde$param.inla$B0[,-1, drop=FALSE] %*% theta))
else
return(exp(spde$param.inla$B0[,1, drop=TRUE]))
}
inla.spde2.theta2phi1 = function(spde, theta)
{
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (spde$n.theta>0)
return(exp(spde$param.inla$B1[,1, drop=TRUE] +
spde$param.inla$B1[,-1, drop=FALSE] %*% theta))
else
return(exp(spde$param.inla$B1[,1, drop=TRUE]))
}
inla.spde2.theta2phi2 = function(spde, theta)
{
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (spde$n.theta>0) {
phi = ((spde$param.inla$B2[,1, drop=TRUE] +
spde$param.inla$B2[,-1, drop=FALSE] %*% theta))
} else {
phi = ((spde$param.inla$B2[,1, drop=TRUE]))
}
if (spde$param.inla$transform == "identity") {
return(phi)
} else if (spde$param.inla$transform == "logit") {
return(cos(pi/(1+exp(-phi))))
} else if (spde$param.inla$transform == "log") {
return(2*exp(phi)-1)
} else {
warning(paste("Unknown link function '",
spde$param.inla$transform,
"' phi2. Using identity link instead.", sep=""))
return(phi)
}
}
inla.spde2.precision =
function(spde, theta=NULL,
phi0=inla.spde2.theta2phi0(spde, theta),
phi1=inla.spde2.theta2phi1(spde, theta),
phi2=inla.spde2.theta2phi2(spde, theta),
...)
{
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (spde$f$model!="spde2") {
stop("'inla.spde2.precision' only supports internal inla models 'spde2'")
}
D0 = Diagonal(spde$n.spde, phi0)
D1 = Diagonal(spde$n.spde, phi1)
D12 = Diagonal(spde$n.spde, phi1*phi2)
Q = (D0 %*% (D1 %*% spde$param.inla$M0 %*% D1 +
D12 %*% spde$param.inla$M1 +
t(spde$param.inla$M1) %*% D12 +
spde$param.inla$M2) %*% D0)
return(Q)
}
inla.spde2.result = function(inla, name, spde, do.transform=TRUE, ...)
{
inla.require.inherits(inla, "inla", "'inla'")
inla.require.inherits(spde, "inla.spde2", "'spde'")
if (!spde$f$model=="spde2") {
stop("'inla.spde2.result' only supports internal inla models 'spde2'")
}
result = list()
## Setup rownames
BLC.names = rownames(spde$param.inla$BLC)
## Values
result$summary.values = inla$summary.random[[name]]
## Marginals for values
if (!is.null(inla$marginals.random[[name]])) {
result$marginals.values = inla$marginals.random[[name]]
}
## Theta
result$summary.hyperpar =
inla.extract.el(inla$summary.hyperpar,
paste("Theta[^ ]+ for ", name, "$", sep=""))
## Marginals for theta
if (!is.null(inla$marginals$hyperpar)) {
result$marginals.hyperpar =
inla.extract.el(inla$marginals.hyperpar,
paste("Theta[^ ]+ for ", name, "$", sep=""))
}
## log-tau/kappa/variance/range
if (!is.null(inla$summary.spde2.blc[[name]])) {
rownames(inla$summary.spde2.blc[[name]]) <- BLC.names
result$summary.theta =
inla.extract.el(inla$summary.spde2.blc[[name]],
"theta\\.[^ ]+$")
result$summary.log.tau =
inla.extract.el(inla$summary.spde2.blc[[name]],
"tau\\.[^ ]+$")
result$summary.log.kappa =
inla.extract.el(inla$summary.spde2.blc[[name]],
"kappa\\.[^ ]+$")
result$summary.log.variance.nominal =
inla.extract.el(inla$summary.spde2.blc[[name]],
"variance.nominal\\.[^ ]+$")
result$summary.log.range.nominal =
inla.extract.el(inla$summary.spde2.blc[[name]],
"range.nominal\\.[^ ]+$")
}
## Marginals for log-tau/kappa/variance/range
if (!is.null(inla$marginals.spde2.blc[[name]])) {
names(inla$marginals.spde2.blc[[name]]) <- BLC.names
result$marginals.theta =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"theta\\.[^ ]+$")
result$marginals.log.tau =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"tau\\.[^ ]+$")
result$marginals.log.kappa =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"kappa\\.[^ ]+$")
result$marginals.log.variance.nominal =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"variance.nominal\\.[^ ]+$")
result$marginals.log.range.nominal =
inla.extract.el(inla$marginals.spde2.blc[[name]],
"range.nominal\\.[^ ]+$")
if (do.transform) {
result$marginals.tau =
lapply(result$marginals.log.tau,
function(x) inla.tmarginal(function(y) exp(y), x))
result$marginals.kappa =
lapply(result$marginals.log.kappa,
function(x) inla.tmarginal(function(y) exp(y), x))
result$marginals.variance.nominal =
lapply(result$marginals.log.variance.nominal,
function(x) inla.tmarginal(function(y) exp(y), x))
result$marginals.range.nominal =
lapply(result$marginals.log.range.nominal,
function(x) inla.tmarginal(function(y) exp(y), x))
}
}
return(result)
}
inla.spde2.models = function()
{
return(c("generic", "matern", "pcmatern"))
}
## spde.common-connections:
inla.spde.precision.inla.spde2 = inla.spde2.precision
inla.spde.result.inla.spde2 = inla.spde2.result
## Deprecated:
##inla.spde.generic2 = inla.spde2.generic
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