1. Converting sf objects to spm
In smile: Spatial Misalignment: Interpolation, Linkage, and Estimation

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(smile)
library(ggplot2)
library(sf)

Often, in an applied setting, it is desirable to change the spatial support of some variables in order to conduct either association or regression analysis. This package provides functions to deal with this problem under two different approaches, a model-based one and a (non-parametric) spatial interpolation.

The purpose of this vignette is to illustrate how to convert sf[@pebesma2018simple] objects to objects support by the smile package. Besides these two packages, we are going to use the ggplot2[@wickham2011ggplot2] package for the data visualization.

This vignette is useful when the user wants to pursue the model-based approach. The methodology is based on the assumption that areal data (e.g., variables measured at census tracts) are composed by averages of a continuous underlying process [@gelfand2001change; @moraga2017geostatistical, @johnson2020dealing; @wilson2020pointless]. In practice, there exists an identifiability problem when estimating some variance parameters, which can be seen either as small scale variation (nugget effect) or measurement errors. When fitting models, the users may chose to ignore the measurement error associated with the problem. Model fitting and prediction are explained in the vignette available on this link.

To illustrate how to convert sf polygons into spm objects, we are going to use the datasets provided by @johnson2020dealing. For this data, we have the Life Expectancy at Birth (LEB) and the Index of Multiple Deprivation (IMD) in Liverpool. These variables are observed at the Middle Super Output Areas (MSOA) and Lower Super Output Areas (LSOA), respectively. After loading our package, the datasets can be loaded by simply running the chunk of code below. Figure \ref{fig:leb-msoa} displays the LEB at the MSOA.

