Exploratory Spatial Data Analysis (ESDA) & Spatial autocorrelation

Spatial autocorrelation is a method of Exploratory Spatial Data Analysis (ESDA). The latter set of methods allow for the study and understanding of the spatial distribution and spatial structure as well as they allow for detecting spatial dependence or autocorrelation in spatial data. More specifically, spatial autocorrelation is the correlation between the values of a single variable that is strictly due to the proximity of these values in geographical space by introducing a deviation from the assumption of independent observations of classical statistics (Griffith, 2003). The most common spatial autocorrelation indicators in the literature are: the Moran’s I, the Geary’s c, and the Getis’ G.

Moran’s I

Moran’s I is one of the oldest statistics used to examine spatial autocorrelation. This global statistic was first proposed by Moran (1948, 1950). Later, Cliff and Ord (1973, 1981) present a comprehensive work on spatial autocorrelation and suggested a formula to calculate the I which is now used in most textbooks and software: $I = \frac{n}{W} \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} w_{ij} z_i z_j} {\sum_{i=1}^{n} z_i^2}$ where n is number of observations, W is the sum of the weights w_ij for all pairs in the system, z_i = x_i − x̄ where x is the value of the variable at location i and x̄ the mean value of the variable in question (Eq. 5.2 Kalogirou, 2003).

The original implementation here (function moransI) allows only nearest neighbour weighting schemes. However, the function moransI.w allows for any weighting scheme if by means of a weighting matrix. Using the package lctools, the latter can be computed by the function w.matrix that currently supports a fixed (straight distance) and an adaptive (number of nearest neighbours) spatial kernels. Resampling and randomization null hypotheses have been tested following the discussion of Goodchild (1986, pp. 24-26).

Case 1: function MoransI

Coords <- cbind(GR.Municipalities@data$X, GR.Municipalities@data$Y)
bw <- 6
mI <- moransI(Coords,bw,GR.Municipalities@data$Income01)
moran.table <- matrix(data=NA,nrow=1,ncol=6)
col.names <- c("Moran's I", "Expected I", "Z resampling", "P-value resampling",
               "Z randomization", "P-value randomization")
colnames(moran.table) <- col.names
moran.table[1,1] <- mI$Morans.I
moran.table[1,2] <- mI$Expected.I
moran.table[1,3] <- mI$z.resampling
moran.table[1,4] <- mI$p.value.resampling
moran.table[1,5] <- mI$z.randomization
moran.table[1,6] <- mI$p.value.randomization

Using table formatting the results can be shown as follows:

Case 2: functions w.matrix and MoransI.w

The above calculation can be repeated as follows:

#adaptive kernel
w.adaptive <- w.matrix(Coords,6, WType='Binary', family='adaptive')

mI.adaptive <- moransI.w(GR.Municipalities@data$Income01, w.adaptive)
mI.adaptive <- t(as.numeric(as.matrix(mI.adaptive[1:6])))
colnames(mI.adaptive) <- col.names

Using table formatting the results can be shown as follows:

An example Moran’s I calculation with a fixed spatial kernel and a 50 km radius as the bandwidth can be computed as follows:

#fixed kernel
w.fixed <- w.matrix(Coords, 50000, WType='Binary', family='fixed')

## The bandwidth  50000  will result  5  observations without neighbours. 
## To avoid this you could set the bandwidth to at least:  146287.7

mI.fixed<- moransI.w(GR.Municipalities@data$Income01, w.fixed)
mI.fixed <- t(as.numeric(as.matrix(mI.fixed[1:6])))

colnames(mI.fixed) <- col.names

The above bandwidth is such that 5 observations will have no neighbours. This is a result of a check that is printed out from the moransI.w function. These observations refer to isolated islands, such as Megisti east of Rhodes, which is 150 km away from its nearest neighbour. Using table formatting the results can be shown as follows:

The above results suggest a strong positive spatial autocorrelation that is statistically significant using either the randomization or resampling hypotheses (Cliff and Ord, 1973; 1981; Goodchild, 1986). In order to examine the sensitivity of the above results, one could try different bandwidth sizes (i.e. number of nearest neighbours). This can be done by either a loop or by using the function moransI.v that computes a number of Moran’s I statistics with different kernel sizes of the same family and optionally shows a scatter plot with the Moran’s I for each kernel size.

bws <- c(3, 4, 6, 9, 12, 18, 24)
moran <- moransI.v(Coords, bws, GR.Municipalities@data$Income01)

The Moran’s I along with significance tests for all spatial kernels are presented in the table below. The Moran’s I statistics for the mean annual recorded income is above 0.5 in all cases suggesting a strong positive spatial autocorrelation. They are also statistically significant in all cases.