---
title: "2. Fitting models and making predictions"
output: rmarkdown::html_vignette
bibliography: biblio.bib
vignette: >
  %\VignetteIndexEntry{2. Fitting models and making predictions}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  \usepackage[utf8]{inputenc}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

---

If interested in the theory involving this calculations, take a look at [this
vignette](https://lcgodoy.me/smile/articles/theory.html).

> These functions will work only for planar projections. If working with a
> spherical projection, we recommend using a _powered exponential_ covariance
> function with $\kappa \in (0, 1)$.

---

```{r setup}
library(smile)
library(ggplot2)
library(sf)
```

Considering the same setup used in the vignette about converting `sf` to `spm`
[objects](https://lcgodoy.me/smile/articles/sf-to-spm.html), 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 @johnson2020dealing.

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"`.
```{R load-data}
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 data^[To
check how the model at the grid level relates to the data at the area level
check the [Theory vignette](https://lcgodoy.me/smile/articles/theory.html)] 
\[
	Y(\mathbf{s}) = \mu + S(\mathbf{s}) + \varepsilon(\mathbf{s}), 
\] 
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](https://lcgodoy.me/smile/articles/sp-cov-functions.html). 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 [@diggle2007model].  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](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/optim.html).
```{R fit-pred-1}
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 AIC^[BIC works for our models too.] associated with each model is given
below. According to this criterion, the best model is the one with $\nu =
.5$.
```{r AIC, echo = TRUE, eval = TRUE}
c("m1" = AIC(fit_msoa1), "m2" = AIC(fit_msoa2), "m3" = AIC(fit_msoa3))
```
We can retrieve the estimated parameters and the associated `(1 - sig)` %
confidence intervals by using the function `summary_spm_fit` as follows
```{R summary-fit1}
summary_spm_fit(fit_msoa1, sig = .05)
```
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.
```{R predict-fit1}
pred_lsoa <- predict_spm(x = liv_lsoa, spm_obj = fit_msoa1, id_var = "lsoa11cd")
```
The 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.
```{R manip-pred-1}
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.
```{R se-ggplot2}
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())
```

# Reference
