Global Moran’s I Coefficient of Spatial Autocorrelation
Moran's I measures the degree of spatial autocorrelation (Moran 1950) in ordinal- and interval-measured data. It is one of the widely used indices that evaluates the extent of spatial autocorrelation between a set of n cells = {*,} located in neighboring areas, where x, is either the rank of the ith cell (ordinal data) or the value of X in the ith cell (interval data). The computation of Moran's l is achieved by dividing the spatial covariation by the total variation. The resultant values range from approximately -1 (perfect dispersion) to 1 (perfect correlation). The positive sign represents positive spatial autocorrelation, while the converse is true for the negative sign, and a zero result represents no spatial autocorrelation (Figure 7.5).
Suppose we have a study region, R, which is subdivided into n cells, where each cell is identified with a spatial feature. Moran's I is calculated as follows:
where w,y= 1 if cells/and /are neighbors, w,_{t}=0 otherwise; c,_{;} = (X, -X)(X, - X), X, and X, are variables at a particular and another location, respectively:
The average of all the n cells is the mean (X), which is used to compute (s^{2}) based on the differences that each X value has from the mean (X).
TASK 7.2 SPATIAL DISTRIBUTION OF LOW BIRTH WEIGHT RATES IN A STUDY REGION A
Figure 7.6 depicts the rates of low birth weight per 1000 children in hypothetical region A. The values in the upper left corner represent the unique identifier for the enumeration spatial units, and values in the center represent the low birth weight rates. Using Moran's I, we can determine the type of areal pattern in this figure. Table 7.2 presents a worktable and results for Moran's I.
и = 4 Xj = 3 Юу_{=}0 110 X_{2} = 2 1 0 0 1 X_{3} = 4 1 0 0 1 X_{4}=701 10
FIGURE 7.6
A regular grid/spatial units of low birth weight example.
Worktable for Deriving Global Moran's I Coefficient for Low Birth Weight Rates
i |
j |
«У |
(X,-X)(X;-X) |
TZWijCy |
i |
2 |
1 |
(3 - 41(2 - 4) |
2 |
i |
3 |
1 |
(3 - 41(4 - 4) |
0 |
i |
4 |
0 |
(3 - 41(7 - 4) |
0 |
2 |
1 |
1 |
(2 - 41(3 - 4) |
2 |
2 |
3 |
0 |
(2 - 41(4 - 4) |
0 |
2 |
4 |
1 |
(2 - 41(7 - 4) |
-6 |
3 |
1 |
1 |
(4 - 41(3 - 4) |
0 |
3 |
2 |
0 |
(4 - 41(2 - 4) |
0 |
3 |
4 |
1 |
(4 - 41(7 - 4) |
0 |
4 |
1 |
0 |
(7 - 41(3 - 4) |
0 |
4 |
2 |
1 |
(7 - 41(2 - 4) |
-6 |
4 |
3 |
1 |
(7 - 41(4 - 4) |
0 |
S>„=8 |
= -8 |
^{a} Weighting scheme based on Rook's case.
sl = (3 — 4) * (3* - 4) = 1
s2 = (2 - 4) * (2 - 4) = 4
s3 = (4 - 4) * (4 - 4) = 0
s4 = (7 - 4) * (7 - 4) = 9
Interpreting Moran’s I and Methodological Flaws
Having computed the Moran 1 statistic, one can proceed to evaluate the statistical significance of the test statistic. As noted earlier, the null hypothesis is one of spatial randomness, meaning that the spatial autocorrelation of the given variable is zero. The statistical significance of Moran's I is based on the normal frequency distribution (Z-score):
where I is the computed Moran's 1 value, £(/) is the expected Moran's I under the null hypothesis of spatial randomness, and S is the standard error of the Moran's I value.
Thus, given a Moran's I value of -0.286 with a Z-score of 0.1597, we fail to reject the null hypothesis and conclude that the areal pattern for low birth rates is statistically insignificant with a weak negative spatial autocorrelation.
