Local Spatial Autocorrelation

Local spatial autocorrelation statistics are used to discover clusters (hot spots or cold spots) and spatial outliers. Local Indicators of Spatial Association (LISA), introduced by Anselin (1995), are local spatial statistics that provide a statistic for each location and assess its contribution to the corresponding global spatial autocorrelation statistic.

Inference is performed through a computational conditional permutation approach by randomly reshuffling the values of the variable to different locations while the value of the observation under consideration is held fixed at its location. The observed statistic for each location is compared to a reference distribution under the null hypothesis of spatial randomness.

All the examples on this page assume that the Guerry dataset is loaded and a polygon contiguity spatial weights object has been built and row-standardized.

using SpatialDatasets
using SpatialDependence
using StableRNGs
using Plots

guerry = sdataset("Guerry")
W = polyneigh(guerry)

Local Moran

Local Moran (Anselin, 1995) is the most used local spatial autocorrelation statistic. It is computed as:

\[I_i = \frac{z_i}{m_2} \sum_{j}w_{ij}z_{j}\]

where $z$ is the variable of interest in deviations from the mean, and $m_2 = \sum_{i}z_i^{2} / (n - 1)$ or $m_2 = \sum_{i}z_i^{2} / n$ is the scaling factor.

Local Moran can be computed with the localmoran function. By default, $9,999$ permutations are calculated for the inference. It is possible to specify a different number of permutations with the permutations optional parameter. For reproducibility, it is possible to specify a custom random number generator with the rng optional parameter. If corrected is set to false the scaling factor is divided by $n$ instead of $n - 1$.

lmguerry = localmoran(guerry.Litercy, W, permutations = 9999, rng = StableRNG(1234567))

Local Moran test of Spatial Autocorrelation
--------------------------------------------

Randomization test with 9999 permutations.
Interesting locations at 0.05 significance level:
       High-High: 18
         Low-Low: 20
        Low-High: 0
        High-Low: 0

In the Local Moran, observations are classified in four categories, depending on the value of the attribute and the value of their neighbors with respect to the mean:

The category to which each observation is assigned can be retrieved with the assignments function:

assignments(lmguerry)

85-element Vector{Symbol}:
 :LH
 :HH
 :LL
 :HH
 :HL
 :LL
 :HH
 :LL
 :HH
 :LL
 ⋮
 :LL
 :LL
 :LH
 :LL
 :LL
 :LL
 :LL
 :HH
 :HH

Local Geary

Local Geary (Anselin, 1995) is computed as:

\[c_i = \frac{1}{m_2} \sum_{j}w_{ij}(z_{i} - z_{j})^2\]

where $z$ is the variable of interest in deviations from the mean, and $m_2 = \sum_{i}z_i^{2} / (n - 1)$ or $m_2 = \sum_{i}z_i^{2} / n$ is the scaling factor.

Local Geary can be computed with the localgeary function. By default, $9,999$ permutations are calculated for the inference. It is possible to specify a different number of permutations with the permutations optional parameter. For reproducibility, it is possible to specify a custom random number generator with the rng optional parameter. If corrected is set to false the scaling factor is divided by $n$ instead of $n - 1$.

lcguerry = localgeary(guerry.Litercy, W, permutations = 9999, rng = StableRNG(1234567))

Local Geary test of Spatial Autocorrelation
--------------------------------------------

Randomization test with 9999 permutations.
Interesting locations at 0.05 significance level:
        Positive: 49
        Negative: 0

In the Local Geary, observations are classified in two categories:

The category to which each observation is assigned can be retrieved with the assignments function:

assignments(lcguerry)

85-element Vector{Symbol}:
 :N
 :P
 :P
 :P
 :P
 :P
 :P
 :P
 :P
 :P
 ⋮
 :P
 :P
 :P
 :P
 :P
 :P
 :P
 :P
 :P

Alternatively, observations can be interpreted together with the quadrants of the moran scatterplot if the parameter categories is set to :moran:

