Spatial analysis with geostan

0.1 Summary

This vignette introduces the geostan R package for spatial analysis. The package is mainly oriented towards areal data, although some models may also be used for other spatial data types, such as network data. The package implements the spatial error/simultaneous spatial autoregressive (SAR) model, conditional autoregressive (CAR) model, and eigenvector spatial filter (ESF) models for spatial regression. A version of ESF modelling also appears in the ecology literature as principle coordinate analysis of neighbor matrices (PCNM) (Griffith and Peres-Neto 2006).

geostan also supports the application of the above regression methods to hierarchical models for count data, as is common in analyses of disease incidence or mortality in small areas (‘disease mapping’). Additional features of the software include models for sampling/measurement error in covariates and methods for handling censored count data, such as when mortality or disease counts have been censored for privacy. The models were built using the Stan modeling language, so all inference is completed using Markov chain Monte Carlo (MCMC) sampling (Stan Development Team 2023; Gabry et al. 2024). The spatial autoregressive models use custom-built Stan functions that speed up MCMC sampling considerably (Donegan 2021).

This post will walk through an example analysis using international data on life expectancy and per capita GDP. Package vignettes can be found with the online documentation, including an introduction to spatial weights matrices, exploratory spatial data analysis, spatial measurement error models, raster regression, and using geostan to build custom spatial models with Stan. A paper in the Journal of Open Source Software reports these and other features and provides the recommended citation when using geostan (Donegan 2022).

0.2 Installation

geostan is currently on CRAN, although that may not always be the case. You can also install directly from geostan’s github repository.

if (!require('devtools')) install.packages('devtools')
devtools::install_github("connordonegan/geostan")

install.packages("geostan")

0.3 Getting started

To begin, load the geostan and sf packages into your R environment, as well as the world data:

library(geostan)
library(sf)
data(world, package = "spData")

The world data contains life expectancy and gross domestic product (GDP) per capita (presumably measured in current $US) for 161 countries as of 2014, gathered from the World Bank. The rest of this post is going to be structured around a bivariate analysis of these variables.

We are going to apply the Robinson map projection for the countries:

world <- st_transform(world, crs = 'ESRI:54030')

At least a couple of the missing values can be filled in using World Bank data, so we will do that:

## https://data.worldbank.org
france <- grep("France", world$name_long)
world$gdpPercap[ france ] <- 43068
world$lifeExp[ france ] <- 82

norway <- grep("Norway", world$name_long)
world$gdpPercap[ norway ] <- 97666
world$lifeExp[ norway ] <- 82.1

And we will also remove Antarctica:

world <- subset(world, name_long != "Antarctica")

Mapping the variables shows the familiar geography of high-, middle-, and low-income countries and a similar geography of longevity:

# store geometry for countries
world_geom <- st_geometry(world)

# show two maps at once, with nice font
ogpar <- par(mfrow = c(2, 1),
             mar = rep(0, 4))

# GDP per capita
pars <- map_pars(world$gdpPercap / 1e3)
plot(world_geom,
     col = pars$col,
     lwd = .2)
legend("bottomleft",
       fill = pars$pal,
       title = 'GDP per capita\n($1,000s)',
       legend = pars$lbls,
       bty = 'n'
)
rm(pars)

# life expectancy
pars <- map_pars(world$lifeExp)
plot(world_geom,
  col = pars$col,
  lwd = .2)
legend("left",
     fill = pars$pal,
     title = 'Life Expectancy',
     legend = pars$lbls,
     bty = 'n'
     )

Choropleth maps of GDP per capita and life expectancy.

par(ogpar)

The map_pars function breaks the variables into quantiles and returns breaks, colors, labels for the maps; it can be found at the end of the post. There will be no discussion of substantive (non-statistical) issues here, for which one can consult any number of texts on global power and inequality (such as Birn, Pillay, and Holz’s Textbook of International Health or Paul Farmer’s Infections and Inequalities).

