gr {geostan} | R Documentation |
The Geary Ratio
Description
An index for spatial autocorrelation. Complete spatial randomness (lack of spatial pattern) is indicated by a Geary Ratio (GR) of 1; positive autocorrelation moves the index towards zero, while negative autocorrelation will push the index towards 2.
Usage
gr(x, w, digits = 3, na.rm = FALSE, warn = TRUE)
Arguments
x |
Numeric vector of length |
w |
An |
digits |
Number of digits to round results to. |
na.rm |
If |
warn |
If |
Details
The Geary Ratio is an index of spatial autocorrelation. The numerator contains a series of sums of squared deviations, which will be smaller when each observation is similar to its neighbors. This term makes the index sensitive to local outliers, which is advantageous for detecting such outliers and for measuring negative autocorrelation. The denominator contains the total sum of squared deviations from the mean value. Hence, under strong positive autocorrelation, the GR approaches zero. Zero spatial autocorrelation is represented by a GR of 1. Negative autocorrelation pushes the GR above 1, towards 2.
GR =
\frac{n-1}{2K} \frac{M}{D}
M = \sum_i \sum_j w_{i,j} (x_i -
x_j)^2
D = \sum_i (x_i - \overline{x})^2
Observations with no neighbors are removed before calculating the GR. (The alternative would be for those observations to contribute zero to the numerator—but zero is not a neutral value, it represents strong positive autocorrelation.)
Value
Returns the Geary ratio (a single numeric value).
Source
Chun, Yongwan, and Daniel A. Griffith. Spatial Statistics and Geostatistics: Theory and Applications for Geographic Information Science and Technology. Sage, 2013.
Geary, R. C. "The contiguity ratio and statistical mapping." The Incorporated Statistician 5, no. 3 (1954): 115-127_129-146.
Unwin, Antony. "Geary's Contiguity Ratio." The Economic and Social Review 27, no. 2 (1996): 145-159.
Examples
data(georgia)
x <- log(georgia$income)
w <- shape2mat(georgia, "W")
gr(x, w)