Intrinsic models in the rSPDE package
Introduction
In this vignette we provide a brief introduction to the intrinsic
models implemented in the rSPDE package.
A fractional intrinsic model
A basic intrinsic model which is implemented in rSPDE is
defined as
\[
(-\Delta)^{\beta/2}(\tau u) = \mathcal{W},
\] where \(\beta > d/2\) and
\(d\) is the dimension of the spatial
domain.
To illustrate these models, we begin by defining a mesh over \([0,2]\times [0, 2]\):
library(fmesher)
bnd <- fm_segm(rbind(c(0, 0), c(2, 0), c(2, 2), c(0, 2)), is.bnd = TRUE)
mesh_2d <- fm_mesh_2d(
boundary = bnd,
cutoff = 0.02,
max.edge = c(0.1)
)
plot(mesh_2d, main = "")
We now use the intrinsic.operators() function to
construct the rSPDE representation of the general
model.
library(rSPDE)
tau <- 0.2
beta <- 1.8
fem <- fm_fem(mesh_2d)
op <- intrinsic.operators(tau = tau, beta = beta, mesh = mesh_2d, m = 2)To see that the rSPDE model is approximating the true
model, we can compare the variogram of the approximation (implemented in
the function variogram in the model object) with the true
variogram (implemented in variogram.intrinsic.spde()) as
follows.
point <- matrix(c(1,1),1,2)
Gamma <- op$variogram(point)
vario <- variogram.intrinsic.spde(point, mesh_2d$loc[,1:2], tau = tau,
beta = beta, L = 2, d = 2)
d = sqrt((mesh_2d$loc[,1]-point[1])^2 + (mesh_2d$loc[,2]-point[2])^2)
plot(d, Gamma, xlim = c(0,0.7), ylim = c(0,3),
ylab = "variogram(h)", xlab = "h")
points(d,vario,col=2)
If we want to increase the accuracy, we can either use a finer mesh
or increase the order of the rational approximation through the argument
m in intrinsic.operators. The default value of
m is 1. We can now use the simulate function
to simulate a realization of the field \(u\):
u <- simulate(op,nsim = 1)
proj <- fm_evaluator(mesh_2d, dims = c(100, 100))
field <- fm_evaluate(proj, field = as.vector(u))
field.df <- data.frame(x1 = proj$lattice$loc[,1],
x2 = proj$lattice$loc[,2],
y = as.vector(field))
library(ggplot2)
library(viridis)
ggplot(field.df, aes(x = x1, y = x2, fill = y)) +
geom_raster() +
scale_fill_viridis()
By default, the field is simulated with a zero-integral constraint.
Fitting the model with R-INLA
Let us now consider a simple Gaussian linear model where the spatial field \(u(\mathbf{s})\) is observed at \(m\) locations, \(\{\mathbf{s}_1 , \ldots , \mathbf{s}_m \}\) under Gaussian measurement noise. For each \(i = 1,\ldots,m,\) we have \[ \begin{align} y_i &= u(\mathbf{s}_i)+\varepsilon_i\\ \end{align}, \] where \(\varepsilon_1,\ldots,\varepsilon_{m}\) are iid normally distributed with mean 0 and standard deviation 0.1.
To generate a data set y from this model, we first draw
some observation locations at random in the domain and then use the
spde.make.A() functions (that wraps the functions
fm_basis(), fm_block() and
fm_row_kron() of the fmesher package) to
construct the observation matrix which can be used to evaluate the
simulated field \(u\) at the
observation locations. After this we simply add the measurment
noise.
n_loc <- 1000
loc_2d_mesh <- matrix(2*runif(n_loc * 2), n_loc, 2)
A <- spde.make.A(
mesh = mesh_2d,
loc = loc_2d_mesh
)
sigma.e <- 0.1
y <- A %*% u + rnorm(n_loc) * sigma.eThe generated data can be seen in the following image.
df <- data.frame(x1 = as.double(loc_2d_mesh[, 1]),
x2 = as.double(loc_2d_mesh[, 2]), y = as.double(y))
ggplot(df, aes(x = x1, y = x2, col = y)) +
geom_point() +
scale_color_viridis()
We will now fit the model using our R-INLA implementation of
the rational SPDE approach. Further details on this implementation can
be found in R-INLA implementation of the
rational SPDE approach.
library(INLA)
rspde.order <- 2
mesh.index <- rspde.make.index(name = "field", mesh = mesh_2d, rspde.order = rspde.order)
Abar <- rspde.make.A(mesh = mesh_2d, loc = loc_2d_mesh, rspde.order = rspde.order)
st.dat <- inla.stack(data = list(y = as.vector(y)), A = Abar, effects = mesh.index)We now create the model object.
Finally, we create the formula and fit the model to the data:
f <- y ~ -1 + f(field, model = rspde_model)
rspde_fit <- inla(f,
data = inla.stack.data(st.dat),
family = "gaussian",
control.predictor = list(A = inla.stack.A(st.dat)))To compare the estimated parameters to the true parameters, we can do the following:
result_fit <- rspde.result(rspde_fit, "field", rspde_model)
summary(result_fit)
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> tau 0.125849 0.0325894 0.0722349 0.122494 0.199424 0.115890
#> nu 0.970906 0.0920042 0.7964780 0.968514 1.156590 0.960622
tau <- op$tau
nu <- op$beta - 1 #beta = nu + d/2
result_df <- data.frame(
parameter = c("tau", "nu", "sigma.e"),
true = c(tau, nu, sigma.e),
mean = c(result_fit$summary.tau$mean,result_fit$summary.nu$mean,
sqrt(1/rspde_fit$summary.hyperpar[1,1])),
mode = c(result_fit$summary.tau$mode, result_fit$summary.nu$mode,
sqrt(1/rspde_fit$summary.hyperpar[1,6]))
)
print(result_df)
#> parameter true mean mode
#> 1 tau 0.2 0.12584912 0.11589026
#> 2 nu 0.8 0.97090581 0.96062230
#> 3 sigma.e 0.1 0.09780298 0.09819109Extreme value models
When used for extreme value statistics, one might want to use a
particular form of the mean value of the latent field \(u\), which is zero at one location \(k\) and is given by the diagonal of \(Q_{-k,-k}^{-1}\) for the remaining
locations. This option can be specified via the
mean.correction argument of
rspde.intrinsic:
We can then fit this model as before:
f <- y ~ -1 + f(field, model = rspde_model2)
rspde_fit <- inla(f,
data = inla.stack.data(st.dat),
family = "gaussian",
control.predictor = list(A = inla.stack.A(st.dat)))To see the posterior distributions of the parameters we can do:
result_fit <- rspde.result(rspde_fit, "field", rspde_model2)
posterior_df_fit <- gg_df(result_fit)
ggplot(posterior_df_fit) + geom_line(aes(x = x, y = y)) +
facet_wrap(~parameter, scales = "free") + labs(y = "Density")
An example with replicates
Let us redo the previous example with replicated data to illustrate
that replicates are handled in the same way as any other
rSPDE model. We start by generating some data with 200
observations per replicate
set.seed(1)
tau <- 0.2
beta <- 1.9
op <- intrinsic.operators(tau = tau, beta = beta, mesh = mesh_2d)
n.rep <- 5
m <- 1000
loc_2d_mesh <- matrix(2*runif(m * 2), m, 2)
A <- spde.make.A(
mesh = mesh_2d,
loc = loc_2d_mesh,
index = rep(1:m, times = n.rep),
repl = rep(1:n.rep, each = m)
)
u <- simulate(op, nsim = n.rep)
y <- as.vector(A %*% as.vector(u)) +
rnorm(m * n.rep) * 0.1We now create the stack, A matrix and index and fit the model:
Abar.rep <- rspde.make.A(
mesh = mesh_2d, loc = loc_2d_mesh, index = rep(1:m, times = n.rep),
repl = rep(1:n.rep, each = m)
)
mesh.index.rep <- rspde.make.index(
name = "field", mesh = mesh_2d,
n.repl = n.rep
)
st.dat.rep <- inla.stack(
data = list(y = y),
A = Abar.rep,
effects = mesh.index.rep
)
rspde_model.rep <- rspde.intrinsic(mesh = mesh_2d, prior.nu.dist = "beta")
f.rep <-
y ~ -1 + f(field,
model = rspde_model.rep,
replicate = field.repl
)
rspde_fit.rep <-
inla(f.rep,
data = inla.stack.data(st.dat.rep),
family = "gaussian",
control.predictor =
list(A = inla.stack.A(st.dat.rep))
)We then compare with the true parameter estimates as before
result_fit <- rspde.result(rspde_fit.rep, "field", rspde_model.rep)
summary(result_fit)
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> tau 0.177503 0.00948496 0.160025 0.177076 0.197253 0.176021
#> nu 0.924729 0.01240760 0.899949 0.924913 0.948665 0.925536
tau <- op$tau
nu <- op$beta - 1 #beta = nu + d/2
result_df <- data.frame(
parameter = c("tau", "nu", "sigma.e"),
true = c(tau, nu, sigma.e),
mean = c(result_fit$summary.tau$mean,result_fit$summary.nu$mean,
sqrt(1/rspde_fit.rep$summary.hyperpar[1,1])),
mode = c(result_fit$summary.tau$mode, result_fit$summary.nu$mode,
sqrt(1/rspde_fit.rep$summary.hyperpar[1,6]))
)
print(result_df)
#> parameter true mean mode
#> 1 tau 0.2 0.17750315 0.1760206
#> 2 nu 0.9 0.92472897 0.9255364
#> 3 sigma.e 0.1 0.09997448 0.1002499To see the posterior distributions of the parameters we can do:
result_fit <- rspde.result(rspde_fit.rep, "field", rspde_model.rep)
posterior_df_fit <- gg_df(result_fit)
ggplot(posterior_df_fit) + geom_line(aes(x = x, y = y)) +
facet_wrap(~parameter, scales = "free") + labs(y = "Density")
A more general model
The rSPDE package also contains a partial implementation
of a more general intrinsic model, which we refer to as an intrinsic
Matérn model. The model is defined as
\[
(-\Delta)^{\beta/2}(\kappa^2-\Delta)^{\alpha/2}(\tau u) = \mathcal{W},
\] where \(\alpha + \beta >
d/2\) and \(d\) is the dimension
of the spatial domain. These models are handled by performing two
rational approximations, one for each fractional operator.
To illustrate this model, we consider the same mesh as before and use
the intrinsic.matern.operators() function to construct the
rSPDE representation of the general model.
bnd <- fm_segm(rbind(c(0, 0), c(2, 0), c(2, 2), c(0, 2)), is.bnd = TRUE)
mesh_2d <- fm_mesh_2d(
boundary = bnd,
cutoff = 0.01,
max.edge = c(0.05)
)
kappa <- 10
tau <- 0.0025
alpha <- 2
beta <- 1
op <- intrinsic.matern.operators(kappa = kappa, tau = tau, alpha = alpha,
beta = beta, mesh = mesh_2d)To see that the rSPDE model is approximating the true
model, we can compare the variogram of the approximation with the true
variogram (implemented in variogram.intrinsic.spde()) as
follows.
point <- matrix(c(1,1),1,2)
Gamma <- op$variogram(point)
vario <- variogram.intrinsic.spde(point, mesh_2d$loc[,1:2], kappa = kappa,
alpha = alpha, tau = tau,
beta = beta, L = 2, d = 2)
d = sqrt((mesh_2d$loc[,1]-point[1])^2 + (mesh_2d$loc[,2]-point[2])^2)
plot(d, Gamma, xlim = c(0,0.5), ylim = c(0,4),
ylab = "variogram(h)", xlab = "h")
lines(sort(d),sort(vario),col=2, lwd = 2)
We can now use the simulate function to simulate a
realization of the field \(u\):
u <- simulate(op,nsim = 1, use_kl = FALSE)
proj <- fm_evaluator(mesh_2d, dims = c(100, 100))
field <- fm_evaluate(proj, field = as.vector(u))
field.df <- data.frame(x1 = proj$lattice$loc[,1],
x2 = proj$lattice$loc[,2],
y = as.vector(field))
library(ggplot2)
library(viridis)
ggplot(field.df, aes(x = x1, y = x2, fill = y)) +
geom_raster() +
scale_fill_viridis()
By default, the field is simulated with a zero-integral constraint.
Fitting the model with R-INLA
We will now fit the model using our R-INLA implementation of
the rational SPDE approach. Further details on this implementation can
be found in R-INLA implementation of the
rational SPDE approach.
We begin by simulating some data as before.
n_loc <- 2000
loc_2d_mesh <- matrix(2*runif(n_loc * 2), n_loc, 2)
A <- spde.make.A(
mesh = mesh_2d,
loc = loc_2d_mesh
)
sigma.e <- 0.1
y <- A %*% u + rnorm(n_loc) * sigma.eThe generated data can be seen in the following image.
df <- data.frame(x1 = as.double(loc_2d_mesh[, 1]),
x2 = as.double(loc_2d_mesh[, 2]), y = as.double(y))
ggplot(df, aes(x = x1, y = x2, col = y)) +
geom_point() +
scale_color_viridis()
To fit the model, we create the \(A\) matrix, the index, and the
inla.stack object. For now, these more general models can
only be estimated with \(\beta = 1\)
and \(\alpha = 1\) or \(\alpha = 2\). For these non-fractional
models, we can use the standard INLA functions to make the required
elements.
mesh.index <- inla.spde.make.index(name = "field", n.spde = mesh_2d$n)
st.dat <- inla.stack(data = list(y = as.vector(y)), A = A, effects = mesh.index)We now create the model object.
