The data model definition
We consider the following linear mixed model for the outcome \[
\mathbf{y} = \mathbf{W}\mathbf{\beta} + \mathbf{A}\mathbf{u} +
\mathbf{e}
\] where \(\beta\) are fixed
effects, or regression coefficients including the intercept, for the
matrix of covariates \(\mathbf{W}\),
\(\mathbf{u}\) is the spatio-temporal
random effect having the matrix \(\mathbf{A}\) the projector matrix from the
discretized domain to the data. The spatio-temporal random effect \(\mathbf{u}\) is defined in a continuous
spacetime domain being discretized considering meshes over time and
space. The difference from Cameletti et al.
(2013) is that we now use the models in Lindgren et al. (2023) for \(\mathbf{u}\).
Define a temporal mesh, with each knot spaced by h,
where h = 1 means one per day.
nt <- max(pdata$time)
h <- 1
tmesh <- inla.mesh.1d(
loc = seq(1, nt + h/2, h),
degree = 1)
tmesh$n
#> [1] 182
Define a spatial mesh, the same used in Cameletti et al. (2013).
smesh <- inla.mesh.2d(
cbind(locs[,2], locs[,3]),
loc.domain = pborders,
max.edge = c(50, 300),
offset = c(10, 140),
cutoff = 5,
min.angle = c(26, 21))
smesh$n
#> [1] 142
Visualize the spatial mesh, the border and the locations.
par(mfrow = c(1,1), mar = c(0,0,1,0))
plot(smesh, asp = 1)
lines(pborders, lwd = 2, col = "green4")
points(locs[, 2:3], pch = 19, col = "blue")
We set the prior for the likelihood precision considering a PC-prior,
Simpson et al. (2017), through the
following probabilistic statements: P(\(\sigma_e > U_{\sigma_e}\)) = \(\alpha_{\sigma_e}\), using \(U_{\sigma_e}\) = 1 and \(\alpha_{\sigma_e} = 0.05\).
lkprec <- list(
prec = list(prior = "pcprec", param = c(1, 0.05)))
With inlabru we can define the likelihood model with
the like() function and use it for fitting models with
different linear predictors later.
lhood <- like(
formula = y ~ .,
family = "gaussian",
control.family = list(
hyper = lkprec),
data = dataf)
The linear predictor, the right-rand side of the formula, can be
defined using the same expression for of the both models that we are
going to fit and is
M <- ~ -1 + Intercept(1) + A + WS + TEMP + HMIX + PREC + EMI +
field(list(space = cbind(UTMX, UTMY),
time = time),
model = stmodel)
The spacetime models
The implementation of the spacetime model uses the
cgeneric interface in INLA, see its
documentation for details. Therefore we have a C code to
mainly build the precision matrix and compute the model parameter priors
and compiled as static library. We have this code included in the
INLAspacetime package but it is also being copied to
the INLA package and compiled with the same compilers
in order to avoid possible mismatches. In order to use it, we have to
define the matrices and vectors needed, including the prior parameter
definitions.
The class of models in Lindgren et al.
(2023) have the spatial range, temporal range and marginal
standard deviation as parameters. We consider the PC-prior, as in Fuglstad et al. (2017), for these parameters
defined from the probability statements: P(\(r_s<U_{r_s}\))=\(\alpha_{r_s}\), P(\(r_t<U_{r_t}\))=\(\alpha_{r_t}\) and P(\(\sigma<U_{\sigma}\))=\(\alpha_{\sigma}\). We consider \(U_{r_s}=100\), \(U_{r_t}=5\) and \(U_{\sigma}=2\). \(\alpha_{r_s}=\alpha_{r_t}=\alpha_{\sigma}=0.05\)
The selection of one of the models in Lindgren
et al. (2023) is by chosing the \(\alpha_t\), \(\alpha_s\) and \(\alpha_e\) as integer numbers. We will
start considering the model \(\alpha_t=1\), \(\alpha_s=0\) and \(\alpha_t=2\), which is a model with
separable spatio-temporal covariance, and then we fit some of the other
models later.
Defining a particular model
We define an object with the needed use the function
stModel.define() where the model is selected considering
the values for \(\alpha_t\), \(\alpha_s\) and \(\alpha_e\) collapsed. In order to
illustrate how it is done, we can set an overall integrate-to-zero
constraint, which is not need but helps model components identification.
It uses the weights based on the mesh node volumes, from both the
temporal and spatial meshes. This can be set automatically when defining
the model by adding constr = TRUE.
model <- "102"
stmodel <- stModel.define(
smesh, tmesh, model,
control.priors = list(
prs = c(150, 0.05),
prt = c(10, 0.05),
psigma = c(5, 0.05)),
constr = TRUE)
Initial values for the hyper-parameters help to fit the models in
less computing time. It is also important to consider in the light that
each dataset has its own parameter scale. For example, we have to
consider that the spatial domain within a box of around \(203.7\) by \(266.5\) kilometers, which we already did
when building the mesh and setting the prior for or \(r_s\).