Getis-Ord Statistics

There are two versions of the Getis-Ord statistics (Getis and Ord, 1992; Ord and Getis, 1995) computed as:

\[G_i = \frac{\sum_{j} w_{ij}x_{j}}{\sum_{j} x_{j}} \qquad \forall j \ne i\]

\[G_i^* = \frac{\sum_{j} w_{ij}x_{j}}{\sum_{j} x_{j}} \qquad \forall j\]

where $x$ is the variable of interest. The $G_i$ statistic excludes the value at the location $x_i$ in the numerator and denominator, whereas the $G_i^*$ statistic includes it. If the contiguity matrix is binary, the self weight is set to $1$, $w_{ii} = 1$, whereas if it is row-standardized, it is set to $1$ divided by the number of neighbors of $i$, including the self.

Getis-Ord Statistics can be computed with the getisord function. The $G_i^*$ statistic is computed if star is set to true, the default, whereas the $G_i$ statistic is calculated if star is false. By default, $9,999$ permutations are calculated for the inference. It is possible to specify a different number of permutations with the permutations optional parameter. For reproducibility, it is possible to specify a custom random number generator with the rng optional parameter.

gosguerry = getisord(guerry.Litercy, W, permutations = 9999, rng = StableRNG(1234567))

Getis-Ord Gi* test of Spatial Autocorrelation
--------------------------------------------

Randomization test with 9999 permutations.
Interesting locations at 0.05 significance level:
            High: 18
             Low: 20

goguerry = getisord(guerry.Litercy, W, permutations = 9999, rng = StableRNG(1234567), star = false)

Getis-Ord Gi test of Spatial Autocorrelation
--------------------------------------------

Randomization test with 9999 permutations.
Interesting locations at 0.05 significance level:
            High: 18
             Low: 20

In the Getis-Ord Statistics, observations are classified in two categories:

The category to which each observation is assigned can be retrieved with the assignments function:

assignments(goguerry)

85-element Vector{Symbol}:
 :H
 :H
 :L
 :H
 :H
 :L
 :H
 :L
 :H
 :L
 ⋮
 :L
 :L
 :H
 :L
 :L
 :L
 :L
 :H
 :H

Significance

The conditional randomization procedure returns a pseudo p-value that can be used to assed the significance of the identified clusters and spatial outliers. The function issignificant returns a vector of boolean values indicating if the local statistics are significant at the desired threshold level:

issignificant(lmguerry, 0.05)

85-element Vector{Bool}:
 0
 1
 1
 0
 0
 0
 1
 0
 1
 0
 ⋮
 0
 0
 0
 0
 0
 1
 1
 1
 0

As multiple tests are performed on the same dataset, the p-values suffer from the problem of multiple comparisons, leading to many false positives (exceeding the nominal Type I error rate). Multiple solutions have been suggested in the literature to address this issue. Some of them are implemented through the adjust parameter. Set the parameter to :bonferroni for the conservative Bonferroni approach. For the False Discovery Rate, set the parameter to :fdr.

issignificant(lmguerry, 0.05, adjust = :fdr)

85-element Vector{Bool}:
 0
 0
 1
 0
 0
 0
 1
 0
 1
 0
 ⋮
 0
 0
 0
 0
 0
 0
 1
 1
 0

For convenience, it is possible to get the category to which each observation is assigned while identifying which ones are not significant. This is done by specifiying a specifying threshold level and an adjustment method in the assignments function. Nonsignificant observations are labeled as :ns.

assignments(lmguerry, 0.05, adjust = :none)

85-element Vector{Symbol}:
 :ns
 :HH
 :LL
 :ns
 :ns
 :ns
 :HH
 :ns
 :HH
 :ns
 ⋮
 :ns
 :ns
 :ns
 :ns
 :ns
 :LL
 :LL
 :HH
 :ns

LISA Cluster Map

A cluster map with the significant locations can be plotted with the plot function if the Plots.jl package is loaded:

plot(guerry, lmguerry)

The threshold significance value and the adjustment can also be set when plotting a cluster map with the sig and adjust parameters:

plot(guerry, lmguerry, sig = 0.05, adjust = :fdr)

If not specified, the default threshold significance value is 0.05 and the adjust parameter is :none.