Finally, we create the formula and fit the model to the data:
f <- y ~ -1 + f(field, model = rspde_model)
rspde_fit <- inla(f,
data = inla.stack.data(st.dat),
family = "gaussian",
control.predictor = list(A = inla.stack.A(st.dat)))We can get a summary of the fit:
summary(rspde_fit)
#> Time used:
#> Pre = 0.142, Running = 19.5, Post = 0.0571, Total = 19.7
#> Random effects:
#> Name Model
#> field CGeneric
#>
#> Model hyperparameters:
#> mean sd 0.025quant 0.5quant
#> Precision for the Gaussian observations 100.64 4.494 92.09 100.53
#> Theta1 for field -5.98 0.048 -6.07 -5.98
#> Theta2 for field 2.35 0.087 2.17 2.35
#> 0.975quant mode
#> Precision for the Gaussian observations 109.79 100.31
#> Theta1 for field -5.88 -5.98
#> Theta2 for field 2.52 2.35
#>
#> Marginal log-Likelihood: 727.87
#> is computed
#> Posterior summaries for the linear predictor and the fitted values are computed
#> (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')To get a summary of the fit of the random field only, we can do the following:
result_fit <- rspde.result(rspde_fit, "field", rspde_model)
summary(result_fit)
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> tau 0.00253262 0.000121583 0.0023034 0.00252838 0.0027804 0.0025204
#> kappa 10.49390000 0.907072000 8.8077100 10.46260000 12.3675000 10.4090000
tau <- op$tau
result_df <- data.frame(
parameter = c("tau", "kappa"),
true = c(tau, kappa), mean = c(result_fit$summary.tau$mean,
result_fit$summary.kappa$mean),
mode = c(result_fit$summary.tau$mode, result_fit$summary.kappa$mode)
)
print(result_df)
#> parameter true mean mode
#> 1 tau 0.0025 0.002532619 0.002520404
#> 2 kappa 10.0000 10.493911521 10.408953673Kriging with R-INLA implementation
Let us now obtain predictions (i.e., do kriging) of the latent field on a dense grid in the region.
We begin by creating the grid of locations where we want to compute
the predictions. To this end, we can use the
rspde.mesh.projector() function. This function has the same
arguments as the function inla.mesh.projector() the only
difference being that the rSPDE version also has an argument
nu and an argument rspde.order. Thus, we
proceed in the same fashion as we would in R-INLA’s standard SPDE
implementation:
projgrid <- inla.mesh.projector(mesh_2d,
xlim = c(0, 2),
ylim = c(0, 2)
)
#> Warning: `inla.mesh.projector()` was deprecated in INLA 23.06.07.
#> ℹ Please use `fmesher::fm_evaluator()` instead.
#> ℹ For more information, see
#> https://inlabru-org.github.io/fmesher/articles/inla_conversion.html
#> ℹ To silence these deprecation messages in old legacy code, set
#> `inla.setOption(fmesher.evolution.warn = FALSE)`.
#> ℹ To ensure visibility of these messages in package tests, also set
#> `inla.setOption(fmesher.evolution.verbosity = 'warn')`.
#> This warning is displayed once per session.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.This lattice contains 100 × 100 locations (the default). Let us now calculate the predictions jointly with the estimation. To this end, first, we begin by linking the prediction coordinates to the mesh nodes through an \(A\) matrix
We now make a stack for the prediction locations. We have no data at
the prediction locations, so we set y= NA. We then join
this stack with the estimation stack.
ef.prd <- list(c(mesh.index))
st.prd <- inla.stack(
data = list(y = NA),
A = list(A.prd), tag = "prd",
effects = ef.prd
)
st.all <- inla.stack(st.dat, st.prd)Doing the joint estimation takes a while, and we therefore turn off
the computation of certain things that we are not interested in, such as
the marginals for the random effect. We will also use a simplified
integration strategy (actually only using the posterior mode of the
hyper-parameters) through the command
control.inla = list(int.strategy = "eb"), i.e. empirical
Bayes:
rspde_fitprd <- inla(f,
family = "Gaussian",
data = inla.stack.data(st.all),
control.predictor = list(
A = inla.stack.A(st.all),
compute = TRUE, link = 1
),
control.compute = list(
return.marginals = FALSE,
return.marginals.predictor = FALSE
),
control.inla = list(int.strategy = "eb")
)We then extract the indices to the prediction nodes and then extract the mean and the standard deviation of the response:
id.prd <- inla.stack.index(st.all, "prd")$data
m.prd <- matrix(rspde_fitprd$summary.fitted.values$mean[id.prd], 100, 100)
sd.prd <- matrix(rspde_fitprd$summary.fitted.values$sd[id.prd], 100, 100)Finally, we plot the results. First the mean:
field.pred.df <- data.frame(x1 = projgrid$lattice$loc[,1],
x2 = projgrid$lattice$loc[,2],
y = as.vector(m.prd))
ggplot(field.pred.df, aes(x = x1, y = x2, fill = y)) +
geom_raster() + scale_fill_viridis()
Then, the marginal standard deviations:
field.pred.sd.df <- data.frame(x1 = proj$lattice$loc[,1],
x2 = proj$lattice$loc[,2],
sd = as.vector(sd.prd))
ggplot(field.pred.sd.df, aes(x = x1, y = x2, fill = sd)) +
geom_raster() + scale_fill_viridis()
Using intrinsic models without R-INLA
Currently, the more general model is only implemented in
R-INLA using fixed integer values of the smoothness
parameters. However, all intrinsic models are implemented in
rSPDE in full generality. In this section, we illustrate
the rSPDE interface. Let us test a model in one
dimension.
Let us start with generating the model
L = 20
x <- seq(from = 0, to = L, length.out = 101)
mesh <- fm_mesh_1d(x)
beta <- 1.1
alpha <- 0
kappa <- 10
tau <- 10
op <- intrinsic.matern.operators(kappa = kappa, tau = tau, alpha = alpha,
beta = beta, mesh = mesh, d = 1)
vario <- variogram.intrinsic.spde(c(L/2), mesh$loc, tau = tau,
beta = beta, alpha = alpha, kappa = kappa, L = L, d = 1)
plot(x, vario, type = "l", col = 2, lwd = 2)
points(x,op$variogram(L/2),col=1)
We now generate some data. The option to use a mean value correction for extremes models is also implemented, so we generate some data using this.
n.rep <- 100
u <- simulate(op,nsim = n.rep, integral.constraint = FALSE, use_kl = TRUE)
drift <- op$mean_correction()
u <- u + matrix(rep(drift, times = n.rep), nrow = op$n, ncol= n.rep)
sigma.e <- 0.01
n.obs <- 300
obs.loc <- runif(n = n.obs, min = 0, max = L)
A <- rSPDE.A1d(x, obs.loc)
Y <- as.matrix(A %*% u + sigma.e * matrix(rnorm(n.obs*n.rep),n.obs,n.rep))Let us now show how to do kriging prediction for this model.
A <- make_A(op, loc = obs.loc)
A.krig <- make_A(op, loc = x)
u.krig <- predict(op,
A = A, Aprd = A.krig, Y = Y[,1], sigma.e = sigma.e,
compute.variances = TRUE
)
plot(obs.loc, Y[,1],
ylab = "u(x)", xlab = "x", main = "Data and prediction",
ylim = c(
min(c(min(u.krig$mean - 2 * sqrt(u.krig$variance)),min(u[,1]))),
max(c(max(u.krig$mean + 2 * sqrt(u.krig$variance)), max(u[,1])))
)
)
lines(x,u[,1],col=3)
lines(x, u.krig$mean)
lines(x, u.krig$mean + 2 * sqrt(u.krig$variance), col = 2)
lines(x, u.krig$mean - 2 * sqrt(u.krig$variance), col = 2)
We now use rspde_lme to fit the parameters based on this
data. Since we generated data with alpha=0, we specify this
in the function to indicate that this parameter should not be fitted but
kept fixed at alpha=0 by setting fix_alpha=0
in model_options. We also specify
mean_correction=TRUE to indicate that we should use the
mean value correction when fitting.
data = data.frame(y = c(Y), loc = rep(obs.loc, n.rep), rep = rep(1:n.rep, each = n.obs))
fit <- rspde_lme(y ~ -1, loc = "loc", repl = "rep", data = data,
model = op, mean_correction = TRUE, parallel = TRUE,
model_options = list(fix_alpha = 0))
#> Warning in rspde_lme(y ~ -1, loc = "loc", repl = "rep", data = data, model =
#> op, : The optimization failed to provide a numerically positive-definite
#> Hessian. You can try to obtain a positive-definite Hessian by setting
#> 'improve_hessian' to TRUE or by setting 'parallel' to FALSE, which allows other
#> optimization methods to be used.
#> Warning in sqrt(diag(inv_fisher)): NaNs produced
rbind(c(fit$coeff$random_effects[c("beta", "tau")], fit$coeff$measurement_error),
c(beta, tau, sigma.e))
#> beta tau std. dev
#> [1,] 1.095676 9.920783 0.009909281
#> [2,] 1.100000 10.000000 0.010000000An example with estimated alpha and beta parameters
In the previous example, we fixed the alpha parameter and only estimated beta. Now, let us demonstrate how to estimate both alpha and beta simultaneously. We will set up a new model with different parameter values:
L = 20
x <- seq(from = 0, to = L, length.out = 101)
mesh <- fm_mesh_1d(x)
beta <- 1.2
alpha <- 0.3
kappa <- 15
tau <- 7
op <- intrinsic.matern.operators(kappa = kappa, tau = tau, alpha = alpha,
beta = beta, mesh = mesh, d = 1)
vario <- variogram.intrinsic.spde(c(L/2), mesh$loc, tau = tau,
beta = beta, alpha = alpha, kappa = kappa, L = L, d = 1)
plot(x, vario, type = "l", col = 2, lwd = 2)
points(x, op$variogram(L/2), col = 1)
We can note here that the variogram of the approximate model is not
particularly close to the variogram of the true continuous model. The
reason for this is that the value of alpha is very small,
and we therefore need a larger order of the rational approximation than
the default value of 2. We can adjust the orders of the rational
approximations through the m_alpha and m_beta
values in intrinsic.matern.operators. Let us increase the
value of m_alpha and decrease the value of
m_beta.
