The World Is Not Random

From Manuel Gimond

Kriging Revisited

Kriging Revisited

• Assumes autocorrelation to facilitate prediction

• Semivariogram decribes how autocorrelation changes with distance

• But what about statistical inference?

Spatial Autocorrelation and Inference

• Attributes (features) are often non-randomly distributed

• Especially true with aggregated data

• Affects estimates of precision (and significance)

Global Moran’s I

\[ I = \frac{n}{W} \cdot \frac{\sum_{i=1}^{n}\sum_{j=1}^{n} w_{ij}(x_i - \bar{x})(x_j - \bar{x})}{\sum_{i=1}^{n}(x_i - \bar{x})^2} \]

where \(W = \sum_{i=1}^{n}\sum_{j=1}^{n} w_{ij}\)

• A first-order estimate of spatial autocorrelation

Moran’s I

…but How Do We Calculate \(w_{i}\)

• How do we define \(I(d)\) for areal data?

• What about \(w_{ij}\)?

• We can use spdep for that!!

Finding Neighbors

• Queen, rook, (and bishop) cases impose neighbors by contiguity

• Weights calculated as a \(1/ num. of neighbors\)

More Formally

\[ E[I] = \frac{-1}{n-1} \]

• When \(n\) is large, E[I] approaches 0

• But large relative to what?

Testing for Spatial Autocorrelation

• \(I\) can be estimated by fitting a model of the lagged average as a function of the measurement

\[ \mu_{lag} = \beta \times measurement \]

where \(\beta = I\)

Comparing observed to expected

• We can generate the expected distribution of Moran’s I coefficients under a Null hypothesis of no spatial autocorrelation

• Using permutation and a loop to generate simulations of Moran’s I

Significance testing

• Pseudo p-value (based on permutations)

• Analytically (sensitive to deviations from assumptions)

• Using Monte Carlo

Local Indicators of Spatial Autocorrelation

• Sometimes we want to know about second-order autocorrelation

• Is there clustering around different locations (but maybe not everywhere)?

Local Moran’s I

\[ I_i = \frac{(x_i - \bar{x})}{\sum_{k=1}^{n}(x_k - \bar{x})^2/n} \sum_{j=1}^{n} w_{ij}(x_j - \bar{x}) \]

• Can identify areas where clustering is great than expected

• Proportional to Global Moran’s I

Autocorrelation and Inference

• Interpolation assumes autocorrelation

• Inference is interested in mechanisms

• Tests for mechanisms get confused by autocorrelation

• Diagnosing autocorrelation helps plan next steps (Weds)