Map of bivariate spatial correlation in R (bivariate LISA)

bivariate_lisa.R

library(stringr)
library(spdep)
library(rgdal)
library(magrittr)
library(ggplot2)
library(sf)



#======================================================

# Load data
  map <- readOGR(dsn="R:/Dropbox/Dout/Data Dout", layer="test_map")
  head(map@data)
  
# Variables to use in the correlation: white and black population in each census track
  x <- map$income
  y <- map$diffaccess

  
  
  
#======================================================
# Programming some functions

# Bivariate Moran's I
moran_I <- function(x, y = NULL, W){
  if(is.null(y)) y = x
  
  xp <- scale(x)[, 1]
  yp <- scale(y)[, 1]
  W[which(is.na(W))] <- 0
  n <- nrow(W)
  
  global <- (xp%*%W%*%yp)/(n - 1)
  local  <- (xp*W%*%yp)
  
  list(global = global, local  = as.numeric(local))
}


# Permutations for the Bivariate Moran's I
simula_moran <- function(x, y = NULL, W, nsims = 2000){
  
  if(is.null(y)) y = x
  
  n   = nrow(W)
  IDs = 1:n
  
  xp <- scale(x)[, 1]
  W[which(is.na(W))] <- 0
  
  global_sims = NULL
  local_sims  = matrix(NA, nrow = n, ncol=nsims)
  
  ID_sample = sample(IDs, size = n*nsims, replace = T)
  
  y_s = y[ID_sample]
  y_s = matrix(y_s, nrow = n, ncol = nsims)
  y_s <- (y_s - apply(y_s, 1, mean))/apply(y_s, 1, sd)
  
  global_sims  <- as.numeric( (xp%*%W%*%y_s)/(n - 1) )
  local_sims  <- (xp*W%*%y_s)
  
  list(global_sims = global_sims,
       local_sims  = local_sims)
}



#======================================================
# Adjacency Matrix (Queen)

nb <- poly2nb(map)
lw <- nb2listw(nb, style = "B", zero.policy = T)
W  <- as(lw, "symmetricMatrix")
W  <- as.matrix(W/rowSums(W))
W[which(is.na(W))] <- 0
  
  
#======================================================
# Calculating the index and its simulated distribution
# for global and local values

m <- moran_I(x, y, W)

# Global Moral
  global_moran <- m[[1]][1]
  #> 0.2218409

# Local values
  m_i <- m[[2]] 

# local simulations
  local_sims <- simula_moran(x, y, W)$local_sims


# global pseudo p-value  
  # get all simulated global moran
  global_sims <- simula_moran(x, y, W)$global_sims

  # Proportion of simulated global values taht are higher (in absolute terms) than the actual index 
  moran_pvalue <- sum(abs(global_sims) > abs( global_moran )) / length(global_sims)
  #> 0

  
# Identifying the significant values 
  alpha <- .05  # for a 95% confidence interval
  probs <- c(alpha/2, 1-alpha/2)
  intervals <- t( apply(local_sims, 1, function(x) quantile(x, probs=probs)))
  sig       <- ( m_i < intervals[,1] )  | ( m_i > intervals[,2] )
  


#======================================================
# Preparing for plotting


# Convert shape file into sf object
map_sf     <- st_as_sf(map)
map_sf$sig <- sig


# Identifying the LISA clusters
  xp <- scale(x)[,1]
  yp <- scale(y)[,1]
  
  
  patterns <- as.character( interaction(xp > 0, W%*%yp > 0) )
  patterns <- patterns %>% 
    str_replace_all("TRUE","High") %>% 
    str_replace_all("FALSE","Low")
  
  patterns[map_sf$sig==0] <- "Not significant"
  map_sf$patterns <- patterns


  # Rename LISA clusters
  map_sf$patterns2 <- factor(map_sf$patterns, levels=c("High.High", "High.Low", "Low.High", "Low.Low", "Not significant"),
                                  labels=c("High income - High access gain", "High income - Low access gain", "Low income - High access gain","Low income - Low access gain", "Not significant"))
  


  

### PLOT

ggplot() +
  geom_sf(data=map_sf, aes(fill=patterns2), color="NA") +
  scale_fill_manual(values = c("red", "pink", "light blue", "dark blue", "grey80")) + 
  guides(fill = guide_legend(title="LISA clusters")) +
  theme_minimal()

lisa_clusters.jpeg

test_map.zip

Thanks for the code! It worked perfectly.
I feel like I can add my 2 cents here... For those having trouble with the non-conformable multiplication (xp %*% W), I had the same issue as I was using a distance-based weights matrix. What worked for me was to use the 'nb2mat' function to transform the listw object into a matrix that the function could read adequately.

So, for example:
W <- nbsmat(listw$neigbours, glist = listw$weights, style = "W", zero.policy = T) #Where listw = the weights list object

Hope this helps. And thanks again for providing the code.

@Tito0411 I recommend using the sfdep::moran_bv() function :)

Hi @JosiahParry ! Thanks for chipping in. I wrote this gist a long time ago in 2016/2017. Lots of things have changed since then and I totally agree the best solution here would be using the sfdep package. BTW, thank you for such a great package !!!