op <- intrinsic.matern.operators(kappa = kappa, tau = tau, alpha = alpha,
beta = beta, mesh = mesh, d = 1, m_alpha = 6,
m_beta = 1)
vario <- variogram.intrinsic.spde(c(L/2), mesh$loc, tau = tau,
beta = beta, alpha = alpha, kappa = kappa, L = L, d = 1)
plot(x, vario, type = "l", col = 2, lwd = 2)
points(x, op$variogram(L/2), col = 1)
We now have a better approximation. Similar to the previous example, we will generate data with the mean value correction for extremes models:
n.rep <- 100
u <- simulate(op, nsim = n.rep, integral.constraint = FALSE, use_kl = TRUE)
drift <- op$mean_correction()
u <- u + matrix(rep(drift, times = n.rep), nrow = op$n, ncol = n.rep)
sigma.e <- 0.015
n.obs <- 300
obs.loc <- runif(n = n.obs, min = 0, max = L)
A <- rSPDE.A1d(x, obs.loc)
Y <- as.matrix(A %*% u + sigma.e * matrix(rnorm(n.obs*n.rep), n.obs, n.rep))Let’s visualize the data and predictions for this model:
A <- make_A(op, loc = obs.loc)
A.krig <- make_A(op, loc = x)
u.krig <- predict(op,
A = A, Aprd = A.krig, Y = Y[,1], sigma.e = sigma.e,
compute.variances = TRUE
)
plot(obs.loc, Y[,1],
ylab = "u(x)", xlab = "x", main = "Data and prediction with fractional alpha and beta",
ylim = c(
min(c(min(u.krig$mean - 2 * sqrt(u.krig$variance)), min(u[,1]))),
max(c(max(u.krig$mean + 2 * sqrt(u.krig$variance)), max(u[,1])))
)
)
lines(x, u[,1], col = 3)
lines(x, u.krig$mean)
lines(x, u.krig$mean + 2 * sqrt(u.krig$variance), col = 2)
lines(x, u.krig$mean - 2 * sqrt(u.krig$variance), col = 2)
Now, we will use rspde_lme to fit the parameters but
this time we will not fix alpha, allowing both alpha and beta to be
estimated. Unlike the previous example where we set
fix_alpha=0, we do not include this constraint:
data = data.frame(y = c(Y), loc = rep(obs.loc, n.rep), rep = rep(1:n.rep, each = n.obs))
op <- intrinsic.matern.operators(kappa = kappa, tau = tau, alpha = 1.3, beta = 1.05, mesh = mesh, d = 1, m_alpha = 3, m_beta = 1)
fit <- rspde_lme(y ~ -1, loc = "loc", repl = "rep", data = data,
model = op, mean_correction = TRUE, parallel = FALSE)
#> alpha = 1.3 , tau = 7 , beta = 1.05 , sigma_e = 0.0158874 , lik = -17733.07 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7.007004 , beta = 1.05 , sigma_e = 0.0158874 , lik = -17858.74 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 6.993003 , beta = 1.05 , sigma_e = 0.0158874 , lik = -17607.56 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.05 , sigma_e = 0.0158874 , lik = -17896.07 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.05 , sigma_e = 0.0158874 , lik = -17570.34 nz = 8 , nz.p = 7
#> alpha = 1.301301 , tau = 7 , beta = 1.05 , sigma_e = 0.0158874 , lik = -18176.24 nz = 8 , nz.p = 7
#> alpha = 1.298701 , tau = 7 , beta = 1.05 , sigma_e = 0.0158874 , lik = -17292.29 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.051051 , sigma_e = 0.0158874 , lik = -17607.87 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.048951 , sigma_e = 0.0158874 , lik = -17858.56 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.05 , sigma_e = 0.0159033 , lik = -17662.66 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.05 , sigma_e = 0.01587152 , lik = -17803.59 nz = 8 , nz.p = 7
#> alpha = 0.5448531 , tau = 5.467411 , beta = 1.343683 , sigma_e = 0.01825015 , lik = 72860.29 nz = 8 , nz.p = 7
#> alpha = 0.5448531 , tau = 5.472881 , beta = 1.343683 , sigma_e = 0.01825015 , lik = 72857.91 nz = 8 , nz.p = 7
#> alpha = 0.5448531 , tau = 5.461946 , beta = 1.343683 , sigma_e = 0.01825015 , lik = 72862.66 nz = 8 , nz.p = 7
#> alpha = 0.5448531 , tau = 5.467411 , beta = 1.343683 , sigma_e = 0.01825015 , lik = 72859.13 nz = 8 , nz.p = 7
#> alpha = 0.5448531 , tau = 5.467411 , beta = 1.343683 , sigma_e = 0.01825015 , lik = 72861.45 nz = 8 , nz.p = 7
#> alpha = 0.5453982 , tau = 5.467411 , beta = 1.343683 , sigma_e = 0.01825015 , lik = 72857.19 nz = 8 , nz.p = 7
#> alpha = 0.5443085 , tau = 5.467411 , beta = 1.343683 , sigma_e = 0.01825015 , lik = 72863.51 nz = 8 , nz.p = 7
#> alpha = 0.5448531 , tau = 5.467411 , beta = 1.343683 , sigma_e = 0.01826841 , lik = 72854.34 nz = 8 , nz.p = 7
#> alpha = 0.5448531 , tau = 5.467411 , beta = 1.343683 , sigma_e = 0.01823191 , lik = 72866.21 nz = 8 , nz.p = 7
#> alpha = 0.5413697 , tau = 5.441431 , beta = 1.34373 , sigma_e = 0.01803617 , lik = 72962.14 nz = 8 , nz.p = 7
#> alpha = 0.5413697 , tau = 5.446875 , beta = 1.34373 , sigma_e = 0.01803617 , lik = 72959.83 nz = 8 , nz.p = 7
#> alpha = 0.5413697 , tau = 5.435992 , beta = 1.34373 , sigma_e = 0.01803617 , lik = 72964.44 nz = 8 , nz.p = 7
#> alpha = 0.5413697 , tau = 5.441431 , beta = 1.34373 , sigma_e = 0.01803617 , lik = 72961.02 nz = 8 , nz.p = 7
#> alpha = 0.5413697 , tau = 5.441431 , beta = 1.34373 , sigma_e = 0.01803617 , lik = 72963.24 nz = 8 , nz.p = 7
#> alpha = 0.5419114 , tau = 5.441431 , beta = 1.34373 , sigma_e = 0.01803617 , lik = 72959.01 nz = 8 , nz.p = 7
#> alpha = 0.5408286 , tau = 5.441431 , beta = 1.34373 , sigma_e = 0.01803617 , lik = 72965.1 nz = 8 , nz.p = 7
#> alpha = 0.5413697 , tau = 5.441431 , beta = 1.345075 , sigma_e = 0.01803617 , lik = 72960.49 nz = 8 , nz.p = 7
#> alpha = 0.5413697 , tau = 5.441431 , beta = 1.342387 , sigma_e = 0.01803617 , lik = 72963.78 nz = 8 , nz.p = 7
#> alpha = 0.5413697 , tau = 5.441431 , beta = 1.34373 , sigma_e = 0.01805421 , lik = 72956.52 nz = 8 , nz.p = 7
#> alpha = 0.5413697 , tau = 5.441431 , beta = 1.34373 , sigma_e = 0.01801814 , lik = 72967.72 nz = 8 , nz.p = 7
#> alpha = 0.5276577 , tau = 5.338741 , beta = 1.343918 , sigma_e = 0.01720503 , lik = 73316.73 nz = 8 , nz.p = 7
#> alpha = 0.5276577 , tau = 5.344082 , beta = 1.343918 , sigma_e = 0.01720503 , lik = 73314.7 nz = 8 , nz.p = 7
#> alpha = 0.5276577 , tau = 5.333405 , beta = 1.343918 , sigma_e = 0.01720503 , lik = 73318.74 nz = 8 , nz.p = 7
#> alpha = 0.5276577 , tau = 5.338741 , beta = 1.343918 , sigma_e = 0.01720503 , lik = 73315.8 nz = 8 , nz.p = 7
#> alpha = 0.5276577 , tau = 5.338741 , beta = 1.343918 , sigma_e = 0.01720503 , lik = 73317.65 nz = 8 , nz.p = 7
#> alpha = 0.5281856 , tau = 5.338741 , beta = 1.343918 , sigma_e = 0.01720503 , lik = 73314.18 nz = 8 , nz.p = 7
#> alpha = 0.5271303 , tau = 5.338741 , beta = 1.343918 , sigma_e = 0.01720503 , lik = 73319.41 nz = 8 , nz.p = 7
#> alpha = 0.5276577 , tau = 5.338741 , beta = 1.345262 , sigma_e = 0.01720503 , lik = 73315.16 nz = 8 , nz.p = 7
#> alpha = 0.5276577 , tau = 5.338741 , beta = 1.342574 , sigma_e = 0.01720503 , lik = 73318.3 nz = 8 , nz.p = 7
#> alpha = 0.5276577 , tau = 5.338741 , beta = 1.343918 , sigma_e = 0.01722224 , lik = 73312.52 nz = 8 , nz.p = 7
#> alpha = 0.5276577 , tau = 5.338741 , beta = 1.343918 , sigma_e = 0.01718783 , lik = 73320.9 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73581.5 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.81 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.03 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.231408 , sigma_e = 0.01335588 , lik = 73583.2 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73581.63 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.233026 , sigma_e = 0.01335588 , lik = 73582.86 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73581.98 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01336925 , lik = 73587.62 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01334254 , lik = 73577.18 nz = 8 , nz.p = 7
#> alpha = 0.219751 , tau = 2.736293 , beta = 0.9660015 , sigma_e = 0.004850002 , lik = 3817.811 nz = 0 , nz.p = 0
#> alpha = 0.219751 , tau = 2.739031 , beta = 0.9660015 , sigma_e = 0.004850002 , lik = 3826.873 nz = 0 , nz.p = 0
#> alpha = 0.219751 , tau = 2.733558 , beta = 0.9660015 , sigma_e = 0.004850002 , lik = 3808.748 nz = 0 , nz.p = 0
#> alpha = 0.219751 , tau = 2.736293 , beta = 0.9660015 , sigma_e = 0.004850002 , lik = 3819.053 nz = 0 , nz.p = 0
#> alpha = 0.219751 , tau = 2.736293 , beta = 0.9660015 , sigma_e = 0.004850002 , lik = 3816.571 nz = 0 , nz.p = 0
#> alpha = 0.2199708 , tau = 2.736293 , beta = 0.9657817 , sigma_e = 0.004850002 , lik = 3821.058 nz = 0 , nz.p = 0
#> alpha = 0.2195313 , tau = 2.736293 , beta = 0.9662212 , sigma_e = 0.004850002 , lik = 3814.57 nz = 0 , nz.p = 0
#> alpha = 0.219751 , tau = 2.736293 , beta = 0.9666876 , sigma_e = 0.004850002 , lik = 3820.556 nz = 0 , nz.p = 0
#> alpha = 0.219751 , tau = 2.736293 , beta = 0.9653161 , sigma_e = 0.004850002 , lik = 3815.087 nz = 0 , nz.p = 0
#> alpha = 0.219751 , tau = 2.736293 , beta = 0.9660015 , sigma_e = 0.004854855 , lik = 3987.035 nz = 0 , nz.p = 0
#> alpha = 0.219751 , tau = 2.736293 , beta = 0.9660015 , sigma_e = 0.004845155 , lik = 3648.209 nz = 0 , nz.p = 0
#> alpha = 0.3506083 , tau = 3.912109 , beta = 1.131163 , sigma_e = 0.0095286 , lik = 66859.44 nz = 8 , nz.p = 7
#> alpha = 0.3506083 , tau = 3.904292 , beta = 1.131163 , sigma_e = 0.0095286 , lik = 66849.87 nz = 8 , nz.p = 7
#> alpha = 0.3506083 , tau = 3.908199 , beta = 1.131163 , sigma_e = 0.0095286 , lik = 66855.93 nz = 8 , nz.p = 7
#> alpha = 0.3506083 , tau = 3.908199 , beta = 1.131163 , sigma_e = 0.0095286 , lik = 66853.39 nz = 8 , nz.p = 7
#> alpha = 0.3502579 , tau = 3.908199 , beta = 1.131513 , sigma_e = 0.0095286 , lik = 66851.69 nz = 8 , nz.p = 7
#> alpha = 0.3506083 , tau = 3.908199 , beta = 1.132145 , sigma_e = 0.0095286 , lik = 66858.89 nz = 8 , nz.p = 7
#> alpha = 0.3506083 , tau = 3.908199 , beta = 1.130181 , sigma_e = 0.0095286 , lik = 66850.42 nz = 8 , nz.p = 7
#> alpha = 0.3506083 , tau = 3.908199 , beta = 1.131163 , sigma_e = 0.009538133 , lik = 66883.57 nz = 8 , nz.p = 7
#> alpha = 0.3506083 , tau = 3.908199 , beta = 1.131163 , sigma_e = 0.009519076 , lik = 66825.65 nz = 8 , nz.p = 7
#> alpha = 0.4240787 , tau = 4.518789 , beta = 1.212173 , sigma_e = 0.01254485 , lik = 73036.27 nz = 8 , nz.p = 7
#> alpha = 0.4240787 , tau = 4.52331 , beta = 1.212173 , sigma_e = 0.01254485 , lik = 73037.92 nz = 8 , nz.p = 7
#> alpha = 0.4240787 , tau = 4.514273 , beta = 1.212173 , sigma_e = 0.01254485 , lik = 73034.61 nz = 8 , nz.p = 7
#> alpha = 0.4240787 , tau = 4.518789 , beta = 1.212173 , sigma_e = 0.01254485 , lik = 73036.89 nz = 8 , nz.p = 7
#> alpha = 0.4240787 , tau = 4.518789 , beta = 1.212173 , sigma_e = 0.01254485 , lik = 73035.64 nz = 8 , nz.p = 7
#> alpha = 0.424503 , tau = 4.518789 , beta = 1.211749 , sigma_e = 0.01254485 , lik = 73037.56 nz = 8 , nz.p = 7
#> alpha = 0.4236548 , tau = 4.518789 , beta = 1.212597 , sigma_e = 0.01254485 , lik = 73035.07 nz = 8 , nz.p = 7
#> alpha = 0.4240787 , tau = 4.518789 , beta = 1.21331 , sigma_e = 0.01254485 , lik = 73037.38 nz = 8 , nz.p = 7
#> alpha = 0.4240787 , tau = 4.518789 , beta = 1.211038 , sigma_e = 0.01254485 , lik = 73035.14 nz = 8 , nz.p = 7
#> alpha = 0.4240787 , tau = 4.518789 , beta = 1.212173 , sigma_e = 0.0125574 , lik = 73044.69 nz = 8 , nz.p = 7
#> alpha = 0.4240787 , tau = 4.518789 , beta = 1.212173 , sigma_e = 0.01253231 , lik = 73027.78 nz = 8 , nz.p = 7
#> alpha = 0.4365099 , tau = 4.619516 , beta = 1.22524 , sigma_e = 0.01307988 , lik = 73430.54 nz = 8 , nz.p = 7
#> alpha = 0.4365099 , tau = 4.624137 , beta = 1.22524 , sigma_e = 0.01307988 , lik = 73431.7 nz = 8 , nz.p = 7
#> alpha = 0.4365099 , tau = 4.614898 , beta = 1.22524 , sigma_e = 0.01307988 , lik = 73429.37 nz = 8 , nz.p = 7
#> alpha = 0.4365099 , tau = 4.619516 , beta = 1.22524 , sigma_e = 0.01307988 , lik = 73431.01 nz = 8 , nz.p = 7
#> alpha = 0.4365099 , tau = 4.619516 , beta = 1.22524 , sigma_e = 0.01307988 , lik = 73430.07 nz = 8 , nz.p = 7
#> alpha = 0.4369467 , tau = 4.619516 , beta = 1.224803 , sigma_e = 0.01307988 , lik = 73431.46 nz = 8 , nz.p = 7
#> alpha = 0.4360736 , tau = 4.619516 , beta = 1.225676 , sigma_e = 0.01307988 , lik = 73429.58 nz = 8 , nz.p = 7
#> alpha = 0.4365099 , tau = 4.619516 , beta = 1.226402 , sigma_e = 0.01307988 , lik = 73431.2 nz = 8 , nz.p = 7
#> alpha = 0.4365099 , tau = 4.619516 , beta = 1.224079 , sigma_e = 0.01307988 , lik = 73429.87 nz = 8 , nz.p = 7
#> alpha = 0.4365099 , tau = 4.619516 , beta = 1.22524 , sigma_e = 0.0130668 , lik = 73424.26 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73581.5 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.81 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.03 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.231408 , sigma_e = 0.01335588 , lik = 73583.2 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73581.63 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.233026 , sigma_e = 0.01335588 , lik = 73582.86 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73581.98 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01336925 , lik = 73587.62 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01334254 , lik = 73577.18 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.675391 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.95 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.675391 , beta = 1.231408 , sigma_e = 0.01335588 , lik = 73584.1 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.675391 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73582.55 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.675391 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73582.9 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.661386 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73580.57 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73581.89 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73581.11 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.666049 , beta = 1.231408 , sigma_e = 0.01335588 , lik = 73582.28 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.666049 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73580.7 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73581.05 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01336925 , lik = 73586.69 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01334254 , lik = 73576.26 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.675391 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73583.72 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73581.89 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73583.2 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.231408 , sigma_e = 0.01335588 , lik = 73583.58 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73582.03 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73582.37 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01336925 , lik = 73588.01 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01334254 , lik = 73577.57 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.675391 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.95 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73581.11 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73581.64 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73581.24 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.233026 , sigma_e = 0.01335588 , lik = 73582.47 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73581.58 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01336925 , lik = 73587.22 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01334254 , lik = 73576.79 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.675391 , beta = 1.231408 , sigma_e = 0.01335588 , lik = 73584.1 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.666049 , beta = 1.231408 , sigma_e = 0.01335588 , lik = 73582.28 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.231408 , sigma_e = 0.01335588 , lik = 73583.58 nz = 8 , nz.p = 7