We can set initial values for the log of the parameters so that it
would take less iterations to converge:
theta.ini <- c(4, 7, 7, 1)
The code to fit the model through inlabru is
fit102 <-
bru(M,
lhood,
options = list(
control.mode = list(theta = theta.ini, restart = TRUE),
control.compute = ctrc))
The computing time is
fit102$cpu
#> Pre Running Post Total
#> 1.629084 264.517733 6.890918 273.037735
Summary of the posterior marginal distributions for the fixed
effects
fit102$summary.fixed[, c(1, 2, 3, 5)]
#> mean sd 0.025quant 0.975quant
#> Intercept 3.73781993 0.237368967 3.269141650 4.205862596
#> A -0.17853994 0.048489756 -0.274754806 -0.083573184
#> WS -0.06027807 0.008449570 -0.076837214 -0.043697859
#> TEMP -0.12143472 0.035183848 -0.190508447 -0.052515214
#> HMIX -0.02387320 0.013208347 -0.049734437 0.002069429
#> PREC -0.05358257 0.008604987 -0.070441470 -0.036692107
#> EMI 0.03692457 0.014957394 0.007196049 0.065953734
For the hyperparameters, we transform the posterior marginal
distributions for the model hyperparameters from the ones computed in
internal scale, \(\log(1/\sigma^2_e)\),
\(\log(r_s)\), \(\log(r_t)\) and \(\log(\sigma)\), to the user scale
parametrization, \(\sigma_e\), \(r_s\), \(r_t\) and \(\sigma\), respectivelly.
post.h <- list(
sigma_e = inla.tmarginal(function(x) exp(-x/2),
fit102$internal.marginals.hyperpar[[1]]),
range_s = inla.tmarginal(function(x) exp(x),
fit102$internal.marginals.hyperpar[[2]]),
range_t = inla.tmarginal(function(x) exp(x),
fit102$internal.marginals.hyperpar[[3]]),
sigma_u = inla.tmarginal(function(x) exp(x),
fit102$internal.marginals.hyperpar[[4]])
)
Then we compute and show the summary of it
shyper <- t(sapply(post.h, function(m)
unlist(inla.zmarginal(m, silent = TRUE))))
shyper[, c(1, 2, 3, 7)]
#> mean sd quant0.025 quant0.975
#> sigma_e 0.1803903 0.003790609 0.1731612 0.1880459
#> range_s 277.0062336 17.461649316 245.8898720 314.3372164
#> range_t 49.4382748 9.030240345 35.8915029 70.9693098
#> sigma_u 1.1259019 0.092690391 0.9726070 1.3339498
However, it is better to look at the posterior marginal itself, and
we will visualize it later.
The model fitted in Cameletti et al.
(2013) includes two more covariates and setup a model for
discrete temporal domain where the temporal correlation is modeled as a
first order autoregression with parameter \(\rho\). In the fitted model here is defined
considering continuous temporal domain with the range parameter \(r_s\). However, the first order
autocorrelation could be taken as \(\rho =
\exp(-h\sqrt{8\nu}/r_s)\), where \(h\) is the temporal resolution used in the
temporal mesh and \(\nu\) is equal
\(0.5\) for the fitted model. We can
compare ou results with Table 3 in Cameletti et
al. (2013) with
c(shyper[c(1, 4, 2), 1],
a = exp(-h * sqrt(8 * 0.5) / shyper[3, 1]))
#> sigma_e sigma_u range_s a
#> 0.1803903 1.1259019 277.0062336 0.9603529
Comparing different models
We now fit the model \(121\) for
\(u\) as well, we use the same code for
building the model matrices
model <- "121"
stmodel <- stModel.define(
smesh, tmesh, model,
control.priors = list(
prs = c(150, 0.05),
prt = c(10, 0.05),
psigma = c(5, 0.05)),
constr = TRUE)
and use the same code for fitting as follows
fit121 <-
bru(M,
lhood,
options = list(
control.mode = list(theta = theta.ini, restart = TRUE),
control.compute = ctrc))
We will join these fits into a list object to make it easier working
with it
results <- list("u102" = fit102, "u121" = fit121)
The computing time for each model fit
sapply(results, function(r) r$cpu)
#> u102 u121
#> Pre 1.629084 0.9067976
#> Running 264.517733 439.8979118
#> Post 6.890918 4.9079370
#> Total 273.037735 445.7126465
and the number of fn-calls during the optimization are
sapply(results, function(r) r$misc$nfunc)
#> u102 u121
#> 222 340
The posterior mode for each parameter in each model (in internal
scale) are
sapply(results, function(r) r$mode$theta)
#> u102 u121
#> Log precision for the Gaussian observations 3.42807601 3.591396
#> Theta1 for field 5.61050104 7.297069
#> Theta2 for field 3.81027598 11.000813
#> Theta3 for field 0.08464823 2.161822
We compute the posterior marginal distribution for the
hyper-parameters in the user-interpretable scale, like we did before for
the first model, with
posts.h2 <- lapply(1:2, function(m) vector("list", 4L))
for(m in 1:2) {
posts.h2[[m]]$sigma_e =
data.frame(
parameter = "sigma_e",
inla.tmarginal(
function(x) exp(-x/2),
results[[m]]$internal.marginals.hyperpar[[1]]))
for(p in 2:4) {
posts.h2[[m]][[p]] <-
data.frame(
parameter = c(NA, "range_s", "range_t", "sigma_u")[p],
inla.tmarginal(
function(x) exp(x),
results[[m]]$internal.marginals.hyperpar[[p]])
)
}
}
Join these all to make visualization easier
posts.df <- rbind(
data.frame(model = "102", do.call(rbind, posts.h2[[1]])),
data.frame(model = "121", do.call(rbind, posts.h2[[2]]))
)
ggplot(posts.df) +
geom_line(aes(x = x, y = y, group = model, color = model)) +
ylab("Density") + xlab("") +
facet_wrap(~parameter, scales = "free")
The comparison of the model parameters of \(\mathbf{u}\) for different models have to
be done in light with the covariance functions as illustrated in Lindgren et al. (2023). The fitted \(\sigma_e\) by the different models are
comparable and we can see that when considering model \(121\) for \(\mathbf{u}\), its posterior marginal are
concentrated in values lower than when considering model \(102\).
