Geary’s C and Local Geary - Geospatial Statistics and Visualisation

Geary’s C¶

Geary’s C is a global measure of spatial autocorrelation used in spatial data analysis to determine the overall degree of spatial dependency in a dataset. It quantifies the degree to which values in a dataset are similar to or different from neighboring values, considering the entire study area. Unlike Moran’s I, Geary’s C focuses on the distance between value of a location and its neighbors rather than their distances to the mean value.

C = \frac{(N-1)\sum_i\sum_j w_{ij}(x_i-x_j)^{2}}{2S_0\sum_i(x_i-\bar{x})^{2}}

(1)

In this equation:

$N$ is the number of observations.
$w_{ij}$ is the spatial weight matrix, representing the spatial relationships between observations $i$ and $j$ .
$x_{k}$ are the observed values of the variable at location k.
$S_{0}$ is the sum of all spatial weights.
$\bar{x}$ is the mean of the observed values.
The lower part (denominator) is a kind of standardizig process to control the range of the resulting values.

Local Geary’s C¶

The Local Geary:

c_i = \frac{\sum_j w_{ij}(x_i - x_j)^2}{2S_0}

(2)

Unlike the global Geary’s C, the local version provides location-specific information about spatial autocorrelation, allowing for the identification of spatial clusters and outliers. Low values of the Local Geary’s C indicate positive spatial autocorrelation (similar values cluster together), while high values indicate negative spatial autocorrelation (dissimilar values are close to each other).

Comparing Global vs. Local¶

\text{global} = a . [\sum_i \text{component}(i)]

(3)

Global¶

C = \frac{(N-1)\sum_i\sum_j w_{ij}(x_i-x_j)^{2}}{2S_0\sum_i(x_i-\bar{x})^{2}}

(4)

C = \frac{(N-1)}{2S_0\sum_i(x_i-\bar{x})^{2}} \times \sum_i\sum_j w_{ij}(x_i-x_j)^{2}

(5)

C = \frac{(N-1)}{\sum_i(x_i-\bar{x})^{2}} \times \sum_i \frac{1}{2S_0}\times \sum_j w_{ij}(x_i-x_j)^{2}

(6)

C = \frac{(N-1)}{\sum_i(x_i-\bar{x})^{2}} \times \sum_i c_i

(7)

Local¶

c_i = \frac{\sum_j w_{ij}(x_i - x_j)^2}{2S_0}

(8)

c_i = \frac{1}{2S_0} \times \sum_j w_{ij}(x_i - x_j)^2

(9)

Interpretation¶

$(x_i - x_j)^2$
attribute dissimilarity
distance in attribute space
Geary’s C: weighted average of distances in attribute space to neighbors in geographic space
positive: similarity, can be high-high, low-low, middle-middle
negative: dissimilarity: no distinction between high-low and low-high
Geary’s C is more sensitive to local patterns of spatial autocorrelation and can better capture smaller-scale variations in the data. spatial non-stationarity: a sub-region of the study area has a different (e.g., lower) average value, for which the similar-values (could be middle-middle) may be overlooked by Moran’s I (and its local variant).

An example¶

Take an example: the donations data (Guerry data)

The data distribution (based on Natural Breaks).

Local Significance Map¶

Local Cluster Map¶

Sensitivity Analysis¶

Local Moran’s I vs. Local Geary’s C¶

Comparison between local Moran’s I and local Geary’s C

Moran’s I vs. Geary’s C¶

Different type of attribute similarity
- cross-product (correlation) vs.
- squared difference (dissimilarity)
power against different alternatives
- the same null hypothesis: spatial randomness
- what should be the ‘alternative’?
Moran’s I and its local variant: Detect similar values based on their deviation from the mean value.
- Moran’s I is more sensitive to global patterns of spatial autocorrelation and overall trends in the data.
Geary’s C and its local variant: Detect similar values based on the absolute differences between pairs of values.
- Geary’s C is more sensitive to local patterns of spatial autocorrelation and smaller-scale variations in the data.

Geospatial Statistics and Visualisation

LISA and Local Moran’s I

Geospatial Statistics and Visualisation

Getis-Ord G statistics