It is important to keep in mind that the Moran's I statistic only provides a measure of spatial autocorrelation for spatial data measured at ordinal and interval scales, and may be sensitive to extreme values in a positive or negative correlation. In some cases, Moran's 1 may not be useful due to its sensitivity to spatial patterning and spatial weight selection.
Global Geary’s C Coefficient of Spatial Autocorrelation
Geary's C is an alternative measure of spatial autocorrelation. It determines the degree of spatial association using the sum of squared differences between pairs of data values as its measure of covariation (Goodchild 1986). The computation of Geary's C results in a value within the range of 0 to +2 (Figure 7.7). When we obtain a zero value, it is interpreted as a strong positive spatial autocorrelation (perfect correlation), a value of 1 indicates a random spatial pattern (no autocorrelation), and a value between 1 and 2 represents a negative spatial autocorrelation (2 is a perfect dispersion).
Suppose we have a study region, R, that is subdivided into n cells, where each cell is identified with a spatial feature. Geary's C can be computed by
where w-,j = 1 if cells i and j are neighbors, ю= 0 otherwise; c,y = (X, - X,)^{2};
FIGURE 7.7
Resultant values of Geary's C.
The average of all n cells is the mean (X), which is used to compute (s^{2}) based on the differences that each X value has from the mean (X).
TASK 7.3 COMPUTING GEARY'S C FOR LOW BIRTH WEIGHTS
We now examine the low birth weight rates used earlier to compute Moran's I. Table 7.3 presents a worktable and results for Geary's C.
TABLE 7.3
Worktable for Deriving Geary's Coefficient for Low Birth Rates
1 |
2 |
1 |
(3 - 2)(3 - 2) |
1 |
1 |
3 |
1 |
(3 - 4)(3 - 4) |
1 |
1 |
4 |
0 |
(3 - 7)(3 - 7) |
0 |
2 |
1 |
1 |
(2 - 3)(2 - 3) |
1 |
2 |
3 |
0 |
(2 - 4)(2 - 4) |
0 |
2 |
4 |
1 |
(2 - 7)(2 - 7) |
25 |
3 |
1 |
1 |
(4 - 3)(4 - 3) |
1 |
3 |
2 |
0 |
(4 - 2)(4 - 2) |
0 |
3 |
4 |
1 |
(4 - 7)(4 - 7) |
9 |
4 |
1 |
0 |
(7 - 3)(7 - 3) |
0 |
4 |
2 |
1 |
(7 - 2)(7 - 2) |
25 |
4 |
3 |
1 |
(7 - 4)(7 - 4) |
9 |
EWjj=8 |
=72 |
^{a} Weighting scheme based on Rook's case.
N = 4 = 3 Wjj = О 11 0
x_{2}=21001 *з=41 001 *_{4}=701 10
Sj=(3 - 4) * (3 - 4)=1 s_{2}=(2 - 4) * (2 - 4)=4 s_{3}=(4 - 4) * (4-4)=0 s_{4}=(7 - 4) * (7 - 4)=9
Interpreting Geary’s C and Methodological Flaws
Geary's C also requires the formulation of a null hypothesis of spatial randomness, which holds true when the spatial autocorrelation of a variable is 1. The statistical significance of Geary's C is also based on the normal frequency distribution (Z-score). For the previous example, Geary's C is 0.964 with a Z-score of 0.1597. Therefore, we do not reject the null hypothesis and conclude that the areal pattern for low birth rates shows a spatial autocorrelation that is statistically random.
Both Moran's l and Geary's C only detect spatial patterns (clusters) of an entire region and are unable to distinguish local patterns. Geary's C is less arranged; therefore, the extremes are less likely to correspond to the positive or negative correlation. Although their calculations are quite similar, Moran's I is based on the cross-product of deviations from the mean for variables at a particular cell and another neighboring cell (location), w'hile Geary's C is a cross-product of actual values of a variable at a particular location and another neighboring cell.