---
title: "3. Areal Interpolation"
output: rmarkdown::html_vignette
bibliography: biblio.bib
vignette: >
  %\VignetteIndexEntry{3. Areal Interpolation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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


In applied settings, estimating socioeconomic and demographic variables for
different administrative regions within a city is often of interest. While areal
interpolation [AI, @goodchild1980areal] easily computes such estimates, it lacks
measures of uncertainty [@bradley2016bayesian]. Our package addresses this by
providing both areal interpolation and methods to approximate the variance of
estimates when the observed data's variance is known, as is common with survey
data.


This demonstration will utilize 2010 Brazilian Census data from Nova Lima. The
objective is to estimate some variables in a separate map of the same
municipality. The census map with the original data will be referred to as the
"source" map, while the map where the data will be estimated will be referred to
as the "target" map. In this example, the target map is artificially
generated. To initiate the process, the data and necessary packages can be
loaded using the code below. We recommend using a planar Coordinate Reference
System (CRS) when using this method.
```{r load-data}
library(sf)
library(smile)
library(ggplot2)

data(nl_ct)
nl_ct <- st_transform(nl_ct, 20823)
```

The code chunk below creates the synthetic target map.

```{r synthetic-data}
set.seed(2024)

## outer polygon
nl_outer <- nl_ct |>
  st_geometry() |>
  st_union() |>
  smile:::st_remove_holes()

## creating `target` data
points_nl <- st_sample(x = nl_outer,
                       size = 40)

nl_vor <- do.call(c, points_nl) |>
  ## voronoi tesselation
  st_voronoi(envelope = nl_outer) |> 
  st_collection_extract(type = "POLYGON") |>
  st_set_crs(st_crs(nl_ct)) |>
  st_intersection(nl_outer) |>
  st_as_sf()

## creating id variable
nl_vor <- transform(nl_vor, id = seq_len(NROW(nl_vor)))
```


Now, to estimate one (or several) variable observed at the `nl_ct` dataset into
the `nl_vor` data we can run the following command:

```{r est-1}
nl_ests <-
    ai(source = nl_ct,
       target = nl_vor,
       vars = c("hh_density",
                "avg_income",
                "avg_age"))

## displaying the result
nl_ests |>
  st_drop_geometry() |>
  head()
```

In the function above, `source` is the map/dataset where the variables to be
estimated are observed, while `target` is the map/dataset where the estimation
is desired. The `vars` argument is a character scalar (or vector) specifying the
variable(s) in `source` to be estimated in `target`. However, this function does
not quantify uncertainty in the estimates.


If the variance of the variable we wish to estimate in the target map is
available in the source map, we can use the `ai_var` function to quantify the
uncertainty around our point estimates. The function is similar to `ai`, but
with a few differences:

1. `vars` must be a single variable name.

1. `my_var` is a character indicating the variable in source containing the
   observed variable's variance.

1. `var_method` specifies the method for approximating variance: Moran's I
   ("MI") for autocorrelation-based approximation, or Cauchy-Schwarz ("CS") for
   an upper bound.

1. The current version only supports one variable at a time.

The `ai_var` function outputs the target dataset with two additional columns:
`est` (estimated variable) and `se_est` (approximate standard error). 

```{r est-2}
nl_est <-
    ai_var(source = nl_ct,
           target = nl_vor,
           vars = "hh_density",
           vars_var = "var_hhd",
           var_method = "MI")

## renaming geometry
st_geometry(nl_est) <- "geometry"
```

Below we use `ggplot2` to plot the observed and estimated "household density" in
the source and target maps, respectively.

```{r ggplot-obs-est, fig.alt="Panel displaying the estimated (on the left) and observed (on the right) household density in Nova Lima."}
viz_dt <-
    rbind(
        transform(nl_ct, source = "Observed",
                  est = hh_density)[c("source", "est")],
        transform(nl_est, source = "Estimated")[c("source", "est")]
    )

ggplot(data = viz_dt,
       aes(fill = est)) +
    geom_sf(color = 1, lwd = .1) +
    scale_fill_viridis_c(option = "H",
                         name = "Household \n density") +
    theme_bw() +
    facet_wrap( ~ source) +
    theme(axis.text = element_blank(),
          axis.ticks = element_blank())
```

Finally, the next map displays the uncertainty about the estimated household
density.

```{r ggplot-obs-est-viz, eval = TRUE, echo = FALSE, fig.alt="Standard error associated to the estimates displayed in the last graph."}
ggplot(data = nl_est,
       aes(fill = se_est)) +
    geom_sf(color = 1, lwd = .1) +
    scale_fill_viridis_c(option = "H",
                         name = "SE(Household \n density)") +
    theme_bw() +
    theme(axis.text = element_blank(),
          axis.ticks = element_blank())
```

# Reference