data(liv_lsoa) # loading the LSOA data
data(liv_msoa) # loading the MSOA data

## 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

```{R read-data, fig.cap="LEB in Liverpool at the MSOA."} ggplot(data = liv_msoa, aes(fill = leb_est)) + geom_sf(color = "black", lwd = .1) + scale_fill_viridis_b(option = "H") + theme_minimal()

<!-- We consider, in a first moment, that this random variable is normally -->
<!-- distributed with a constant mean and a certain (spatial) covariance -->
<!-- matrix. Now, let $A_1$ and $A_2$ be two distinct MSOA's. Under the --> <!--
before mentioned assumption that the observed random variable is driven --> <!--
by an underlying Gaussian random field with isotropic covariance --> <!--
function $C(\lVert x - y \rVert \, ; \, \theta)$, where $\lVert x - y --> <!--
\rVert$ is the distance between two points in the coordinates $x$ and --> <!--
$y$, respectively. Then, the covariance between the (averaged) variable --> <!--
observed at $A_1$ and $A_2$ is define as \[ \Sigma_{12} = Cov(A_1, A_2) --> <!--
= \frac{1}{\lvert A_1 \rvert \lvert A_2 \rvert} \int_{A_1} \int_{A_2} --> <!--
C(\lVert x - y \rVert; \theta) \, dy \, dx.  \] In order to approximate --> <!--
the continuous surface on which these covariances have to be computed, --> <!--
we generate a fine grid over the whole study region (or a fine grid --> <!--
within each MSOA) and approximate such covariance by numerical integration. -->

Now, suppose we are interested in estimating the LEB at the LSOA, so we will be
able to conduct association analysis between LEB and IMD at the MSOA level. We
assume, that the LEB, denoted $Y(\cdot)$, is driven by a stationary and
isotropic continuous Gaussian process over the region of study, such that, the
observed data at the $i$-th MSOA area (denoted $A_i$) is an average of this
underlying process. If we knew the parameters and covariance function associated
with this process, then the LEB at the $i$-th MSOA would be as follow
\[
    Y(A_i) = \frac{1}{\lvert A_i \rvert} \int_{A_i} Y(\mathbf{s}) \, {\rm d}
    \mathbf{s},
\]
where $\lvert A_i \rvert$ stands for the area associated with the region $A_i$.

Similarly,
\begin{align*}
{\rm E}[Y(A_i)] & = \frac{1}{\lvert A_i \rvert} \int_{A_i} {\rm
E}[Y(\mathbf{s})] \, {\rm d} \mathbf{s} \\
& = \frac{1}{\lvert A_i \rvert} \int_{A_i} \mu \, {\rm d} \mathbf{s} \\
& = \mu,
\end{align*}
and
\begin{align*}
{\rm Cov}[Y(A_i), Y(A_j)] & = \frac{1}{\lvert A_i \rvert \lvert A_j \rvert}
\int_{A_i \times A_j} {\rm Cov}[Y(\mathbf{s}, Y(\mathbf{s}')] \, {\rm d}
\mathbf{s} \, {\rm d} \mathbf{s'} \\
& = \frac{1}{\lvert A_i \rvert \lvert A_j \rvert} \int_{A_i \times A_j} {\rm C}(
\lVert \mathbf{s} - \mathbf{s}' \rVert ; \boldsymbol{\theta}) \, {\rm d}
\mathbf{s} \, {\rm d} \mathbf{s'},
\end{align*}
where $\lVert \mathbf{s} - \mathbf{s}' \rVert$ is the Euclidean distance between
the coordinates $\mathbf{s}$ and $\mathbf{s}'$, and ${\rm C}( \lVert
\mathbf{s} - \mathbf{s}' \rVert ; \boldsymbol{\theta})$ is an isotropic
covariance function depending on the parameter $\boldsymbol{\theta}$.

In practice, however, the integrals in the equation above are hard to be
evaluated analytically. A common workaround is to evaluate them either
numerically or by Monte Carlo integration [@gelfand2001change]. When using the
function `sf_to_spm`, the parameter `"type"` controls the method of
integration. The options are `"regular"` (or `"hexagonal"`) for numerical
integration, or `"random"` for Monte Carlo integration. Regardless of the
`"type"` chosen, we have to generate a grid of points over the study
region. When doing so, we may chose whether we want to approximate the integral
within each area with the same precision or if we want the precision to vary
according to the size of the polygon. This is controlled by the parameter
`"by_polygon"`, which is a boolean scalar. When set to `TRUE`, all the integrals
will be estimated with the same precision, regardless of the size of the
polygon. On the other hand, if this parameter is set to `FALSE`, the grid of
points will be generated over the whole study region and, afterwards, the points
will be attributed to the areas they are contained in. This way, larger polygons
will contain more points and, thus, the respective integrals will have a smaller
numerical error. Lastly, there exists a parameter called `"npts"`. This
parameter controls the number of points used to compute this integrals. We may
either input a vector with the same length as the number of areas or a
scalar. When inputting a scalar, this scalar will stand for the number of points
over the whole city if `by_polygon = FALSE` and the number of points per polygon
(area) otherwise.

<!-- Firstly, we are going to show how does it work with a single `sf` -->
<!-- object. Besides the `sf` objects, this function has 5 additional arguments. The -->
<!-- first one, called `n_pts`, controls the number of points we are going to -->
<!-- generate to create a grid within the study region. We can input either a single -->
<!-- (integer) value or a vector with length equal to the number of rows of the `sf` -->
<!-- objected being analyzed. The next parameter is called `type`. This one is string -->
<!-- scalar that assumes either `"random"`, `"regular"`, or `"hexagonal"`. Finally, -->
<!-- the parameter `by_polygon` is a logical that if set to `TRUE`, will generate -->
<!-- `n_pts` for each polygon within the study region^[This parameter has to be set -->
<!-- to `TRUE` when `n_pts` is a vector.]. The last two parameters are `poly_ids` and -->
<!-- `var_ids`. Both are strings, the first one is a scalar representing the variable -->
<!-- that uniquely identify each polygon within the study region, while the second -->
<!-- one indicates which (numerical) variables we want to analyze. For example, in -->
<!-- the code chunk below, we are transforming the `sf` object `liv_msoa` into a -->
<!-- `smile` compatible object. We are going to generate a grid of 1000 points -->
<!-- within the study region, with `type = "regular"`, and `by_polygon = FALSE`. The -->
<!-- id variable is called `"msoa11cd"` and the numerical variable that we are -->
<!-- interested in is called `"leb_est"`. The object returned by the function is of -->
<!-- class `"sspm"`^[when we use more than one partition, the resulting class is -->
<!-- `"mspm"`]. Under the hood, this object is a 5 positions named list. The first -->
<!-- position is called `"var"` and stores the numerical variable (or variables) to -->
<!-- be analyzed. The second position is named `"dists"` and stores the pairwise -->
<!-- distances between points belonging to different polygons, this object is -->
<!-- important to calculate the covariance matrix associated with the polygons. The -->
<!-- third positions stores the name of the id variable, while the fourth and fifth -->
<!-- positions contain the grid of points generated and the polygon associated with -->
<!-- the data. All these objects are used in functions that will be explained further -->
<!-- in other vignettes.  -->

If we wish to estimate the LEB in the LSOA areas, we will need to create a `spm`
object associated with this variable, fit the model, and then compute the
predictions. The chunk of code below shows how to convert the `liv_msoa` (of
class `sf`) to a `spm` object. In this case, we are generating a grid of 1000
points over the whole city of Liverpool, then we will be attributing each of
these points to the area they are contained in. Also, the `"poly_ids"` argument
is a string indicating the variable in the `liv_msoa` dataset that contains the
unique identifier associated with each area. The argument `"var_ids"` is a
string as well but this indicates the "response variable". That is, the variable
that for which we will be interested in fitting a model to. 
```{R sf-to-spm1}
msoa_spm <-
    sf_to_spm(sf_obj = liv_msoa, n_pts = 1000,
              type = "regular", by_polygon = FALSE,
              poly_ids = "msoa11cd", var_ids = "leb_est")