We can look at the posterior mean of \(u\) from both models and see that under
model ‘121’ there is a wider spread.
par(mfrow = c(1, 1), mar = c(3, 3, 0, 0.0), mgp = c(2, 1, 0))
uu.hist <- lapply(results, function(r)
hist(r$summary.random$field$mean,
-60:60/20, plot = FALSE))
ylm <- range(uu.hist[[1]]$count, uu.hist[[2]]$count)
plot(uu.hist[[1]], ylim = ylm,
col = rgb(1, 0.1, 0.1, 1.0), border = FALSE,
xlab = "u", main = "")
plot(uu.hist[[2]], add = TRUE, col = rgb(0.1, 0.1, 1, 0.5), border = FALSE)
legend("topleft", c("separable", "non-separable"),
fill = rgb(c(1,0.1), 0.1, c(0.1, 1), c(1, 0.5)),
border = 'transparent', bty = "n")
We can also check fitting statistics such as DIC, WAIC, the negative
of the log of the probability ordinates (LPO) and its cross-validated
version (LCPO), summarized as the mean.
t(sapply(results, function(r) {
c(DIC = mean(r$dic$local.dic, na.rm = TRUE),
WAIC = mean(r$waic$local.waic, na.rm = TRUE),
LPO = -mean(log(r$po$po), na.rm = TRUE),
LCPO = -mean(log(r$cpo$cpo), na.rm = TRUE))
}))
#> DIC WAIC LPO LCPO
#> u102 -0.3783417 -0.2633347 -0.4095321 -0.1143291
#> u121 -0.4113311 -0.3522373 -0.5076508 -0.1098770
The automatic group-leave-out cross validation
One may be interested in evaluating the model prediction. The
leave-one-out strategy was already available in INLA
since several years ago, see Held, Schrodle, and
Rue (2010) for details. Recently, an automatic group cross
validation strategy was implemented, see Liu and
Rue (2023) for details.
g5cv <- lapply(
results, inla.group.cv, num.level.sets = 5,
strategy = "posterior", size.max = 50)
We can inspect the detected observations that have the posterior
linear predictor correlated with each one, including itself. For 100th
observation under model “102” we have
g5cv$u102$group[[100]]
#> $idx
#> [1] 52 76 98 100 106 124
#>
#> $corr
#> [1] 0.2515164 0.4248860 0.2970854 1.0000000 0.3271075 0.4248860
and for the result under model “121” we have
g5cv$u121$group[[100]]
#> $idx
#> [1] 52 76 98 100 124 148
#>
#> $corr
#> [1] 0.1681806 0.3746131 0.1681806 1.0000000 0.3770463 0.1649602
which has intersection but are not the same, for the model setup
used.
We can check which are these observations in the dataset
dataf[g5cv$u102$group[[100]]$idx, ]
#> UTMX UTMY time A WS TEMP HMIX PREC
#> 52 437.36 4973.34 3 -0.8881265 0.982687762 1.077344 -0.6959771 1.94505331
#> 76 437.36 4973.34 4 -0.8881265 0.674216318 1.297701 0.9479287 3.70037355
#> 98 423.48 4950.69 5 -0.7563919 -0.578948923 1.387847 -0.4413098 0.08590275
#> 100 437.36 4973.34 5 -0.8881265 0.770613644 1.441935 0.2530248 0.16426526
#> 106 416.65 4985.65 5 0.3225334 -0.000564966 1.397863 1.1608209 0.63052220
#> 124 437.36 4973.34 6 -0.8881265 0.751334179 1.618220 1.0845756 0.03692618
#> EMI y
#> 52 -0.01821338 2.564949
#> 76 -0.01542101 2.708050
#> 98 -0.27115486 2.484907
#> 100 0.01622576 2.890372
#> 106 -0.61950205 2.708050
#> 124 0.02599903 2.995732
and found that most are at the same locations in nearby time.
We can compute the negative of the mean of the log score so that
lower number is better
sapply(g5cv, function(r) -mean(log(r$cv), na.rm = TRUE))
#> u102 u121
#> 0.04266194 0.07031282