By conventional methods, the correlation coefficient for life expectancy and log GDP per capita is 0.81:

log_x <- log10( world$gdpPercap )
y <- world$lifeExp
cor.test(log_x, y)

## 
##  Pearson's product-moment correlation
## 
## data:  log_x and y
## t = 17.394, df = 160, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.7478143 0.8561778
## sample estimates:
##       cor 
## 0.8087528

The conventional assessment is based on the proposition that we have 161 independent observations. The visible geography of the variables, and any level of social awareness, indicates that these are not independent observations. Rather, there are various functional regions of countries that share basic political-economic conditions. A lot, but not all, of the variation can be described as variation across continents and regions. We will want to account for this dependence using a spatial model (for background see Chun and Griffith 2012; Donegan 2024). The first step will be to construct a spatial weights matrix.

0.4 Adjacency matrix

This section will illustrate use of two geostan functions for creating an revising a spatial weights matrix: shape2mat and edges. The shape2mat function may be helpful for some users but one can always do this using spdep or other methods, especially if shape2mat does not provide the exact method you’re looking for.

We are going to start by removing the 15 countries that are missing values:

## remove missing values
world <- subset(world, !is.na(gdpPercap) & !is.na(lifeExp))

## leaving 162 observations
nrow(world)

## [1] 162

Now we can apply the shape2mat function to obtain an adjacency matrix that encodes spatial adjacency relations for countries into a binary N-by-N matrix. The function uses spdep to find adjacency relations and returns results as a sparse matrix (using the Matrix package):

A <- shape2mat(world, "B", method = "rook")

## Warning in spdep::poly2nb(shape, queen = queen, snap = snap): some observations have no neighbours;
## if this seems unexpected, try increasing the snap argument.

## Warning in spdep::poly2nb(shape, queen = queen, snap = snap): neighbour object has 19 sub-graphs;
## if this sub-graph count seems unexpected, try increasing the snap argument.

## Contiguity condition: rook

## Number of neighbors per unit, summary:

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   2.000   3.000   3.605   5.000  13.000

## 
## Spatial weights, summary:

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##       1       1       1       1       1       1

Visualizing the connections in the matrix is important for uncovering unexpected results. geostan’s edges function converts the matrix into a list of nodes and edges that we can plot. For this we need to supply the function with the adjacency matrix, A, and the associated spatial object, world:

# edges with geometry
E <- edges(A, shape = world) 
graph <- st_geometry(E)

ogpar <- par(mar = rep(0, 4))
# plot countries
plot(world_geom, lwd = .1)
# add graph nodes
plot(graph, add = TRUE, type = 'p')
# add graph edges
plot(graph, add = TRUE, type = 'l')

par(ogpar)

This reveals quite a few unexpected results. French Guiana is stored in the world data as part of France (a multi-part polygon); this is correct of course but it leads to Brazil and Suriname being listed as neighbors of France, which is not sensible. Besides removing those connections, there are a number of island nations that we might want to connect to nearby places.

To connect Mozambique to Madagascar, we just replace the zeroes with ones in the slots that correspond to those countries. First we grab their index positions in the matrix:

moz_idx <- grep("Mozambique", world$name_long)
mad_idx <- grep("Madagascar", world$name_long)

And then we assign the correct slots in the matrix a value of 1 (or TRUE), remembering that the adjacency matrix is symmetric:

A[moz_idx, mad_idx] <- A[mad_idx, moz_idx] <- TRUE

This can become tedious but it is important. Before moving on, we will make a series of adjustments. This will be made a bit easier with this convenience function:

connect <- function(country_a, country_b,
    names_vec = world$name_long, matrix = A, add = TRUE) {
  stopifnot( country_a %in% names_vec )
  stopifnot( country_b %in% names_vec )
  a_idx <- which(names_vec == country_a)
  b_idx <- which( names_vec == country_b)
  matrix[a_idx, b_idx] <- matrix[b_idx, a_idx] <- add
  return( matrix )
}

The following are at least reasonable changes to make; they also ensure that every country has at least one neighbor:

A <- connect("Mozambique", "Madagascar")
A <- connect("Australia", "New Zealand")
A <- connect("Philippines", "Malaysia")
A <- connect("Japan", "Republic of Korea")
A <- connect("Fiji", "Vanuatu")
A <- connect("Solomon Islands", "Vanuatu")
A <- connect("Solomon Islands", "Papua New Guinea")
A <- connect("Australia", "Papua New Guinea")
A <- connect("Haiti", "Jamaica")
A <- connect("Bahamas", "United States")
A <- connect("Dominican Republic", "Puerto Rico")
A <- connect("Trinidad and Tobago", "Venezuela")
A <- connect("Sri Lanka", "India")
A <- connect("Cyprus", "Turkey")
A <- connect("Cyprus", "Lebanon")
A <- connect("Norway", "Iceland")

## remove connections between South American and France
A <- connect("Suriname", "France", add = FALSE)
A <- connect("Brazil", "France", add = FALSE)

We should look at the revised adjacency matrix:

graph <- st_geometry( edges(A, shape = world) )
ogpar <- par(mar = rep(0, 4))
plot(world_geom, lwd = .1)
plot(graph, add = TRUE, type = 'p')
plot(graph, add = TRUE, type = 'l')

par(ogpar)

Sometimes it can help to examine the edge list interactively using a proper geographic information system like QGIS. For those who are familiar with (non-R) GIS software, you can save the edges list as a GeoPackage and then open it up in your GIS to examine the connections ‘by hand’ with a base map or other data:

E <- edges(A, shape = world)
st_write(E, "world.gpkg", layer = "edge list")

0.5 Non-spatial regression

Fitting regression models with geostan is similar to using base R’s glm function: the user provides a model formula, data, and the model family or distribution. We can fit a normal linear model using the stan_glm function:

fit_lm <- stan_glm(lifeExp ~ log(gdpPercap), data = world, iter = 800, quiet = TRUE)

The iter argument will be discussed below; we used iter = 800 (lower than the default) to keep this demo running quickly. We can examine parameter estimates by printing to the console:

print(fit_lm)

## Spatial Model Results 
## Formula: lifeExp ~ log(gdpPercap)
## Likelihood:  gaussian 
## Link:  identity 
## Spatial method:  none 
## Observations:  162 
## 
## Inference for Stan model: foundation.
## 4 chains, each with iter=800; warmup=400; thin=1; 
## post-warmup draws per chain=400, total post-warmup draws=1600.
## 
##                  mean se_mean    sd   2.5%    20%    50%    80%  97.5% n_eff
## intercept      20.649   0.135 2.922 15.000 18.134 20.668 23.160 26.351   465
## log(gdpPercap)  5.501   0.015 0.316  4.872  5.234  5.495  5.773  6.101   456
## sigma           4.901   0.010 0.274  4.391  4.672  4.887  5.137  5.484   786
##                 Rhat
## intercept      1.005
## log(gdpPercap) 1.005
## sigma          1.001
## 
## Samples were drawn using NUTS(diag_e) at Sat Nov 16 12:25:25 2024.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

The output printed to the console provides a summary of the posterior probability distributions of the model parameters. The distributions can also be visualized using plot(fit_lm):

plot(fit_lm)

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

The mean of the distribution is reported in the mean column. For those who are more familiar with concepts from sampling theory, the mean may be understood as the estimate of the parameter. Each distribution’s standard deviation is found in the sd column; this describes the width of the posterior distribution. The sd is analogous to the standard error of the estimate. The quantiles also summarize the width of the posterior distributions; the 2.5% and 97.5% values form a 95% credible interval for the parameter value.

fdf <- fitted(fit_lm)
head(fdf)

##               mean        sd     2.5%      20%      50%      80%    97.5%
## fitted[1] 70.23709 0.3728245 69.49739 69.92302 70.22702 70.56993 70.93080
## fitted[2] 63.46833 0.5744286 62.35143 62.95940 63.47290 63.96314 64.58705
## fitted[3] 79.34752 0.5991503 78.17483 78.85867 79.34689 79.85888 80.51738
## fitted[4] 80.37454 0.6466143 79.12227 79.84581 80.36746 80.92272 81.65540
## fitted[5] 76.03422 0.4643730 75.13398 75.63451 76.04287 76.42418 76.94885
## fitted[6] 67.89454 0.4134008 67.06976 67.54904 67.88968 68.25030 68.66991

rdf <- resid(fit_lm)
moran_plot(rdf$mean, A)

0.6 Spatial regression

Options for spatial regression models currently include conditional autoregressive (CAR), simultaneous autoregressive (SAR/spatial error), and eigenvector spatial filtering (ESF). For count data, common variations on the intrinsic autoregressive (ICAR) model are also available.

