If interested in the theory involving this calculations, take a look at this vignette.
These functions will work only for planar projections. If working with a spherical projection, we recommend using a powered exponential covariance function with \(\kappa \in (0, 1)\).
library(smile)
library(ggplot2)
library(sf)
#> Linking to GEOS 3.11.0, GDAL 3.5.3, PROJ 9.1.0; sf_use_s2() is TRUEConsidering the same setup used in the vignette about converting
sf to spm objects, we
are going to predict the Life Expectation at Birth (LEF) score (observed
at the Middle Super Output Areas (MSOA), in Liverpool, into the Lower
Super Output Area (LSOA). In this second level, we have the Life
Expectancy at Birth for each area. This data was taken from Johnson, Diggle, and Giorgi (2020).
The code below loads the data and converts the LSOA sf
object into a spm object. In particular, we are going to
use numerical integration (based on a 500 points grid) with the
precision of the integrals varying according to the size of each MSOA
region. The name of the variable we are interested in, in the
liv_msoa, is "leb_est".
data(liv_msoa)
data(liv_lsoa)
## workaround for compatibility with different PROJ versions
st_crs(liv_msoa) <-
st_crs(liv_msoa)$input
st_crs(liv_lsoa) <-
st_crs(liv_lsoa)$input
## msoa from sf to spm
spm_msoa <-
sf_to_spm(sf_obj = liv_msoa,
n_pts = 500,
type = "regular",
by_polygon = FALSE,
poly_ids = "msoa11cd",
var_ids = "leb_est")Next, we assume the following model, at the grid level, drives the data1 \[ Y(\mathbf{s}) = \mu + S(\mathbf{s}) + \varepsilon(\mathbf{s}), \] where \(S(\cdot) \sim {\rm GP}(0, {\rm C}(\cdot ; \, \boldsymbol{\theta}))\), and \(\varepsilon(\cdot) \overset{{\rm i.i.d.}}{\sim} {\rm N}(0, \tau)\), with \(S(\cdot) \perp \varepsilon(\cdot)\). The form of the covariance functions considered here are written as follows \({\rm C}(\cdot ; \, \boldsymbol{\theta} ) = \sigma^2 \rho(\cdot ; \, \boldsymbol{\theta})\). Where \(\rho(\cdot ; \, \boldsymbol{\theta})\) is a stationary and isotropic correlation function. To check the families of correlation functions available in our package, check this link. Another important detail is that, if we make \(\nu = \tau / \sigma^2\), the estimation process is slightly easier. The package allows for the two parametrizations.
For this problem we are going to consider the the Matérn covariance
function. The model will be fitted with \(\nu
= .5, 1.5, 2.5\), this the parameter controls the smoothness of
the spatial process. Also, for simplicity, we are going to ignore \(\tau\). Next, we evaluate the AIC
associated with three models. The function fit_spm fits the
models and needs starting values for \(\phi\) (and \(\nu\) or \(\tau\), if you decide not to ignore the
small scale variation), the covariance model to be used is informed as a
string in the argument model, the available options are
c("matern", "pexp", "gaussian", "spherical"). Notice that,
when not inputting an initial value for \(\nu\) (or \(\tau\)), we are forcing this parameter to
be 0. Also, the theta_st argument (which takes the initial
values for the parameters), must be a named vector. If we input initial
values for \(\mu\), \(\sigma^2\), \(\phi\), and \(\nu\) (or \(\tau\)), the likelihood is optimized
numerically for all the parameters. If we omit \(\mu\) and \(\sigma^2\), then a profile likelihood
approach is used to find numerically for the \(\hat{\phi}\) and, if wished, \(\hat{\nu}\). While, \(\hat{\mu}\) and \(\hat{\sigma}^2\) have closed form
expressions (Diggle and Ribeiro 2007).
