Autocorrelation

spatial autocorrelation: correlation between \(Z(s_i)\) and \(Z(s_j)\) (the same attribute)

positive spatial autocorrelation: closer = similar attribute values; create a visual clustering in 3D - \((x,y,Z(s = (x,y)))\).

What is the degree to which data are autocorrelated?

Moran’s I:

For continuous attribute \(Z\) with \(E[Z(s)] = \mu\) as well as constant variance:

\[I = \frac{n}{(n-1)s^2 w_{..}} \sum_{i=1}^n \sum_{j=1}^n w_{ij} (Z(s_i) - \bar{Z})(Z(s_j)-\bar{Z})\] \[\textrm{ where } w_{..} = \sum_i \sum_j w_{ij}\]

• If \(I > \frac{-1}{n-1}\): a location tends to be connected to locations with similar attribute values.

• If \(I < \frac{-1}{n-1}\): attribute values at locations connected to a particular location tend to be dissimilar

• Local Moran (Anselin 1995): \[I_i = n (Z_i - \bar{Z}) \frac{\sum_j w_{ij} (Z_j - \bar{Z})}{\sum_i (Z_i - \bar{Z})^2}\]

Moran’s I:

Can <- st_read("scottish_lip_cancer.shp")

## Reading layer `scottish_lip_cancer' from data source 
##   `G:\My Drive\My Documents\Work\Conferences and Seminars\Courses\3MC 2024\Inger's Notes\scottish_lip_cancer.shp' 
##   using driver `ESRI Shapefile'
## Simple feature collection with 56 features and 11 fields
## Geometry type: MULTIPOLYGON
## Dimension:     XY
## Bounding box:  xmin: -8.621389 ymin: 54.62722 xmax: -0.7530556 ymax: 60.84444
## Geodetic CRS:  GCS_Assumed_Geographic_1

plot(Can)

## Warning: plotting the first 10 out of 11 attributes; use max.plot = 11 to plot
## all

nb <- poly2nb(Can, queen=TRUE)
weights<-nb2listw(nb, style="W")
moran.test(Can$SMR, weights)

## 
##  Moran I test under randomisation
## 
## data:  Can$SMR  
## weights: weights    
## 
## Moran I statistic standard deviate = 6.9173, p-value = 2.301e-12
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic       Expectation          Variance 
##       0.599660901      -0.018181818       0.007977681

moran.mc(Can$SMR, weights, nsim = 999)

## 
##  Monte-Carlo simulation of Moran I
## 
## data:  Can$SMR 
## weights: weights  
## number of simulations + 1: 1000 
## 
## statistic = 0.59966, observed rank = 1000, p-value = 0.001
## alternative hypothesis: greater

m <- localmoran(Can$SMR, weights)
image(m)

nb <- poly2nb(Can, queen=FALSE)
weights<-nb2listw(nb, style="W")
moran.test(Can$SMR, weights)

## 
##  Moran I test under randomisation
## 
## data:  Can$SMR  
## weights: weights    
## 
## Moran I statistic standard deviate = 6.9173, p-value = 2.301e-12
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic       Expectation          Variance 
##       0.599660901      -0.018181818       0.007977681

moran.mc(Can$SMR, weights, nsim = 999)

## 
##  Monte-Carlo simulation of Moran I
## 
## data:  Can$SMR 
## weights: weights  
## number of simulations + 1: 1000 
## 
## statistic = 0.59966, observed rank = 1000, p-value = 0.001
## alternative hypothesis: greater

m <- localmoran(Can$SMR, weights)
image(m)

Geary’s C:

For continuous attribute \(Z\) with \(E[Z(s)] = \mu\) as well as constant variance we have a type of autocorrelation measure:

\[C = \frac{1}{2S^2 w_{..}} \sum_{i=1}^n \sum_{j=1}^n w_{ij} \left(Z(s_i)-Z(s_j) \right)^2\]

If \(C>1\) the locations are connected to locations with dissimilar values and vice versa for \(C<1\).

The constant mean and variance is important: if this is not true the similarity/dissimilarity is more likely due to the heterogeneous mean and variance.

Geary’s C:

g <- geary(x=Can$SMR,listw=weights,n=56,n1=55,Szero(weights))
g$C

## [1] 0.4632456