Ctrl+K

Moran’s I and Geary’s C

11. Moran’s I and Geary’s C#

11.1. Global Spatial Autocorrelation Measures#

  1. One statistic for the whole pattern

  2. Test for clustering not for clusters (location)

11.1.1. Moran’s I (1948)#

img

  • The most commonly used for many spatial autocorrelation statistics

 I=[∑i∑jWijzizjS0∑izi2N]=N⋅∑izi∑jwijzjs0∑izi2

  • S0=∑i∑jWij, Zi=yi−mx: deviations from mean

    • ∑jwijzj is the independent variable in this regression, called spatial lag

  • Moran's I depends on spatial weights, relative magnitude for same weights

    img

  • Similar to Person Correlation, which is (CrossProduct)/Sd, Here is autocorrelation,

11.1.2. Inference#

  1. How to assess whether the computed value of Moran's Index is significantly different from a value for a spatially random distribution

  • Assume normal distribution

  • Compare value to a reference distribution obtained from a series of randomly permuted patterns

    1. Positive and significant = clustering of like value

    2. Different Process can result in the same pattern

      1. True contagion: Evidence of clustering due to spatial interaction

        • Spatial interaction can through network other than geographical distance

      2. Apparent contagion: Evidence of clustering due to spatial heterogeneity

    3. Negative and significant = alternating values

  1. Moran Z-value

z=Observed I−Mean(I)sd

  • Z are comparable across variables and across spatial weights

    • Moran’s I are not comparable if spatial weight matrix are not same

    • Spatial weight matrix is for spatial similarity measuring

11.2. Geary’s C (1954)#

  1. Squared difference as measure of dissimilarity

    • Values between 0 and 2

  2. Similar to notion of variogram (geo-statistics)

C=(N−1)∑i∑jwij(xi−xj)22s0∑izi2

  • S0=∑i∑jWij

11.2.1. Interpretation#

  1. Positive spatial autocorrelation: (c<1) or (z<0)

  2. Negative spatial autocorrelation: (c>1) or (z>0)

11.2.2. Inference#

  • Same to Moran's I

image.png

Contents