5.2 Spatial neighborhood matrices

Packages needed:

library(tidyverse)
library(sf)
library(spdep)
library(SpatialEpi)
library(INLA)
library(cowplot)

A spatial model is a particular type of model as it takes consideration of autocorrelation between spatial points and so some weights are considered to identify the points’ proximity.

\[y=\sum_{i=1}^n{w_if(x_i)}\]

\(\mathit{W}\) is the proximity matrix or the inverse distance between areas and spatially connects areas \(i\) and \(j\), its elements \((i,j)\) , are:

\[i=(1,...,n)\] \[j=(1,...,m)\]

\[W=\begin{bmatrix} (1,1) & (1,2) & ... & (1,j)\\ (2,1) & ... & ... & (2,j)\\ ... & ... & ... & ...\\ (i,1) & ...& ... &(i,j)\\ \end{bmatrix}\]

The most simple cases are:

\[w_{i,j}=\left\{\begin{matrix} 1 & \text{ if i and j share same boundaries}\\ 0 & \text{ otherwise} \end{matrix}\right.\]

As an alternative, a standardized matrix can be used, adjusted for the total number of neighbors in each area:

\[w_{std,i,j}=\frac{w_{i,j}}{\sum_{j=1}^n{w_{ij}}}\]

Let’s have a look at the data: Lung Cancer in Pennsylvania

# install.packages("SpatialEpi")
# library(SpatialEpi)

map <- pennLC$spatial.polygon
plot(map)

And construct the neighbors list for each county:

# install.packages("spdep")
# library(spdep)
nb <- spdep::poly2nb(map)
nb%>%class
[1] "nb"
head(nb,3)
[[1]]
[1] 21 28 67

[[2]]
[1]  3  4 10 63 65

[[3]]
[1]  2 10 16 32 33 65
d <- data.frame(county = names(map), 
                neigh = rep(0, length(map)))

rownames(d) <- names(map)

map <- SpatialPolygonsDataFrame(map, d, match.ID = TRUE)
map$neigh[nb[[2]]] <- 1
map$neigh[nb[[44]]] <- 1
map$neigh[nb[[58]]] <- 1
map%>%as_tibble()%>%head
# A tibble: 6 × 2
  county    neigh
  <chr>     <dbl>
1 adams         0
2 allegheny     0
3 armstrong     1
4 beaver        1
5 bedford       0
6 berks         0
coord <- coordinates(map)
map$long <- coord[, 1]
map$lat <- coord[, 2]
map$ID <- 1:dim(map@data)[1]
map@proj4string
## Coordinate Reference System:
## Deprecated Proj.4 representation: +proj=longlat
map%>%as_tibble()%>%head
## # A tibble: 6 × 5
##   county    neigh  long   lat    ID
##   <chr>     <dbl> <dbl> <dbl> <int>
## 1 adams         0 -77.2  39.9     1
## 2 allegheny     0 -80.0  40.5     2
## 3 armstrong     1 -79.5  40.8     3
## 4 beaver        1 -80.3  40.7     4
## 5 bedford       0 -78.5  40.0     5
## 6 berks         0 -75.9  40.4     6
# install.packages("sf")
# library(sf)
mapsf <- sf::st_as_sf(map)
# ggplot(mapsf)+
#   geom_sf()

ggplot(mapsf) + 
  geom_sf(aes(fill = as.factor(neigh))) +
  geom_text(aes(long, lat, label = ID), color = "white") +
  guides(fill = "none") +
  theme_bw()
Neighbors of areas 2, 44 and 58 of Pennsylvania

Figure 5.1: Neighbors of areas 2, 44 and 58 of Pennsylvania