All of the spatial models require at least a spatial weights matrix as input. All additional requirements for data preparation are handled by geostan’s prep_ functions: prep_car_data, prep_icar_data, prep_sar_data.

The make_EV function is used to create Moran’s eigenvectors for ESF regression; if you want to create your own eigenvectors (say, following the PCNM method) you can provide those directly to the ESF model (see ?stan_esf).

For the CAR model, we always provide the binary adjacency matrix as input to prep_car_data. See the prep_car_data documentation for options. Here we will fit an intercept-only CAR model to the life expectancy data:

cars <- prep_car_data(A, quiet = TRUE)
fit_car <- stan_car(lifeExp ~ 1, data = world, car_parts = cars, iter = 800, quiet = TRUE)

print(fit_car)

## Spatial Model Results 
## Formula: lifeExp ~ 1
## Likelihood:  auto_gaussian 
## Link:  identity 
## Spatial method:  CAR 
## Residual Moran Coefficient:  -0.355875 
## Observations:  162 
## 
## Inference for Stan model: foundation.
## 4 chains, each with iter=800; warmup=400; thin=1; 
## post-warmup draws per chain=400, total post-warmup draws=1600.
## 
##             mean se_mean    sd   2.5%    20%    50%    80%  97.5% n_eff  Rhat
## intercept 70.071   0.082 2.627 64.539 67.941 70.172 72.102 75.217  1037 1.000
## car_rho    0.980   0.000 0.011  0.955  0.973  0.982  0.989  0.995   988 1.000
## car_scale  7.824   0.013 0.439  7.005  7.457  7.793  8.176  8.761  1156 0.999
## 
## Samples were drawn using NUTS(diag_e) at Sat Nov 16 12:25:28 2024.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Notice that using iter = 1000 was more than adequate for inference in this case.

The CAR model has a spatial dependence parameter car_rho. This parameter does not have an interpretation similar to a correlation coefficient, and it is often near 1; this is not a problem unless one misinterprets it or desires a value similar to the correlation coefficient. The spatial dependence parameter in the SAR model does provide that kind of interpretation.

theta <- spatial(fit_car)$mean

pars <- map_pars(theta)
ogpar <- par(mar = rep(0, 4))
plot(st_geometry(world),
  col = pars$col,
  lwd = .2)
legend("left",
     fill = pars$pal,
     title = 'Spatial trend (LE)',
     legend = pars$lbls,
     bty = 'n'
     )

par(ogpar)

# lifeExpt detrended
dy <- resid(fit_car)$mean

# log per capita GDP detrended
fit_carx <- stan_car(log(gdpPercap) ~ 1, data = world, car = cars, iter = 1e3, quiet = TRUE)
dx <- resid(fit_carx)$mean

# adjusted correlation
cor.test(dx, dy)

## 
##  Pearson's product-moment correlation
## 
## data:  dx and dy
## t = 9.1402, df = 160, p-value = 2.692e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4743301 0.6785994
## sample estimates:
##       cor 
## 0.5856886

W <- row_standardize(A)
sars <- prep_sar_data(W)

fit_sar <- stan_sar(lifeExp ~ log(gdpPercap), data = world, sar_parts = sars, centerx = TRUE,
                    iter = 800, quiet = TRUE)

plot(fit_sar)

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

world <- transform(world,
                  sx = scale(log(gdpPercap), scale = T, center = T),
                  sy = scale(lifeExp, scale = T, center = T)
                   )

fit_scaled <- stan_sar(sy ~ sx, data = world, sar_parts = sars, iter = 800, quiet = TRUE)
print(fit_scaled)