Additionally, the commented part of the code below shows how to proceed
to estimate \(\nu\), an equivalently
\(\tau\), from the data. The argument
apply_exp is a workaround for parameters that cannot be
negative. The arguments opt_method and control
are passed to the function optim in order to optimize the
log likelihood associated with the data. opt_method
controls the optimization algorithm to be used, while
control inputs control arguments for such optimization
algorithms, for more details see optim.
theta_st_msoa <- c("phi" = 1)
## 1) it is important to NAME the initial values for each parameter
## 2) to estimate "nu" from the data we only need to provide an initial value for such
## parameter
## 3) uncomment the code below to do so.
## 4) Note that it is possible to set the boundaries for the parameter space on
## which we want to optmize the likelihood.
## theta_st_msoa <- c("phi" = 1, "nu" = 1)
## fit_msoa1 <-
## fit_spm(x = spm_msoa,
## theta_st = theta_st_msoa,
## model = "matern",
## nu = .5,
## lower = c(1e-16, 1e-16),
## upper = c(Inf, 1),
## opt_method = "L-BFGS-B",
## control = list(maxit = 500))
fit_msoa1 <-
fit_spm(x = spm_msoa,
theta_st = theta_st_msoa,
model = "matern",
nu = .5,
apply_exp = TRUE,
opt_method = "L-BFGS-B",
control = list(maxit = 500))
fit_msoa2 <-
fit_spm(x = spm_msoa,
theta_st = theta_st_msoa,
model = "matern",
nu = 1.5,
apply_exp = TRUE,
opt_method = "L-BFGS-B",
control = list(maxit = 500))
fit_msoa3 <-
fit_spm(x = spm_msoa,
theta_st = theta_st_msoa,
model = "matern",
nu = 2.5,
apply_exp = TRUE,
opt_method = "L-BFGS-B",
control = list(maxit = 500))The AIC2 associated with each model is given below. According to this criterion, the best model is the one with \(\nu = .5\).
c("m1" = AIC(fit_msoa1), "m2" = AIC(fit_msoa2), "m3" = AIC(fit_msoa3))
#> m1 m2 m3
#> 296.4401 297.2103 297.6494We can retrieve the estimated parameters and the associated
(1 - sig) % confidence intervals by using the function
summary_spm_fit as follows
summary_spm_fit(fit_msoa1, sig = .05)
#>
#> optimization algorithm converged: yes
#>
#> par estimate se ci
#> 1 mu 75.9402456 0.7550275 [74.460; 77.420]
#> 2 sigsq 23.5580009 4.3091156 [15.112; 32.004]
#> 3 phi 0.7564532 0.1930406 [0.378; 1.135]Finally, we have almost everything we need to perform predictions at
the LSOA areas. The function predict_spm calculates the
predictions associated to the spm_obj (which contains the
results associated with a fitted model) into a sf object
given as an input for the argument x. We need to specify
whether we want to “aggregate” (i.e. average) the predicted surface over
the polygons associated with the sf data. The number of
points and the type of integration for prediction need to be set as
well. In this case we are specifying a finer grid (3500 points) because
the LSOA areas are smaller than the MSOA areas.
pred_lsoa <- predict_spm(x = liv_lsoa, spm_obj = fit_msoa1, id_var = "lsoa11cd")
#> Warning: st_centroid assumes attributes are constant over geometriesThe resulting object is a list with entries "mu_pred",
"sig_pred", "pred_grid",
"pred_agg". The first two positions correspond to the mean
and covariance matrix at the predicted locations, respectively.
"pred_grid" can be seen as the predicted surface over the
study region, while "pred_agg" contains the (integrals)
averages of such surface within each LSOA area. The chunk of code below
plots the predicted life expectancy at the LSOA areas.
ggplot(data = pred_lsoa$pred_agg,
aes(fill = mu_pred)) +
geom_sf(color = 1,
lwd = .1) +
scale_fill_viridis_c(option = "H") +
guides(fill = "none") +
theme_bw() +
theme(axis.text = element_blank(),
axis.ticks = element_blank())
Next, we plot the standard errors associated with the predictions.
ggplot(data = pred_lsoa$pred_agg,
aes(fill = se_pred)) +
geom_sf(color = 1,
lwd = .1) +
scale_fill_viridis_c(option = "H") +
guides(fill = "none") +
theme_bw() +
theme(axis.text = element_blank(),
axis.ticks = element_blank())
To check how the model at the grid level relates to the data at the area level check the Theory vignette↩︎
BIC works for our models too.↩︎