#> alpha = 0.4437481 , tau = 4.670718 , beta = 1.230964 , sigma_e = 0.01335588 , lik = 73583.97 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.232583 , sigma_e = 0.01335588 , lik = 73583.63 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.230234 , sigma_e = 0.01335588 , lik = 73582.75 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.231408 , sigma_e = 0.01336925 , lik = 73588.39 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.231408 , sigma_e = 0.01334254 , lik = 73577.95 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.675391 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73582.55 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.666049 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73580.7 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73582.03 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01335588 , lik = 73581.24 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4419767 , tau = 4.670718 , beta = 1.232736 , sigma_e = 0.01335588 , lik = 73580.85 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.233469 , sigma_e = 0.01335588 , lik = 73582.08 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.231119 , sigma_e = 0.01335588 , lik = 73581.18 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01336925 , lik = 73586.82 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01334254 , lik = 73576.39 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.675391 , beta = 1.233026 , sigma_e = 0.01335588 , lik = 73583.77 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.233026 , sigma_e = 0.01335588 , lik = 73582.47 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.232583 , sigma_e = 0.01335588 , lik = 73583.63 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.233469 , sigma_e = 0.01335588 , lik = 73582.08 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.234203 , sigma_e = 0.01335588 , lik = 73583.29 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.233026 , sigma_e = 0.01336925 , lik = 73588.06 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.233026 , sigma_e = 0.01334254 , lik = 73577.61 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.675391 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73582.9 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73581.05 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73582.37 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01335588 , lik = 73581.58 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.230234 , sigma_e = 0.01335588 , lik = 73582.75 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.231119 , sigma_e = 0.01335588 , lik = 73581.18 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01336925 , lik = 73587.17 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01334254 , lik = 73576.73 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.675391 , beta = 1.231851 , sigma_e = 0.01336925 , lik = 73588.53 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01336925 , lik = 73586.69 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01336925 , lik = 73588.01 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01336925 , lik = 73587.22 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.231408 , sigma_e = 0.01336925 , lik = 73588.39 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01336925 , lik = 73586.82 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.233026 , sigma_e = 0.01336925 , lik = 73588.06 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01336925 , lik = 73587.17 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01338262 , lik = 73592.76 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.675391 , beta = 1.231851 , sigma_e = 0.01334254 , lik = 73578.09 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.666049 , beta = 1.231851 , sigma_e = 0.01334254 , lik = 73576.26 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01334254 , lik = 73577.57 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01334254 , lik = 73576.79 nz = 8 , nz.p = 7
#> alpha = 0.4433046 , tau = 4.670718 , beta = 1.231408 , sigma_e = 0.01334254 , lik = 73577.95 nz = 8 , nz.p = 7
#> alpha = 0.4424189 , tau = 4.670718 , beta = 1.232294 , sigma_e = 0.01334254 , lik = 73576.39 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.230677 , sigma_e = 0.01334254 , lik = 73576.73 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.01335588 , lik = 73582.42 nz = 8 , nz.p = 7
#> alpha = 0.4428615 , tau = 4.670718 , beta = 1.231851 , sigma_e = 0.0133292 , lik = 73571.88 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.05 , sigma_e = 0.0158874 , lik = -17733.07 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 10.59236 , beta = 1.05 , sigma_e = 0.0158874 , lik = -85529.65 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.05 , sigma_e = 0.0158874 , lik = -113218 nz = 8 , nz.p = 7
#> alpha = 1.967153 , tau = 7 , beta = 1.05 , sigma_e = 0.0158874 , lik = -686266.8 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.588854 , sigma_e = 0.0158874 , lik = 14321.98 nz = 8 , nz.p = 7
#> alpha = 1.3 , tau = 7 , beta = 1.05 , sigma_e = 0.02404073 , lik = 4075.185 nz = 8 , nz.p = 7
#> alpha = 0.8591097 , tau = 8.261443 , beta = 1.239216 , sigma_e = 0.01875041 , lik = 53325.03 nz = 8 , nz.p = 7
#> alpha = 0.5677458 , tau = 8.975008 , beta = 1.346251 , sigma_e = 0.02036994 , lik = 66817.93 nz = 8 , nz.p = 7
#> alpha = 0.9333136 , tau = 9.124953 , beta = 1.368743 , sigma_e = 0.02071026 , lik = 57961.47 nz = 8 , nz.p = 7
#> alpha = 1.013927 , tau = 8.539796 , beta = 1.280969 , sigma_e = 0.01938217 , lik = 48660.12 nz = 8 , nz.p = 7
#> alpha = 0.8174502 , tau = 5.681083 , beta = 1.521861 , sigma_e = 0.02302707 , lik = 65624.75 nz = 8 , nz.p = 7
#> alpha = 0.917977 , tau = 6.638524 , beta = 1.387007 , sigma_e = 0.0209866 , lik = 59766.11 nz = 8 , nz.p = 7
#> alpha = 0.6789979 , tau = 7.907861 , beta = 1.765425 , sigma_e = 0.0267124 , lik = 65037.29 nz = 8 , nz.p = 7
#> alpha = 0.7987064 , tau = 7.670414 , beta = 1.550366 , sigma_e = 0.02345837 , lik = 62920.59 nz = 8 , nz.p = 7
#> alpha = 0.5236468 , tau = 8.303161 , beta = 2.173268 , sigma_e = 0.01841299 , lik = 65703.77 nz = 8 , nz.p = 7
#> alpha = 0.6573016 , tau = 7.956226 , beta = 1.811892 , sigma_e = 0.01968249 , lik = 66912.05 nz = 8 , nz.p = 7
#> alpha = 0.9673848 , tau = 7.399379 , beta = 1.569721 , sigma_e = 0.01867872 , lik = 54207.16 nz = 8 , nz.p = 7
#> alpha = 0.535687 , tau = 8.267797 , beta = 1.532143 , sigma_e = 0.02581877 , lik = 67381.58 nz = 8 , nz.p = 7
#> alpha = 0.4446683 , tau = 6.444929 , beta = 1.893523 , sigma_e = 0.02543298 , lik = 68328.64 nz = 8 , nz.p = 7
#> alpha = 0.3069308 , tau = 5.416414 , beta = 2.323293 , sigma_e = 0.02818403 , lik = 66294.81 nz = 8 , nz.p = 7
#> alpha = 0.5160151 , tau = 6.852568 , beta = 1.448372 , sigma_e = 0.01933498 , lik = 70539.08 nz = 8 , nz.p = 7
#> alpha = 0.4498414 , tau = 6.378969 , beta = 1.362043 , sigma_e = 0.01644974 , lik = 72397.07 nz = 8 , nz.p = 7
#> alpha = 0.3375273 , tau = 9.991002 , beta = 1.745771 , sigma_e = 0.01960587 , lik = 70698.09 nz = 8 , nz.p = 7
#> alpha = 0.4210629 , tau = 8.6759 , beta = 1.646647 , sigma_e = 0.02041029 , lik = 70196.44 nz = 8 , nz.p = 7
#> alpha = 0.3947101 , tau = 6.601383 , beta = 2.015082 , sigma_e = 0.02182697 , lik = 69152.05 nz = 8 , nz.p = 7
#> alpha = 0.4322621 , tau = 7.128283 , beta = 1.81767 , sigma_e = 0.02145322 , lik = 70451.04 nz = 8 , nz.p = 7
#> alpha = 0.2883105 , tau = 7.12646 , beta = 1.611069 , sigma_e = 0.02338127 , lik = 70460.5 nz = 8 , nz.p = 7
#> alpha = 0.3542717 , tau = 7.325414 , beta = 1.63847 , sigma_e = 0.02239603 , lik = 70661.68 nz = 8 , nz.p = 7
#> alpha = 0.2998363 , tau = 6.533834 , beta = 1.837895 , sigma_e = 0.01683863 , lik = 72799.26 nz = 8 , nz.p = 7
#> alpha = 0.2243214 , tau = 5.80841 , beta = 1.968903 , sigma_e = 0.01359854 , lik = 71302.61 nz = 8 , nz.p = 7
#> alpha = 0.3085306 , tau = 8.427908 , beta = 1.494912 , sigma_e = 0.01449469 , lik = 72605.41 nz = 8 , nz.p = 7
#> alpha = 0.3380514 , tau = 7.881237 , beta = 1.582368 , sigma_e = 0.0166823 , lik = 72578.16 nz = 8 , nz.p = 7
#> alpha = 0.277337 , tau = 8.147923 , beta = 1.439661 , sigma_e = 0.01468927 , lik = 73017.06 nz = 8 , nz.p = 7
#> alpha = 0.2221458 , tau = 8.711198 , beta = 1.292757 , sigma_e = 0.01215498 , lik = 72177.39 nz = 8 , nz.p = 7
#> alpha = 0.3068223 , tau = 8.273429 , beta = 1.507943 , sigma_e = 0.01188648 , lik = 70273.25 nz = 8 , nz.p = 7
#> alpha = 0.3417622 , tau = 7.551714 , beta = 1.604346 , sigma_e = 0.01911576 , lik = 72069.96 nz = 8 , nz.p = 7
#> alpha = 0.3236537 , tau = 5.423546 , beta = 1.37178 , sigma_e = 0.01344128 , lik = 73507.12 nz = 8 , nz.p = 7
#> alpha = 0.3169322 , tau = 3.995953 , beta = 1.221808 , sigma_e = 0.0111293 , lik = 71613.86 nz = 8 , nz.p = 7
#> alpha = 0.3127577 , tau = 6.28553 , beta = 1.400122 , sigma_e = 0.01197442 , lik = 72023.72 nz = 8 , nz.p = 7
#> alpha = 0.3342682 , tau = 7.213061 , beta = 1.549633 , sigma_e = 0.0170062 , lik = 72846.53 nz = 8 , nz.p = 7
#> alpha = 0.2110021 , tau = 7.816001 , beta = 1.654859 , sigma_e = 0.01410154 , lik = 73382.63 nz = 8 , nz.p = 7
#> alpha = 0.2549647 , tau = 7.428927 , beta = 1.597197 , sigma_e = 0.01465513 , lik = 73383.3 nz = 8 , nz.p = 7
#> alpha = 0.2850498 , tau = 5.62468 , beta = 1.609984 , sigma_e = 0.01607418 , lik = 73640.6 nz = 8 , nz.p = 7
#> alpha = 0.2739883 , tau = 4.595014 , beta = 1.669228 , sigma_e = 0.01692735 , lik = 73497.64 nz = 8 , nz.p = 7
#> alpha = 0.2874415 , tau = 6.833157 , beta = 1.253847 , sigma_e = 0.01358188 , lik = 73492.79 nz = 8 , nz.p = 7
#> alpha = 0.2904913 , tau = 6.757064 , beta = 1.375617 , sigma_e = 0.01433167 , lik = 73535.14 nz = 8 , nz.p = 7
#> alpha = 0.2437552 , tau = 6.03011 , beta = 1.406945 , sigma_e = 0.01255842 , lik = 73099.89 nz = 8 , nz.p = 7
#> alpha = 0.2637784 , tau = 6.306288 , beta = 1.441252 , sigma_e = 0.01354731 , lik = 73485.18 nz = 8 , nz.p = 7
#> alpha = 0.287961 , tau = 4.818243 , beta = 1.515387 , sigma_e = 0.01407613 , lik = 73775.2 nz = 8 , nz.p = 7
#> alpha = 0.2934247 , tau = 3.705182 , beta = 1.55537 , sigma_e = 0.01377923 , lik = 73510.19 nz = 8 , nz.p = 7
#> alpha = 0.3288539 , tau = 4.44421 , beta = 1.327084 , sigma_e = 0.01388326 , lik = 73805.49 nz = 8 , nz.p = 7
#> alpha = 0.3734775 , tau = 3.437391 , beta = 1.1953 , sigma_e = 0.0135127 , lik = 73640.23 nz = 8 , nz.p = 7
#> alpha = 0.3471757 , tau = 4.551506 , beta = 1.428498 , sigma_e = 0.01516642 , lik = 73874.9 nz = 8 , nz.p = 7
#> alpha = 0.3982946 , tau = 3.866748 , beta = 1.41424 , sigma_e = 0.01604716 , lik = 73806.28 nz = 8 , nz.p = 7
#> alpha = 0.2910016 , tau = 4.933947 , beta = 1.528767 , sigma_e = 0.01604253 , lik = 73766.61 nz = 8 , nz.p = 7
#> alpha = 0.2988421 , tau = 5.052039 , beta = 1.48868 , sigma_e = 0.01534846 , lik = 73847.9 nz = 8 , nz.p = 7
#> alpha = 0.3279187 , tau = 3.525357 , beta = 1.579719 , sigma_e = 0.01546408 , lik = 73738.08 nz = 8 , nz.p = 7
#> alpha = 0.3181325 , tau = 4.14803 , beta = 1.524796 , sigma_e = 0.01517285 , lik = 73853.04 nz = 8 , nz.p = 7
#> alpha = 0.3491833 , tau = 3.749393 , beta = 1.309679 , sigma_e = 0.01347308 , lik = 73631.6 nz = 8 , nz.p = 7
#> alpha = 0.2998845 , tau = 5.082339 , beta = 1.531939 , sigma_e = 0.01538025 , lik = 73814.58 nz = 8 , nz.p = 7
#> alpha = 0.3513085 , tau = 4.471364 , beta = 1.399846 , sigma_e = 0.0159406 , lik = 73803.49 nz = 8 , nz.p = 7
#> alpha = 0.3342719 , tau = 4.55567 , beta = 1.429731 , sigma_e = 0.01545252 , lik = 73864.96 nz = 8 , nz.p = 7
#> alpha = 0.3096408 , tau = 4.895952 , beta = 1.65241 , sigma_e = 0.0168694 , lik = 73513.57 nz = 8 , nz.p = 7
#> alpha = 0.3239416 , tau = 4.55308 , beta = 1.40192 , sigma_e = 0.01457617 , lik = 73888.11 nz = 8 , nz.p = 7
#> alpha = 0.3117407 , tau = 4.81575 , beta = 1.493219 , sigma_e = 0.01525975 , lik = 73852.42 nz = 8 , nz.p = 7
#> alpha = 0.3574089 , tau = 4.043344 , beta = 1.419256 , sigma_e = 0.01490018 , lik = 73901.94 nz = 8 , nz.p = 7
#> alpha = 0.3908653 , tau = 3.617243 , beta = 1.380406 , sigma_e = 0.01468097 , lik = 73908.47 nz = 8 , nz.p = 7
#> alpha = 0.3750653 , tau = 3.782936 , beta = 1.371431 , sigma_e = 0.01475678 , lik = 73915.5 nz = 8 , nz.p = 7
#> alpha = 0.4113993 , tau = 3.35283 , beta = 1.306856 , sigma_e = 0.01451155 , lik = 73911.04 nz = 8 , nz.p = 7