## Spatial Model Results 
## Formula: sy ~ sx
## Likelihood:  auto_gaussian 
## Link:  identity 
## Spatial method:  SAR (SEM) 
## Residual Moran Coefficient:  -0.0513575 
## Observations:  162 
## 
## Inference for Stan model: foundation.
## 4 chains, each with iter=800; warmup=400; thin=1; 
## post-warmup draws per chain=400, total post-warmup draws=1600.
## 
##            mean se_mean    sd   2.5%    20%   50%   80% 97.5% n_eff  Rhat
## intercept 0.020   0.003 0.119 -0.224 -0.075 0.020 0.115 0.254  1362 1.002
## sx        0.587   0.002 0.060  0.470  0.538 0.585 0.639 0.703  1389 0.998
## sar_rho   0.699   0.002 0.057  0.572  0.656 0.702 0.743 0.801  1215 1.000
## sar_scale 0.437   0.001 0.027  0.389  0.415 0.436 0.460 0.493  1458 1.000
## 
## Samples were drawn using NUTS(diag_e) at Sat Nov 16 12:25:36 2024.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

gdp <- range(world$gdpPercap)
min_gdp <- gdp[1]
max_gdp <- gdp[2]
pdf <- data.frame(gdpPercap = seq(min_gdp, max_gdp, length.out = 200))

preds <- predict(fit_sar, newdata = pdf)

head(preds)

##   gdpPercap     mean       sd     2.5%      20%      50%      80%    97.5%
## 1  597.1352 60.09070 1.532659 57.03877 58.85966 60.09465 61.27172 63.22281
## 2 1201.4715 62.90496 1.337918 60.29475 61.84333 62.89485 63.95095 65.68105
## 3 1805.8079 64.54508 1.240928 62.13350 63.57647 64.52757 65.52520 67.18923
## 4 2410.1442 65.70707 1.181702 63.34402 64.77785 65.68221 66.63414 68.17773
## 5 3014.4805 66.60768 1.142126 64.26974 65.72997 66.57256 67.50285 68.99647
## 6 3618.8169 67.34316 1.114365 65.08381 66.49068 67.30675 68.21869 69.67265

# scale GDP
preds <- transform(preds, gdpPercap = gdpPercap / 1e3)

yrange <- c(min(preds$`2.5%`), max(preds$`97.5%`))
plot(preds$gdpPercap, preds$mean,
     t = 'l',
     ylim = yrange,
     axes = F, 
     xlab = "GDP per capita ($1,000s)",
     ylab = "Life expectancy")
axis(1)
axis(2)

# add credible intervals
lines(preds$gdpPercap, preds$`2.5%`, lty = 3)
lines(preds$gdpPercap, preds$`97.5%`, lty = 3)

0.7 Future work and support

You can submit any questions, requests, or issues on the package issues page or the discussions page. geostan is still actively being developed so users are encouraged to check the package news page for updates.

If you are interesting contributing to the package you are encouraged to send an e-mail to the author or use the discussions page. You can submit a pull request with any bug fixes. Contributions that would make the package more useful to fields other than geostan’s current focus (human geography and public health), such as ecology, would be especially welcome.

0.8 Appendix

# function for getting colors, breaks, and labels for mapping
map_pars <- function(x, 
                     brks = quantile(x, probs = seq(0, 1, by = 0.2), na.rm = TRUE), 
                     cols = c("#A8554EFF", "gray95", "#5D74A5FF")) {

  # put x values into bins
  x_cut <- cut(x, breaks = brks, include.lowest = TRUE)
  
  # labels for each bin
  lbls <- levels( cut(x, brks, include.lowest = TRUE) )
  
  # colors 
  rank <- as.numeric( x_cut )  
  max_rank <- max( rank , na.rm = TRUE )
  pal_fun <- colorRampPalette( cols )
  pal <- pal_fun( max_rank )
  colors <-  pal[ rank ]

  # return list
  ls <- list(brks = brks, lbls = lbls, pal = pal, col = colors)
  return( ls )
}