Finally, the Figure below displays the grids associated with each of the possible combinations of the parameters type and by_polygon when calling the sf_to_spm function. ```{R sf-to-spm2, echo = FALSE, warning = FALSE, message = FALSE} set.seed(123)

msoa_spm1 <- sf_to_spm(sf_obj = liv_msoa, n_pts = 305, type = "random", by_polygon = FALSE, poly_ids = "msoa11cd", var_ids = "leb_est")

msoa_spm2 <- sf_to_spm(sf_obj = liv_msoa, n_pts = 305, type = "regular", by_polygon = FALSE, poly_ids = "msoa11cd", var_ids = "leb_est")

msoa_spm3 <- sf_to_spm(sf_obj = liv_msoa, n_pts = 305, type = "hexagonal", by_polygon = FALSE, poly_ids = "msoa11cd", var_ids = "leb_est")

msoa_spm4 <- sf_to_spm(sf_obj = liv_msoa, n_pts = 305/61, type = "random", by_polygon = TRUE, poly_ids = "msoa11cd", var_ids = "leb_est")

msoa_spm5 <- sf_to_spm(sf_obj = liv_msoa, n_pts = 305/61, type = "regular", by_polygon = TRUE, poly_ids = "msoa11cd", var_ids = "leb_est")

msoa_spm6 <- sf_to_spm(sf_obj = liv_msoa, n_pts = 305/61, type = "hexagonal", by_polygon = TRUE, poly_ids = "msoa11cd", var_ids = "leb_est")

to_plot <- rbind( transform(msoa_spm1$grid, type = "random", by_polygon = FALSE), transform(msoa_spm2$grid, type = "regular", by_polygon = FALSE), transform(msoa_spm3$grid, type = "hexagonal", by_polygon = FALSE), transform(msoa_spm4$grid, type = "random", by_polygon = TRUE), transform(msoa_spm5$grid, type = "regular", by_polygon = TRUE), transform(msoa_spm6$grid, type = "hexagonal", by_polygon = TRUE), deparse.level = 1 )

rm(list = ls(pattern = "^msoa_spm")); invisible(gc(verbose = FALSE, full = TRUE))

ggplot(data = to_plot, aes(color = msoa11cd)) + guides(color = "none") + geom_sf(pch = 15) + scale_color_viridis_d(option = "H") + facet_grid(by_polygon ~ type, labeller = label_both) + theme_bw() + theme(axis.text = element_blank()) ```

For details on fitting models and making precitions, see this vignette.