#> alpha = 0.3925473 , tau = 4.232977 , beta = 1.283137 , sigma_e = 0.01467716 , lik = 73901.02 nz = 8 , nz.p = 7
#> alpha = 0.3724526 , tau = 4.211578 , beta = 1.342981 , sigma_e = 0.01479954 , lik = 73909.14 nz = 8 , nz.p = 7
#> alpha = 0.3901113 , tau = 3.735017 , beta = 1.339558 , sigma_e = 0.01416476 , lik = 73861.21 nz = 8 , nz.p = 7
#> alpha = 0.3474338 , tau = 4.334982 , beta = 1.407897 , sigma_e = 0.01512 , lik = 73897.22 nz = 8 , nz.p = 7
#> alpha = 0.3757234 , tau = 3.666253 , beta = 1.334887 , sigma_e = 0.01441428 , lik = 73898.49 nz = 8 , nz.p = 7
#> alpha = 0.3683736 , tau = 3.869953 , beta = 1.358185 , sigma_e = 0.01459874 , lik = 73910.49 nz = 8 , nz.p = 7
#> alpha = 0.4239157 , tau = 3.434255 , beta = 1.336174 , sigma_e = 0.01500726 , lik = 73929.35 nz = 8 , nz.p = 7
#> alpha = 0.4849375 , tau = 2.982608 , beta = 1.292623 , sigma_e = 0.01522756 , lik = 73931.48 nz = 8 , nz.p = 7
#> alpha = 0.4516476 , tau = 3.105947 , beta = 1.287531 , sigma_e = 0.01450856 , lik = 73921.96 nz = 8 , nz.p = 7
#> alpha = 0.422978 , tau = 3.375922 , beta = 1.320219 , sigma_e = 0.01465906 , lik = 73925.77 nz = 8 , nz.p = 7
#> alpha = 0.4143438 , tau = 3.620841 , beta = 1.298415 , sigma_e = 0.01493351 , lik = 73938.69 nz = 8 , nz.p = 7
#> alpha = 0.4266068 , tau = 3.622642 , beta = 1.257914 , sigma_e = 0.0150614 , lik = 73944.33 nz = 8 , nz.p = 7
#> alpha = 0.4590919 , tau = 2.928024 , beta = 1.296377 , sigma_e = 0.01491816 , lik = 73937.88 nz = 8 , nz.p = 7
#> alpha = 0.435705 , tau = 3.20658 , beta = 1.309634 , sigma_e = 0.01488842 , lik = 73937.32 nz = 8 , nz.p = 7
#> alpha = 0.3989772 , tau = 3.585045 , beta = 1.33481 , sigma_e = 0.01476008 , lik = 73927.22 nz = 8 , nz.p = 7
#> alpha = 0.5103225 , tau = 2.853673 , beta = 1.230589 , sigma_e = 0.01509283 , lik = 73944.25 nz = 8 , nz.p = 7
#> alpha = 0.4725092 , tau = 3.062037 , beta = 1.262038 , sigma_e = 0.01500811 , lik = 73947.22 nz = 8 , nz.p = 7
#> alpha = 0.4730106 , tau = 3.075304 , beta = 1.257404 , sigma_e = 0.01533712 , lik = 73935.58 nz = 8 , nz.p = 7
#> alpha = 0.4599733 , tau = 3.147851 , beta = 1.273624 , sigma_e = 0.01516472 , lik = 73942.34 nz = 8 , nz.p = 7
#> alpha = 0.5308287 , tau = 2.74912 , beta = 1.239722 , sigma_e = 0.01539784 , lik = 73920.4 nz = 8 , nz.p = 7
#> alpha = 0.4284988 , tau = 3.354821 , beta = 1.306769 , sigma_e = 0.014917 , lik = 73938.85 nz = 8 , nz.p = 7
#> alpha = 0.4156446 , tau = 3.463549 , beta = 1.265587 , sigma_e = 0.01480262 , lik = 73945.06 nz = 8 , nz.p = 7
#> alpha = 0.4319794 , tau = 3.336492 , beta = 1.272631 , sigma_e = 0.01490773 , lik = 73946.68 nz = 8 , nz.p = 7
#> alpha = 0.4284831 , tau = 3.717314 , beta = 1.253814 , sigma_e = 0.0151054 , lik = 73934.57 nz = 8 , nz.p = 7
#> alpha = 0.4512406 , tau = 3.108052 , beta = 1.285522 , sigma_e = 0.01496475 , lik = 73943.97 nz = 8 , nz.p = 7
#> alpha = 0.46867 , tau = 3.146748 , beta = 1.233543 , sigma_e = 0.0151259 , lik = 73949.87 nz = 8 , nz.p = 7
#> alpha = 0.4901464 , tau = 3.047602 , beta = 1.195872 , sigma_e = 0.01523145 , lik = 73947.19 nz = 8 , nz.p = 7
#> alpha = 0.4398838 , tau = 3.353175 , beta = 1.251612 , sigma_e = 0.01486356 , lik = 73949.68 nz = 8 , nz.p = 7
#> alpha = 0.4448224 , tau = 3.300622 , beta = 1.257062 , sigma_e = 0.01493829 , lik = 73949.56 nz = 8 , nz.p = 7
#> alpha = 0.443848 , tau = 3.500817 , beta = 1.227004 , sigma_e = 0.01502136 , lik = 73948.89 nz = 8 , nz.p = 7
#> alpha = 0.4456847 , tau = 3.398202 , beta = 1.241308 , sigma_e = 0.01500719 , lik = 73949.85 nz = 8 , nz.p = 7
#> alpha = 0.4777654 , tau = 2.927668 , beta = 1.245745 , sigma_e = 0.01490346 , lik = 73951.08 nz = 8 , nz.p = 7
#> alpha = 0.5056013 , tau = 2.631903 , beta = 1.243484 , sigma_e = 0.01482511 , lik = 73949.08 nz = 8 , nz.p = 7
#> alpha = 0.4912245 , tau = 3.016822 , beta = 1.219396 , sigma_e = 0.01505536 , lik = 73950.63 nz = 8 , nz.p = 7
#> alpha = 0.475692 , tau = 3.093747 , beta = 1.233424 , sigma_e = 0.01501831 , lik = 73951.47 nz = 8 , nz.p = 7
#> alpha = 0.450297 , tau = 3.300799 , beta = 1.221174 , sigma_e = 0.01495874 , lik = 73954.53 nz = 8 , nz.p = 7
#> alpha = 0.4395856 , tau = 3.427074 , beta = 1.201566 , sigma_e = 0.01493411 , lik = 73955.54 nz = 8 , nz.p = 7
#> alpha = 0.4835636 , tau = 3.040606 , beta = 1.209454 , sigma_e = 0.01513284 , lik = 73951.77 nz = 8 , nz.p = 7
#> alpha = 0.472253 , tau = 3.115905 , beta = 1.220384 , sigma_e = 0.01506507 , lik = 73953.09 nz = 8 , nz.p = 7
#> alpha = 0.4884527 , tau = 2.89802 , beta = 1.211365 , sigma_e = 0.01501111 , lik = 73954.48 nz = 8 , nz.p = 7
#> alpha = 0.4773906 , tau = 3.015701 , beta = 1.219208 , sigma_e = 0.01501013 , lik = 73954.36 nz = 8 , nz.p = 7
#> alpha = 0.4695606 , tau = 3.116682 , beta = 1.227993 , sigma_e = 0.01505594 , lik = 73952.75 nz = 8 , nz.p = 7
#> alpha = 0.4600538 , tau = 3.337281 , beta = 1.193092 , sigma_e = 0.01513107 , lik = 73952.02 nz = 8 , nz.p = 7
#> alpha = 0.4644191 , tau = 3.229796 , beta = 1.205929 , sigma_e = 0.01507385 , lik = 73953.88 nz = 8 , nz.p = 7
#> alpha = 0.4576476 , tau = 3.212964 , beta = 1.194115 , sigma_e = 0.01503754 , lik = 73953.97 nz = 8 , nz.p = 7
#> alpha = 0.4620935 , tau = 3.182736 , beta = 1.203747 , sigma_e = 0.01503273 , lik = 73954.75 nz = 8 , nz.p = 7
#> alpha = 0.4606601 , tau = 3.216187 , beta = 1.189805 , sigma_e = 0.01499071 , lik = 73955.13 nz = 8 , nz.p = 7
#> alpha = 0.4628693 , tau = 3.191017 , beta = 1.199192 , sigma_e = 0.01500699 , lik = 73955.27 nz = 8 , nz.p = 7
#> alpha = 0.4543715 , tau = 3.247788 , beta = 1.188898 , sigma_e = 0.0149585 , lik = 73954.19 nz = 8 , nz.p = 7
#> alpha = 0.4587773 , tau = 3.214303 , beta = 1.196639 , sigma_e = 0.01498507 , lik = 73955.04 nz = 8 , nz.p = 7
#> alpha = 0.4597824 , tau = 3.127242 , beta = 1.199291 , sigma_e = 0.01491451 , lik = 73952.65 nz = 8 , nz.p = 7
#> alpha = 0.4632556 , tau = 3.203846 , beta = 1.204264 , sigma_e = 0.01503385 , lik = 73954.96 nz = 8 , nz.p = 7
#> alpha = 0.4279955 , tau = 3.627967 , beta = 1.190335 , sigma_e = 0.01498592 , lik = 73955.7 nz = 8 , nz.p = 7
#> alpha = 0.4006337 , tau = 4.059236 , beta = 1.179053 , sigma_e = 0.01497334 , lik = 73954.24 nz = 8 , nz.p = 7
#> alpha = 0.4387485 , tau = 3.481041 , beta = 1.193179 , sigma_e = 0.0149457 , lik = 73955.84 nz = 8 , nz.p = 7
#> alpha = 0.427522 , tau = 3.64052 , beta = 1.187821 , sigma_e = 0.01490237 , lik = 73955.61 nz = 8 , nz.p = 7
#> alpha = 0.4282356 , tau = 3.574811 , beta = 1.188154 , sigma_e = 0.01490947 , lik = 73955.38 nz = 8 , nz.p = 7
#> alpha = 0.4367342 , tau = 3.478225 , beta = 1.192224 , sigma_e = 0.01494047 , lik = 73955.63 nz = 8 , nz.p = 7
#> alpha = 0.4239822 , tau = 3.677446 , beta = 1.193697 , sigma_e = 0.01494019 , lik = 73955.72 nz = 8 , nz.p = 7
#> alpha = 0.4324254 , tau = 3.555752 , beta = 1.194571 , sigma_e = 0.01495139 , lik = 73955.85 nz = 8 , nz.p = 7
#> alpha = 0.408952 , tau = 3.868164 , beta = 1.1888 , sigma_e = 0.01489623 , lik = 73955.45 nz = 8 , nz.p = 7
#> alpha = 0.4218119 , tau = 3.686472 , beta = 1.191676 , sigma_e = 0.01492384 , lik = 73955.81 nz = 8 , nz.p = 7
#> alpha = 0.4235624 , tau = 3.708391 , beta = 1.183384 , sigma_e = 0.01496481 , lik = 73955.62 nz = 8 , nz.p = 7
#> alpha = 0.4275126 , tau = 3.635968 , beta = 1.187888 , sigma_e = 0.01495713 , lik = 73955.81 nz = 8 , nz.p = 7
#> alpha = 0.4227039 , tau = 3.719257 , beta = 1.190804 , sigma_e = 0.01496511 , lik = 73955.51 nz = 8 , nz.p = 7
#> alpha = 0.4331836 , tau = 3.536978 , beta = 1.191892 , sigma_e = 0.01494663 , lik = 73955.81 nz = 8 , nz.p = 7
#> alpha = 0.4334185 , tau = 3.529704 , beta = 1.193378 , sigma_e = 0.01490406 , lik = 73955.62 nz = 8 , nz.p = 7
#> alpha = 0.4293448 , tau = 3.603148 , beta = 1.191096 , sigma_e = 0.01496541 , lik = 73955.83 nz = 8 , nz.p = 7
#> alpha = 0.442897 , tau = 3.442065 , beta = 1.191622 , sigma_e = 0.01498271 , lik = 73955.73 nz = 8 , nz.p = 7
#> alpha = 0.4269872 , tau = 3.62379 , beta = 1.191721 , sigma_e = 0.01493854 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.428792 , tau = 3.622513 , beta = 1.191496 , sigma_e = 0.01495664 , lik = 73955.8 nz = 8 , nz.p = 7
#> alpha = 0.4328279 , tau = 3.551698 , beta = 1.192471 , sigma_e = 0.01494212 , lik = 73955.85 nz = 8 , nz.p = 7
#> alpha = 0.4300742 , tau = 3.580121 , beta = 1.191813 , sigma_e = 0.01494258 , lik = 73955.85 nz = 8 , nz.p = 7
#> alpha = 0.4272498 , tau = 3.629874 , beta = 1.189803 , sigma_e = 0.01494783 , lik = 73955.85 nz = 8 , nz.p = 7
#> alpha = 0.4296977 , tau = 3.58961 , beta = 1.193147 , sigma_e = 0.01494497 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4281644 , tau = 3.613454 , beta = 1.19141 , sigma_e = 0.01495197 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4278886 , tau = 3.623151 , beta = 1.191609 , sigma_e = 0.01494759 , lik = 73955.85 nz = 8 , nz.p = 7
#> alpha = 0.4295268 , tau = 3.59083 , beta = 1.191763 , sigma_e = 0.01494383 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.431634 , tau = 3.558066 , beta = 1.194412 , sigma_e = 0.01494074 , lik = 73955.85 nz = 8 , nz.p = 7
#> alpha = 0.4283417 , tau = 3.611787 , beta = 1.190953 , sigma_e = 0.01494606 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4242993 , tau = 3.660866 , beta = 1.191107 , sigma_e = 0.01494803 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4264156 , tau = 3.633263 , beta = 1.191455 , sigma_e = 0.01494655 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4298729 , tau = 3.591786 , beta = 1.191768 , sigma_e = 0.01495481 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4291496 , tau = 3.59976 , beta = 1.191757 , sigma_e = 0.01495074 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4271811 , tau = 3.628357 , beta = 1.191725 , sigma_e = 0.01495228 , lik = 73955.85 nz = 8 , nz.p = 7
#> alpha = 0.4289392 , tau = 3.600175 , beta = 1.191754 , sigma_e = 0.01494595 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4286019 , tau = 3.602661 , beta = 1.192858 , sigma_e = 0.01495001 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4287321 , tau = 3.598107 , beta = 1.193812 , sigma_e = 0.01495199 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4289546 , tau = 3.596693 , beta = 1.19298 , sigma_e = 0.01494332 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4287569 , tau = 3.600876 , beta = 1.192587 , sigma_e = 0.01494548 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4270494 , tau = 3.625126 , beta = 1.191019 , sigma_e = 0.01495053 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4296977 , tau = 3.58961 , beta = 1.193147 , sigma_e = 0.01494497 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4275073 , tau = 3.61793 , beta = 1.192157 , sigma_e = 0.01494828 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4287705 , tau = 3.601418 , beta = 1.192306 , sigma_e = 0.01494798 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4278249 , tau = 3.613876 , beta = 1.191938 , sigma_e = 0.01495027 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4286794 , tau = 3.601769 , beta = 1.192723 , sigma_e = 0.01494775 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4288757 , tau = 3.60121 , beta = 1.192308 , sigma_e = 0.01495038 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4276782 , tau = 3.613849 , beta = 1.192484 , sigma_e = 0.01494734 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4270807 , tau = 3.620184 , beta = 1.192572 , sigma_e = 0.01494582 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4273431 , tau = 3.619411 , beta = 1.191821 , sigma_e = 0.01494602 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4276575 , tau = 3.615217 , beta = 1.19208 , sigma_e = 0.01494702 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4280522 , tau = 3.608715 , beta = 1.192798 , sigma_e = 0.01494447 , lik = 73955.86 nz = 8 , nz.p = 7
#> alpha = 0.4278817 , tau = 3.612585 , beta = 1.192153 , sigma_e = 0.01494882 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4268809 , tau = 3.625191 , beta = 1.191785 , sigma_e = 0.01494742 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4273298 , tau = 3.619321 , beta = 1.192019 , sigma_e = 0.0149475 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4276255 , tau = 3.615834 , beta = 1.192192 , sigma_e = 0.01494785 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4262631 , tau = 3.6319 , beta = 1.192098 , sigma_e = 0.01494683 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4250149 , tau = 3.647238 , beta = 1.191991 , sigma_e = 0.01494625 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4268592 , tau = 3.623842 , beta = 1.192178 , sigma_e = 0.01494654 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4264766 , tau = 3.627852 , beta = 1.192171 , sigma_e = 0.01494588 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4263555 , tau = 3.629522 , beta = 1.192325 , sigma_e = 0.01494692 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4257059 , tau = 3.636696 , beta = 1.192446 , sigma_e = 0.01494686 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4252639 , tau = 3.641843 , beta = 1.192367 , sigma_e = 0.01494434 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4259169 , tau = 3.634507 , beta = 1.192315 , sigma_e = 0.01494546 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4252495 , tau = 3.641158 , beta = 1.19262 , sigma_e = 0.01494484 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4257686 , tau = 3.635686 , beta = 1.19247 , sigma_e = 0.0149455 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4247668 , tau = 3.648711 , beta = 1.192087 , sigma_e = 0.01494613 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4253441 , tau = 3.641559 , beta = 1.192209 , sigma_e = 0.01494605 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4252143 , tau = 3.640807 , beta = 1.192605 , sigma_e = 0.01494481 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4246909 , tau = 3.645268 , beta = 1.192858 , sigma_e = 0.0149438 