Spatial Autocorrelation

14. Spatial Autocorrelation#

Characterize overall attribute pattern in space

14.1. Analyze and Map Spatial Variations#

  • Spatial variation of the location of objects/events:

    • Spatial descriptive statistics (mean center, standard deviation ellipse).

    • Point pattern analysis (Randomness and cluster mapping).

  • Spatial variation of attributes associated with objects/events:

    • Spatial autocorrelation (Moran’s I).

  • Spatial variation of relationships:

    • How relationships vary in space.

14.2. Characterize Attribute Spatial Pattern#

  • Location randomness (Point patterns): Quadrat analysis and Nearest neighbor distance.

  • Attribute randomness (Spatial autocorrelation):

    • How different is an observed attribute pattern from a random distribution?

    • Positive spatial autocorrelation: Similar values cluster together.

    • Negative spatial autocorrelation: Dissimilar values cluster together (checkerboard).

14.3. First Law of Geography#

  • Waldo Tobler (1970): “Everything is related to everything else, but near things are more related than distant things.”

  • Spatial autocorrelation is a quantification of this law.

14.4. Moran’s I#

  • The most common measure of spatial autocorrelation.

  • Compares the attribute value at each location with the values at its neighboring locations.

  • Formula components:

    • \(z_i\): Deviation of an attribute for feature \(i\) from its mean (\(x_i - \bar{x}\)).

    • \(w_{i,j}\): Spatial weight between feature \(i\) and \(j\) (neighborhood definition).

    • \(S_0\): Aggregate of all spatial weights.

14.5. Interpreting Moran’s I#

  • Range is generally between -1.0 and +1.0.

  • Positive value: Clustered pattern (Positive autocorrelation).

  • Zero: Random pattern.

  • Negative value: Dispersed/Checkerboard pattern (Negative autocorrelation).

14.6. Spatial Weights (\(w_{i,j}\))#

  • Define the neighborhood relationship for the autocorrelation calculation.

  • Binary weights: 1 if \(j\) is a neighbor of \(i\), 0 otherwise.

  • Distance-based weights: \(1/d\) or \(1/d^2\) (inverse distance).

  • Adjacency-based:

    • Rook’s Case: Share an edge.

    • Queen’s Case: Share an edge or a corner.

14.7. Significance Testing#

  • Null Hypothesis: The attribute being analyzed is randomly distributed among the features in the study area.

  • Z-score and p-value are used to determine if the observed Moran’s I index is statistically significant.

  • If p < 0.05, we reject the null hypothesis of randomness.

14.8. Spatial Lag#

  • The “spatial lag” of a variable is the weighted average of the values of its neighbors.

  • \(Lag(x_i) = \sum (w_{i,j} \times x_j)\)

  • Moran’s I can be seen as the correlation between a variable and its spatial lag.

14.9. Correlograms#

  • A plot of spatial autocorrelation (Moran’s I) against different distance “lags” (neighborhood sizes).

  • Typically, autocorrelation is highest at short distances and decreases as distance increases.

  • The point where it hits zero or stays flat indicates the “range” of spatial influence.

14.10. Semivariance as a Measure of Difference#

  • While Moran’s I measures similarity, semivariance measures difference.

  • Computed from sample pairs at a certain distance \(h\) (the lag).

  • \(\gamma(h) = \frac{1}{2N(h)} \sum [z(x_i) - z(x_i + h)]^2\)

14.11. The Variogram (or Semivariogram)#

  • A plot of semivariance versus distance (lag).

  • Nugget: Measurement error or micro-scale variation at zero distance.

  • Sill: The point where the variance flattens out.

  • Range: The distance at which locations are no longer spatially autocorrelated.

14.12. Summary#

  • Spatial autocorrelation helps distinguish between meaningful spatial patterns and random noise.

  • Moran’s I focuses on global similarity across a map.

  • Variograms focus on the scale and distance of spatial dependence, often used as a precursor to spatial interpolation.

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