• Some processes are essentially spatial
  • Weather patterns
  • Epidemic spreading
  • Global temperature patterns
• Need to summarise and visualise these processes to help understand them
  • Here we can do this in R with tmap, sf and other helpers
  • Need to think about space and time

• Quite a number of differences over borders: \(I\) = -0.314

• Mixture of differences/similarities across borders: \(I\) = -0.03

• Mainly same sign across borders: \(I\) = 0.515

• You need my function morans_I in the file morans_i.R
• The files are in the usual folder on Teams
• Also need the adjacency information - thats in adjacency.csv
• read all of this in to R…

source('morans_I.R')
adj <- read_csv('adjacency.csv')
adj

## # A tibble: 7 x 2
##   County1      County2
##   <chr>        <chr>  
## 1 Armagh       Antrim 
## 2 Down         Antrim 
## 3 Londonderry  Antrim 
## 4 Tyrone       Antrim 
## 5 Antrim       Armagh 
## 6 Down         Armagh 
## 7 Louth County Armagh

• If the levels of rainfall were scattered randomly amoungs the counties, there would be some degree of clustering
• Think of the star constellations in session 1!
• So, are the 0.5-ish levels seen for rainfall indicative of clustering or just a random pattern?
• To test this, start by assuming levels are just randomly scattered
• then…

1. Randomly assign each rainfall value to a county
2. Compute the Moran’s-\(I\) for that random assignment
3. Note the value
4. Repeat from step 1 a large number of times (eg 1,000)
5. Compare the actual Moran’s-\(I\) to these
6. The number of random values exceeding the actual value is an estimate of the chance that the actual value or something more extreme could have happened if pattern really was random.
7. Very low values of this suggest pattern is clustered, not random

• In R this is done using a loop

# make a list for randomised Moran's I values
# start by filling with zeroes then fill in values
random_m_I <- rep(0,10000) 

## Loop to calculate the randomised Moran's I's
for (i in 1:10000) {
  test <- mutate(counties,mean_rain=sample(mean_rain)) # Randomise
  random_m_I[i] <- morans_I(test,adj,County,mean_rain) # calculate I
}

## Quick summary of results
fivenum(random_m_I)

## [1] -0.31701185 -0.09599609 -0.03478540  0.03124031  0.42912956

hist(random_m_I,xlim=c(-0.7,0.7),
     main="Randomised Values of Moran's I",
     xlab="Moran's I")
abline(v=m_I,col='brown',lwd=2)
text(0.5,1000,'Actual Value',srt=90,col='brown')

• Note ylim not xlim here!!

boxplot(random_m_I, horizontal=TRUE,
     main="Randomised Values of Moran's I",
     xlab="Moran's I",ylim=c(-0.7,0.7))
abline(v=m_I,col='brown',lwd=2)
text(0.5,1,'Actual Value',srt=90,col='brown')

dens <- density(random_m_I)
plot(dens,
     main="Randomised Values of Moran's I",
     xlab="Moran's I",xlim=c(-0.7,0.7))
abline(v=m_I,col='brown',lwd=2)
text(0.5,2,'Actual Value',srt=90,col='brown')

length(which(random_m_I > m_I))/10000

## [1] 0

• Effectively zero!
• In reality not exactly zero BUT
  • In 10,000 random simulations none exceed the actual value
  • Possibly in 10 million dimulations some may have
  • Pretty strong evidence for clustering of values
  • Positive spatial autocorrelation

• which is easier to read?

Ordinary R:

high_counties <- st_drop_geometry(arrange(filter(counties,mean_rain > 80),
                                          mean_rain))

Tidyverse:

library(tidyverse)
counties %>% filter(mean_rain > 80) %>% 
  arrange(mean_rain) %>% 
  st_drop_geometry() -> high_counties

• Generally tidyverse most useful for processing data tables
• Should you use ‘->’ or ‘<-’ in tidyverse expressions?
  • I don’t mind, its a personal choice!
  • But try to stick to one or the other!