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.425279 , tau = 3.640717 , beta = 1.192505 , sigma_e = 0.01494592 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4249605 , tau = 3.643827 , beta = 1.192601 , sigma_e = 0.01494615 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4258307 , tau = 3.635132 , beta = 1.192193 , sigma_e = 0.01494707 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4256853 , tau = 3.636637 , beta = 1.1923 , sigma_e = 0.01494651 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.424348 , tau = 3.651391 , beta = 1.192649 , sigma_e = 0.0149465 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4232877 , tau = 3.663218 , beta = 1.192885 , sigma_e = 0.0149468 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4241486 , tau = 3.653114 , beta = 1.192551 , sigma_e = 0.01494549 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4233721 , tau = 3.66135 , beta = 1.192603 , sigma_e = 0.0149448 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4237211 , tau = 3.656154 , beta = 1.192947 , sigma_e = 0.0149458 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4241263 , tau = 3.6525 , beta = 1.192763 , sigma_e = 0.01494587 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4232544 , tau = 3.663067 , beta = 1.192686 , sigma_e = 0.01494744 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4222778 , tau = 3.674249 , beta = 1.192725 , sigma_e = 0.01494876 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4212278 , tau = 3.684527 , beta = 1.193302 , sigma_e = 0.01494586 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4223738 , tau = 3.672116 , beta = 1.193028 , sigma_e = 0.01494616 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4210607 , tau = 3.687123 , beta = 1.193069 , sigma_e = 0.01494678 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4191243 , tau = 3.708964 , beta = 1.193293 , sigma_e = 0.0149471 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4210566 , tau = 3.685858 , beta = 1.192955 , sigma_e = 0.01494625 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4199455 , tau = 3.697231 , beta = 1.192988 , sigma_e = 0.01494597 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4191227 , tau = 3.70952 , beta = 1.192905 , sigma_e = 0.01494731 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4202676 , tau = 3.696106 , beta = 1.192918 , sigma_e = 0.01494693 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.417779 , tau = 3.723672 , beta = 1.193363 , sigma_e = 0.01494931 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4191703 , tau = 3.707993 , beta = 1.193178 , sigma_e = 0.01494819 , lik = 73955.87 nz = 8 , nz.p = 7
#> alpha = 0.4169383 , tau = 3.73352 , beta = 1.193073 , sigma_e = 0.01494922 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.4142469 , tau = 3.764605 , beta = 1.193079 , sigma_e = 0.01495074 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.4138423 , tau = 3.767789 , beta = 1.193503 , sigma_e = 0.01494741 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.409688 , tau = 3.815448 , beta = 1.193849 , sigma_e = 0.01494674 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.4131754 , tau = 3.775023 , beta = 1.193191 , sigma_e = 0.01494893 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.4102327 , tau = 3.808493 , beta = 1.193121 , sigma_e = 0.01494985 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.4114946 , tau = 3.794142 , beta = 1.193029 , sigma_e = 0.01494693 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.4071727 , tau = 3.843135 , beta = 1.193487 , sigma_e = 0.01494878 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.4013261 , tau = 3.911737 , beta = 1.193693 , sigma_e = 0.01494952 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.3990697 , tau = 3.943882 , beta = 1.193579 , sigma_e = 0.01495155 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.4041893 , tau = 3.880718 , beta = 1.193495 , sigma_e = 0.01495015 , lik = 73955.88 nz = 8 , nz.p = 7
#> alpha = 0.4006016 , tau = 3.920651 , beta = 1.193745 , sigma_e = 0.01494654 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.4039702 , tau = 3.881043 , beta = 1.193607 , sigma_e = 0.01494759 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.4014417 , tau = 3.910723 , beta = 1.192995 , sigma_e = 0.01495046 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.397381 , tau = 3.959249 , beta = 1.192534 , sigma_e = 0.01495231 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.3941385 , tau = 4.000265 , beta = 1.193512 , sigma_e = 0.01495242 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.3984076 , tau = 3.947708 , beta = 1.193437 , sigma_e = 0.01495105 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.3907523 , tau = 4.042834 , beta = 1.193495 , sigma_e = 0.01494997 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.3955339 , tau = 3.98293 , beta = 1.193459 , sigma_e = 0.01494994 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.3931757 , tau = 4.009066 , beta = 1.193211 , sigma_e = 0.01494959 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.387782 , tau = 4.074823 , beta = 1.192999 , sigma_e = 0.01494931 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3915694 , tau = 4.029929 , beta = 1.192708 , sigma_e = 0.01495432 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.3938082 , tau = 4.002327 , beta = 1.192978 , sigma_e = 0.01495237 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.3879171 , tau = 4.076271 , beta = 1.192427 , sigma_e = 0.01495248 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3912268 , tau = 4.0345 , beta = 1.192768 , sigma_e = 0.01495174 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3866101 , tau = 4.09124 , beta = 1.192296 , sigma_e = 0.01495152 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3895263 , tau = 4.054875 , beta = 1.1926 , sigma_e = 0.0149514 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3858806 , tau = 4.098813 , beta = 1.191832 , sigma_e = 0.01495325 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.3882716 , tau = 4.06953 , beta = 1.19225 , sigma_e = 0.01495243 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3805405 , tau = 4.168892 , beta = 1.192539 , sigma_e = 0.01495093 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3723898 , tau = 4.277841 , beta = 1.19237 , sigma_e = 0.01495023 , lik = 73955.89 nz = 8 , nz.p = 7
#> alpha = 0.378765 , tau = 4.191834 , beta = 1.191955 , sigma_e = 0.01495029 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3714602 , tau = 4.289927 , beta = 1.191316 , sigma_e = 0.01494925 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3808622 , tau = 4.162076 , beta = 1.192393 , sigma_e = 0.01494931 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.377383 , tau = 4.205653 , beta = 1.192344 , sigma_e = 0.01494773 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3762097 , tau = 4.2242 , beta = 1.192568 , sigma_e = 0.01494749 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3791895 , tau = 4.18499 , beta = 1.192512 , sigma_e = 0.01494872 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3749088 , tau = 4.240036 , beta = 1.192604 , sigma_e = 0.01494727 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3778005 , tau = 4.202338 , beta = 1.19255 , sigma_e = 0.01494834 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3687622 , tau = 4.32534 , beta = 1.191647 , sigma_e = 0.01494867 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.382937 , tau = 4.136058 , beta = 1.192715 , sigma_e = 0.01494915 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.3767243 , tau = 4.214389 , beta = 1.192319 , sigma_e = 0.01494634 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3748306 , tau = 4.237323 , beta = 1.1922 , sigma_e = 0.01494405 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3769128 , tau = 4.209458 , beta = 1.193008 , sigma_e = 0.01494448 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3759901 , tau = 4.218297 , beta = 1.193532 , sigma_e = 0.01494157 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3704493 , tau = 4.296349 , beta = 1.192293 , sigma_e = 0.01494375 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3643591 , tau = 4.378809 , beta = 1.191986 , sigma_e = 0.01494106 , lik = 73955.9 nz = 8 , nz.p = 7
#> alpha = 0.370637 , tau = 4.290953 , beta = 1.192446 , sigma_e = 0.01494219 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3664333 , tau = 4.344936 , beta = 1.192366 , sigma_e = 0.01493893 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3731546 , tau = 4.255516 , beta = 1.192326 , sigma_e = 0.01494161 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3735924 , tau = 4.251641 , beta = 1.192396 , sigma_e = 0.01494303 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.369214 , tau = 4.30901 , beta = 1.192572 , sigma_e = 0.01493927 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3712396 , tau = 4.282935 , beta = 1.192527 , sigma_e = 0.01494138 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3702996 , tau = 4.29516 , beta = 1.192867 , sigma_e = 0.01494189 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3680546 , tau = 4.324374 , beta = 1.193186 , sigma_e = 0.0149408 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3656557 , tau = 4.358591 , beta = 1.191956 , sigma_e = 0.01494042 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3740664 , tau = 4.246256 , beta = 1.192764 , sigma_e = 0.01494346 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3734843 , tau = 4.250455 , beta = 1.192908 , sigma_e = 0.01494103 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3712058 , tau = 4.284829 , beta = 1.192448 , sigma_e = 0.01494307 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.369394 , tau = 4.30853 , beta = 1.192819 , sigma_e = 0.01494177 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3725383 , tau = 4.265792 , beta = 1.192505 , sigma_e = 0.01494271 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3722548 , tau = 4.270193 , beta = 1.192688 , sigma_e = 0.01494395 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3727634 , tau = 4.263835 , beta = 1.192768 , sigma_e = 0.01494523 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3708441 , tau = 4.289122 , beta = 1.192782 , sigma_e = 0.01494311 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3712669 , tau = 4.283278 , beta = 1.192713 , sigma_e = 0.01494301 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3721998 , tau = 4.269447 , beta = 1.192947 , sigma_e = 0.01494273 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3686151 , tau = 4.317628 , beta = 1.192689 , sigma_e = 0.01494204 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3699704 , tau = 4.299673 , beta = 1.192712 , sigma_e = 0.0149424 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3717581 , tau = 4.276125 , beta = 1.193126 , sigma_e = 0.01494339 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3714775 , tau = 4.279827 , beta = 1.192956 , sigma_e = 0.01494309 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3712115 , tau = 4.282406 , beta = 1.192957 , sigma_e = 0.01494261 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3711838 , tau = 4.281971 , beta = 1.193078 , sigma_e = 0.01494241 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3725365 , tau = 4.265308 , beta = 1.192885 , sigma_e = 0.01494395 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.371976 , tau = 4.272752 , beta = 1.192881 , sigma_e = 0.01494343 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3704687 , tau = 4.291276 , beta = 1.193141 , sigma_e = 0.01494168 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3709145 , tau = 4.285995 , beta = 1.193028 , sigma_e = 0.01494224 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3729569 , tau = 4.258521 , beta = 1.193287 , sigma_e = 0.01494294 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.372208 , tau = 4.268772 , beta = 1.193144 , sigma_e = 0.0149428 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3707263 , tau = 4.288398 , beta = 1.193133 , sigma_e = 0.01494264 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3710941 , tau = 4.283652 , beta = 1.193087 , sigma_e = 0.01494266 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3711465 , tau = 4.281423 , beta = 1.193196 , sigma_e = 0.01494209 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3709811 , tau = 4.282221 , beta = 1.193315 , sigma_e = 0.01494158 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3699951 , tau = 4.297594 , beta = 1.193026 , sigma_e = 0.01494209 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3688935 , tau = 4.312078 , beta = 1.192964 , sigma_e = 0.01494174 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3689956 , tau = 4.309356 , beta = 1.19332 , sigma_e = 0.01494079 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3697385 , tau = 4.300175 , beta = 1.193212 , sigma_e = 0.01494145 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3702046 , tau = 4.294311 , beta = 1.193094 , sigma_e = 0.01494245 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3700726 , tau = 4.295829 , beta = 1.19307 , sigma_e = 0.01494284 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3696864 , tau = 4.300169 , beta = 1.193075 , sigma_e = 0.01494157 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3699461 , tau = 4.297223 , beta = 1.19309 , sigma_e = 0.01494184 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3707573 , tau = 4.287214 , beta = 1.192947 , sigma_e = 0.01494291 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3705023 , tau = 4.290451 , beta = 1.193014 , sigma_e = 0.01494255 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3706728 , tau = 4.288351 , beta = 1.193066 , sigma_e = 0.01494235 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3689923 , tau = 4.310888 , beta = 1.192858 , sigma_e = 0.01494258 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3695297 , tau = 4.303503 , beta = 1.192943 , sigma_e = 0.01494246 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.368793 , tau = 4.312956 , beta = 1.192905 , sigma_e = 0.01494242 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3678567 , tau = 4.325312 , beta = 1.192822 , sigma_e = 0.01494245 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3701548 , tau = 4.294485 , beta = 1.192952 , sigma_e = 0.01494331 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3698391 , tau = 4.298876 , beta = 1.192955 , sigma_e = 0.01494292 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3691849 , tau = 4.308232 , beta = 1.192772 , sigma_e = 0.0149438 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3688049 , tau = 4.313746 , beta = 1.192612 , sigma_e = 0.01494478 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3676 , tau = 4.32896 , beta = 1.192776 , sigma_e = 0.01494347 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3660315 , tau = 4.349984 , beta = 1.192684 , sigma_e = 0.01494375 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3681699 , tau = 4.320762 , beta = 1.192801 , sigma_e = 0.01494427 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3683753 , tau = 4.318292 , beta = 1.192816 , sigma_e = 0.01494385 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3663391 , tau = 4.345958 , beta = 1.192477 , sigma_e = 0.01494459 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3672689 , tau = 4.333371 , beta = 1.192627 , sigma_e = 0.01494415 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3647429 , tau = 4.368086 , beta = 1.192396 , sigma_e = 0.01494463 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3660884 , tau = 4.349568 , beta = 1.19254 , sigma_e = 0.0149443 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3663147 , tau = 4.346666 , beta = 1.192425 , sigma_e = 0.01494623 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3666996 , tau = 4.341318 , beta = 1.192524 , sigma_e = 0.01494528 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3643841 , tau = 4.371594 , beta = 1.19255 , sigma_e = 0.01494448 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3621937 , tau = 4.400808 , beta = 1.192507 , sigma_e = 0.01494432 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3626309 , tau = 4.39666 , beta = 1.192244 , sigma_e = 0.014945 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3640078 , tau = 4.377561 , beta = 1.192388 , sigma_e = 0.01494482 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3638624 , tau = 4.378764 , beta = 1.192455 , sigma_e = 0.01494518 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3644176 , tau = 4.371446 , beta = 1.192477 , sigma_e = 0.01494496 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.362628 , tau = 4.395604 , beta = 1.192504 , sigma_e = 0.01494513 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3635522 , tau = 4.38314 , beta = 1.1925 , sigma_e = 0.014945 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3611878 , tau = 4.414614 , beta = 1.192581 , sigma_e = 0.01494306 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3624627 , tau = 4.397528 , beta = 1.192546 , sigma_e = 0.01494385 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3600534 , tau = 4.430468 , beta = 1.192263 , sigma_e = 0.01494557 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3615387 , tau = 4.410209 , beta = 1.192373 , sigma_e = 0.01494512 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3604766 , tau = 4.423765 , beta = 1.192519 , sigma_e = 0.0149448 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3587239 , tau = 4.44705 , beta = 1.192576 , sigma_e = 0.01494479 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.358575 , tau = 4.45001 , beta = 1.192495 , sigma_e = 0.01494428 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3598896 , tau = 4.432091 , beta = 1.19249 , sigma_e = 0.01494451 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.3581816 , tau = 4.454827 , beta = 1.192446 , sigma_e = 0.01494399 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3592881 , tau = 4.439947 , beta = 1.192464 , sigma_e = 0.01494428 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3566459 , tau = 4.475997 , beta = 1.192357 , sigma_e = 0.01494533 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3580913 , tau = 4.456249 , beta = 1.19241 , sigma_e = 0.01494496 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3576693 , tau = 4.460924 , beta = 1.192693 , sigma_e = 0.01494352 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3564832 , tau = 4.47623 , beta = 1.192903 , sigma_e = 0.01494249 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3533028 , tau = 4.521619 , beta = 1.192569 , sigma_e = 0.01494403 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3555049 , tau = 4.491109 , beta = 1.192567 , sigma_e = 0.0149441 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3547601 , tau = 4.500271 , beta = 1.192645 , sigma_e = 0.01494398 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3528679 , tau = 4.525613 , beta = 1.19271 , sigma_e = 0.01494383 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3530329 , tau = 4.523841 , beta = 1.192796 , sigma_e = 0.01494419 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3543131 , tau = 4.506488 , beta = 1.192713 , sigma_e = 0.01494414 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3531119 , tau = 4.521635 , beta = 1.193067 , sigma_e = 0.0149424 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3539921 , tau = 4.510182 , beta = 1.192892 , sigma_e = 0.01494313 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3488591 , tau = 4.581447 , beta = 1.192998 , sigma_e = 0.01494199 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3512996 , tau = 4.547472 , beta = 1.19291 , sigma_e = 0.01494269 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3534073 , tau = 4.516179 , beta = 1.193191 , sigma_e = 0.01494221 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3534595 , tau = 4.513461 , beta = 1.193502 , sigma_e = 0.01494129 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3490629 , tau = 4.577113 , beta = 1.193065 , sigma_e = 0.01494327 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3454108 , tau = 4.628403 , beta = 1.193107 , sigma_e = 0.01494366 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3518265 , tau = 4.53731 , beta = 1.193181 , sigma_e = 0.01494346 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3520903 , tau = 4.532238 , beta = 1.193316 , sigma_e = 0.01494385 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3496153 , tau = 4.567514 , beta = 1.193116 , sigma_e = 0.01494433 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.34788 , tau = 4.590627 , beta = 1.193132 , sigma_e = 0.01494529 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3476429 , tau = 4.592126 , beta = 1.193514 , sigma_e = 0.01494297 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3449789 , tau = 4.626654 , beta = 1.19385 , sigma_e = 0.01494236 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3446739 , tau = 4.631051 , beta = 1.194039 , sigma_e = 0.01494275 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3467044 , tau = 4.604464 , beta = 1.19372 , sigma_e = 0.01494302 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3406501 , tau = 4.691539 , beta = 1.193413 , sigma_e = 0.01494587 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3344205 , tau = 4.783196 , beta = 1.193253 , sigma_e = 0.01494817 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.346658 , tau = 4.599891 , beta = 1.194021 , sigma_e = 0.01494439 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3463458 , tau = 4.607002 , beta = 1.193793 , sigma_e = 0.01494421 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3378512 , tau = 4.728336 , beta = 1.193884 , sigma_e = 0.01494435 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3309491 , tau = 4.829544 , beta = 1.194018 , sigma_e = 0.0149446 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3379625 , tau = 4.723713 , beta = 1.194425 , sigma_e = 0.01494253 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.340415 , tau = 4.690084 , beta = 1.194122 , sigma_e = 0.01494322 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3384336 , tau = 4.719733 , beta = 1.193714 , sigma_e = 0.01494497 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.335356 , tau = 4.764709 , beta = 1.193525 , sigma_e = 0.01494608 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3343296 , tau = 4.780108 , beta = 1.193738 , sigma_e = 0.01494686 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3369607 , tau = 4.741274 , beta = 1.193787 , sigma_e = 0.01494573 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3293709 , tau = 4.855981 , beta = 1.193685 , sigma_e = 0.01494562 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.3420217 , tau = 4.668024 , beta = 1.193819 , sigma_e = 0.01494456 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3354167 , tau = 4.758906 , beta = 1.194357 , sigma_e = 0.01494343 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3328303 , tau = 4.792952 , beta = 1.194805 , sigma_e = 0.0149422 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3314758 , tau = 4.819912 , beta = 1.194117 , sigma_e = 0.01494429 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3340814 , tau = 4.781483 , beta = 1.194064 , sigma_e = 0.01494435 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3339109 , tau = 4.786304 , beta = 1.193419 , sigma_e = 0.01494705 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3349192 , tau = 4.770579 , beta = 1.193673 , sigma_e = 0.01494592 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3328584 , tau = 4.798667 , beta = 1.193868 , sigma_e = 0.01494586 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3303897 , tau = 4.834224 , beta = 1.193841 , sigma_e = 0.01494661 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3354908 , tau = 4.761545 , beta = 1.193843 , sigma_e = 0.01494551 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3327573 , tau = 4.797885 , beta = 1.194389 , sigma_e = 0.01494424 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3334051 , tau = 4.789569 , beta = 1.194174 , sigma_e = 0.0149447 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3325879 , tau = 4.799602 , beta = 1.194442 , sigma_e = 0.01494392 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3331692 , tau = 4.79233 , beta = 1.194251 , sigma_e = 0.01494442 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3330575 , tau = 4.793 , beta = 1.194129 , sigma_e = 0.01494551 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3333132 , tau = 4.790118 , beta = 1.194113 , sigma_e = 0.01494522 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3306941 , tau = 4.825762 , beta = 1.194449 , sigma_e = 0.01494437 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3283215 , tau = 4.858194 , beta = 1.194732 , sigma_e = 0.0149438 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3279504 , tau = 4.868484 , beta = 1.194075 , sigma_e = 0.01494659 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3242797 , tau = 4.924215 , beta = 1.193894 , sigma_e = 0.01494817 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3277592 , tau = 4.86904 , beta = 1.19423 , sigma_e = 0.01494607 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3249722 , tau = 4.909268 , beta = 1.194234 , sigma_e = 0.01494675 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3247393 , tau = 4.913388 , beta = 1.194132 , sigma_e = 0.01494728 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3268266 , tau = 4.88284 , beta = 1.194175 , sigma_e = 0.01494657 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3215784 , tau = 4.961693 , beta = 1.194218 , sigma_e = 0.0149469 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.315988 , tau = 5.048253 , beta = 1.194161 , sigma_e = 0.0149476 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.3206887 , tau = 4.970863 , beta = 1.194675 , sigma_e = 0.01494592 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3159456 , tau = 5.040624 , beta = 1.195014 , sigma_e = 0.01494557 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.317787 , tau = 5.019812 , beta = 1.193921 , sigma_e = 0.01494944 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3203885 , tau = 4.97891 , beta = 1.194144 , sigma_e = 0.01494803 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3151917 , tau = 5.054344 , beta = 1.194557 , sigma_e = 0.01494722 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3089997 , tau = 5.149918 , beta = 1.194666 , sigma_e = 0.01494754 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.312045 , tau = 5.10323 , beta = 1.194756 , sigma_e = 0.01494663 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3151713 , tau = 5.055092 , beta = 1.194634 , sigma_e = 0.0149468 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3067992 , tau = 5.187293 , beta = 1.194753 , sigma_e = 0.01494712 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3112449 , tau = 5.116344 , beta = 1.194692 , sigma_e = 0.01494703 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.306013 , tau = 5.195886 , beta = 1.194988 , sigma_e = 0.01494702 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2985152 , tau = 5.317096 , beta = 1.195169 , sigma_e = 0.01494708 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3015609 , tau = 5.267025 , beta = 1.195319 , sigma_e = 0.01494548 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3061615 , tau = 5.19347 , beta = 1.195102 , sigma_e = 0.01494612 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3019777 , tau = 5.265077 , beta = 1.194576 , sigma_e = 0.01494817 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3123942 , tau = 5.095824 , beta = 1.194944 , sigma_e = 0.01494622 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3069907 , tau = 5.178823 , beta = 1.195097 , sigma_e = 0.01494639 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3048855 , tau = 5.210347 , beta = 1.195284 , sigma_e = 0.01494607 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3033738 , tau = 5.235454 , beta = 1.195211 , sigma_e = 0.01494655 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3055187 , tau = 5.20208 , beta = 1.195114 , sigma_e = 0.01494657 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2995038 , tau = 5.300063 , beta = 1.19507 , sigma_e = 0.0149471 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3091205 , tau = 5.146133 , beta = 1.195012 , sigma_e = 0.01494644 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.306779 , tau = 5.181401 , beta = 1.194974 , sigma_e = 0.01494733 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3030855 , tau = 5.237973 , beta = 1.195508 , sigma_e = 0.01494582 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3045534 , tau = 5.215819 , beta = 1.195306 , sigma_e = 0.01494625 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3048691 , tau = 5.208412 , beta = 1.195419 , sigma_e = 0.01494586 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3042988 , tau = 5.214687 , beta = 1.195632 , sigma_e = 0.01494528 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3001372 , tau = 5.284133 , beta = 1.195503 , sigma_e = 0.01494621 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2957439 , tau = 5.354514 , beta = 1.195677 , sigma_e = 0.0149461 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2980286 , tau = 5.317612 , beta = 1.195845 , sigma_e = 0.01494483 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3045676 , tau = 5.215123 , beta = 1.19521 , sigma_e = 0.0149467 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3022497 , tau = 5.250989 , beta = 1.195242 , sigma_e = 0.01494669 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3024585 , tau = 5.247732 , beta = 1.195309 , sigma_e = 0.01494647 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3015145 , tau = 5.259746 , beta = 1.195549 , sigma_e = 0.01494602 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.3019783 , tau = 5.253662 , beta = 1.195465 , sigma_e = 0.01494615 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2988719 , tau = 5.302622 , beta = 1.195526 , sigma_e = 0.01494654 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2959098 , tau = 5.34937 , beta = 1.195614 , sigma_e = 0.01494677 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2953431 , tau = 5.361556 , beta = 1.195443 , sigma_e = 0.0149471 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2976963 , tau = 5.322853 , beta = 1.195457 , sigma_e = 0.01494679 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2920207 , tau = 5.414298 , beta = 1.195695 , sigma_e = 0.01494642 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2859424 , tau = 5.51672 , beta = 1.195798 , sigma_e = 0.01494628 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2906009 , tau = 5.43972 , beta = 1.195553 , sigma_e = 0.01494708 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2934044 , tau = 5.392597 , beta = 1.19556 , sigma_e = 0.01494685 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2869108 , tau = 5.500729 , beta = 1.195815 , sigma_e = 0.01494661 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2983398 , tau = 5.312341 , beta = 1.19544 , sigma_e = 0.01494667 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2942492 , tau = 5.377343 , beta = 1.195435 , sigma_e = 0.01494743 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2935046 , tau = 5.388795 , beta = 1.195312 , sigma_e = 0.01494809 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2897886 , tau = 5.4511 , beta = 1.195571 , sigma_e = 0.01494743 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2919032 , tau = 5.416074 , beta = 1.195554 , sigma_e = 0.01494724 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2905144 , tau = 5.436965 , beta = 1.195653 , sigma_e = 0.01494712 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2881297 , tau = 5.475066 , beta = 1.195736 , sigma_e = 0.01494713 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2903196 , tau = 5.438586 , beta = 1.195624 , sigma_e = 0.01494749 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2910877 , tau = 5.427052 , beta = 1.19561 , sigma_e = 0.01494733 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2856738 , tau = 5.518928 , beta = 1.195506 , sigma_e = 0.01494785 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2869451 , tau = 5.494652 , beta = 1.195406 , sigma_e = 0.01494878 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2882057 , tau = 5.474453 , beta = 1.195484 , sigma_e = 0.01494819 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2885224 , tau = 5.466896 , beta = 1.19551 , sigma_e = 0.01494808 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2888384 , tau = 5.462943 , beta = 1.195526 , sigma_e = 0.01494791 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2830492 , tau = 5.560417 , beta = 1.195775 , sigma_e = 0.01494734 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.277962 , tau = 5.648267 , beta = 1.195904 , sigma_e = 0.01494696 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2832662 , tau = 5.55855 , beta = 1.195568 , sigma_e = 0.01494788 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2850133 , tau = 5.528313 , beta = 1.195593 , sigma_e = 0.01494778 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2825089 , tau = 5.572304 , beta = 1.195689 , sigma_e = 0.01494744 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2872429 , tau = 5.49008 , beta = 1.195576 , sigma_e = 0.0149478 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2868018 , tau = 5.497674 , beta = 1.195664 , sigma_e = 0.01494747 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2857365 , tau = 5.513307 , beta = 1.195732 , sigma_e = 0.01494762 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2822533 , tau = 5.573844 , beta = 1.195827 , sigma_e = 0.01494705 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793233 , tau = 5.624215 , beta = 1.195964 , sigma_e = 0.01494649 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2818269 , tau = 5.580934 , beta = 1.195721 , sigma_e = 0.0149476 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2803401 , tau = 5.60915 , beta = 1.195655 , sigma_e = 0.01494748 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2843778 , tau = 5.537113 , beta = 1.195719 , sigma_e = 0.01494758 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.277754 , tau = 5.652134 , beta = 1.195819 , sigma_e = 0.01494721 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2800964 , tau = 5.611178 , beta = 1.19578 , sigma_e = 0.01494736 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2803883 , tau = 5.603594 , beta = 1.195965 , sigma_e = 0.01494692 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.281105 , tau = 5.592299 , beta = 1.195868 , sigma_e = 0.01494716 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2795451 , tau = 5.619071 , beta = 1.195833 , sigma_e = 0.01494729 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2804171 , tau = 5.60435 , beta = 1.195821 , sigma_e = 0.0149473 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2763381 , tau = 5.67598 , beta = 1.195894 , sigma_e = 0.01494693 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2823462 , tau = 5.571508 , beta = 1.195779 , sigma_e = 0.01494742 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.634292 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.628661 , beta = 1.195446 , sigma_e = 0.01494767 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.628661 , beta = 1.196005 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.196701 , sigma_e = 0.01494767 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.194751 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.195726 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.195726 , sigma_e = 0.01493273 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.6118 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.617415 , beta = 1.195446 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.617415 , beta = 1.196005 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.196701 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.194751 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.195726 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.195726 , sigma_e = 0.01493273 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.195446 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196005 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.196701 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.194751 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01496262 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01493273 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.195446 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196005 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.196701 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.194751 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01496262 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01493273 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.628661 , beta = 1.195446 , sigma_e = 0.01494767 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.617415 , beta = 1.195446 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.195446 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.195446 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2799311 , tau = 5.623035 , beta = 1.195166 , sigma_e = 0.01494767 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.196422 , sigma_e = 0.01494767 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.194472 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.195446 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.195446 , sigma_e = 0.01493273 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.628661 , beta = 1.196005 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.617415 , beta = 1.196005 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196005 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196005 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2788137 , tau = 5.623035 , beta = 1.196284 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196981 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.19503 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196005 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196005 , sigma_e = 0.01493273 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.196701 , sigma_e = 0.01494767 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.196701 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.196701 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.196701 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.196422 , sigma_e = 0.01494767 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196981 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.197678 , sigma_e = 0.01494767 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.196701 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.196701 , sigma_e = 0.01493273 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.194751 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.194751 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.194751 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.194472 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.19503 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.193777 , sigma_e = 0.01494767 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.194751 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.194751 , sigma_e = 0.01493273 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.195726 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.195726 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01496262 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01496262 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.195446 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196005 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.196701 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.194751 , sigma_e = 0.01496262 , lik = 73955.91 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01497759 , lik = 73955.84 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.628661 , beta = 1.195726 , sigma_e = 0.01493273 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.617415 , beta = 1.195726 , sigma_e = 0.01493273 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01493273 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01493273 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2796513 , tau = 5.623035 , beta = 1.195446 , sigma_e = 0.01493273 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2790926 , tau = 5.623035 , beta = 1.196005 , sigma_e = 0.01493273 , lik = 73955.93 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.196701 , sigma_e = 0.01493273 , lik = 73955.92 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.194751 , sigma_e = 0.01493273 , lik = 73955.94 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.01494767 , lik = 73955.95 nz = 8 , nz.p = 7
#> alpha = 0.2793718 , tau = 5.623035 , beta = 1.195726 , sigma_e = 0.0149178 , lik = 73955.87 nz = 8 , nz.p = 7
#> Warning in rspde_lme(y ~ -1, loc = "loc", repl = "rep", data = data, model =
#> op, : All optimization methods failed to provide a numerically
#> positive-definite Hessian. The optimization method with largest likelihood was
#> chosen. You can try to obtain a positive-definite Hessian by setting
#> 'improve_hessian' to TRUE.
# Compare estimated and true parameter values
rbind(c(fit$coeff$random_effects[c("alpha", "beta", "tau", "kappa")], fit$coeff$measurement_error),
c(alpha, beta, tau, kappa, sigma.e))
#> alpha beta tau kappa std. dev
#> [1,] 0.2793718 1.195726 5.623035 42.82057 0.01494767
#> [2,] 0.3000000 1.200000 7.000000 15.00000 0.01500000