Spatial Weight Matrices
Spatial Weight Matrices can be created from different geometries (polygons, multi polygons, or points) or raw data in Matrix format. By default, spatial weights will be row-standardized so that all rows sum to 1.
The package implements GeoInterface.jl 1.0 traits for geospatial vector data. Therefore, any geospatial data loaded with a package that implements this feature, such as Shapefile.jl or GeoDataFrames.jl, should, in principle, be compatible with this package.
Polygon contiguity
The polyneigh function creates a spatial weights object that identifies as neighbors those polygons that are physically contiguous.
using SpatialDependence
using SpatialDatasets
guerry = sdataset("Guerry")
W = polyneigh(guerry.geometry)Spatial Weights
Observations: 85
Transformation: row
Minimum nunmber of neighbors: 2
Maximum nunmber of neighbors: 8
Average number of neighbors: 4.9412
Median number of neighbors: 5.0
Islands (isloated): 0
Density: 5.8131%
By default, the queen contiguity criterion will be used for contiguity. It is possible to specify a Rook criterion by setting the parameter criterion to :Rook.
W = polyneigh(guerry.geometry, criterion = :Rook)When the geometry has imperfections and polygons do not touch completely, a tolerance parameter for contiguity can be set with tol.
Points distance threshold
For points geometry, a spatial weights object that identifies as neighbors those points that are below or equal to a specific threshold is computed with the dnearneigh function:
using SpatialDependence
using SpatialDatasets
boston = sdataset("Bostonhsg")
W = dnearneigh(boston.geometry, threshold = 4.0)Spatial Weights
Observations: 506
Transformation: row
Minimum nunmber of neighbors: 1
Maximum nunmber of neighbors: 178
Average number of neighbors: 72.2253
Median number of neighbors: 56.0
Islands (isloated): 0
Density: 14.2738%
The dnearneigh function can also be used with two vectors of coordinates:
W = dnearneigh(boston.x, boston.y, threshold = 4.0)Spatial Weights
Observations: 506
Transformation: row
Minimum nunmber of neighbors: 1
Maximum nunmber of neighbors: 178
Average number of neighbors: 72.2253
Median number of neighbors: 56.0
Islands (isloated): 0
Density: 14.2738%
Points K nearest neighbors
For points geometry, a spatial weights object that identifies the k-nearest neighbors is computed with the knearneigh function:
using SpatialDependence
using SpatialDatasets
boston = sdataset("Bostonhsg")
W = knearneigh(boston.geometry, k = 5)Spatial Weights
Observations: 506
Transformation: row
Minimum nunmber of neighbors: 5
Maximum nunmber of neighbors: 5
Average number of neighbors: 5.0
Median number of neighbors: 5.0
Islands (isloated): 0
Density: 0.9881%
The knearneigh function can also be used with two vectors of coordinates:
W = knearneigh(boston.x, boston.y, k = 5)Spatial Weights
Observations: 506
Transformation: row
Minimum nunmber of neighbors: 5
Maximum nunmber of neighbors: 5
Average number of neighbors: 5.0
Median number of neighbors: 5.0
Islands (isloated): 0
Density: 0.9881%
Polygon centroids
Polygon centroids and mean centers are computed with the centroid and meancenter functions.
cx, cy = centroid(guerry.geometry);cmx, cmy = meancenter(guerry.geometry);Polygon centroids can be used to calculate neighbors based on distance thresholds or k nearest neighbors.
knearneigh(cx, cy, k = 5)Spatial Weights
Observations: 85
Transformation: row
Minimum nunmber of neighbors: 5
Maximum nunmber of neighbors: 5
Average number of neighbors: 5.0
Median number of neighbors: 5.0
Islands (isloated): 0
Density: 5.8824%
Spatial Weights from a Matrix
It is possible to create a spatial weights object by supplying a square Matrix. All values that are different from 0 will be neighbors.
W = SpatialWeights([0 1 0; 1 0 1; 0 1 0])Spatial Weights
Observations: 3
Transformation: row
Minimum nunmber of neighbors: 1
Maximum nunmber of neighbors: 2
Average number of neighbors: 1.3333
Median number of neighbors: 1.0
Islands (isloated): 0
Density: 44.4444%
By default, the spatial weights will be row-standardized. However, it is possible to keep the original values in the Matrix as weighs by setting the standardize optional parameter to false.
W = SpatialWeights([0 1 0; 1 0 1; 0 1 0], standardize = false)Spatial Weights transformation
By default, Spatial weight objects are row-standardized, with rows summing 1. It is possible to specify a different transformation with the wtransform! and wtransform functions. wtransform! applies an in-place transformation to the spatial weights object. In contrast,wtransform returns a transformed copy of the spatial weights object.
Available transoformations are:
| Code | Transformation |
|---|---|
:binary | Binary coding |
:row | Row standardization |
wtransform!(W, :binary)Wbin = wtransform(W, :binary)Spatial Weights
Observations: 85
Transformation: binary
Minimum nunmber of neighbors: 2
Maximum nunmber of neighbors: 8
Average number of neighbors: 4.9412
Median number of neighbors: 5.0
Islands (isloated): 0
Density: 5.8131%
Spatial Weights information
There is a set of functions to obtain information about spatial weights objects.
The function nobs returns the number of observations in the spatial weights object.
nobs(W)85The number of neighbors of each observation is returned with the cardinalities function:
cardinalities(W)85-element Vector{Int64}:
4
6
6
4
3
7
3
3
5
5
⋮
5
6
3
6
4
6
6
6
5A vector of neighbors for particular observations is obtained with the neighbors function:
neighbors(W, 1)4-element Vector{Int64}:
36
37
67
69And a vector of weights with the weights function:
weights(W, 1)4-element Vector{Float64}:
0.25
0.25
0.25
0.25The number of islands - observatiosn with no neighbors - is obtained with the nislands function, and the vector of islands with the islands function.
nislands(W)0islands(W)Int64[]Spatial weights objects implement the following methods to obtain descriptive statistics of the number of neighbors: minimum, maximum, mean, median.
Connectivity Histogram
The plot function with a spatial weights object plots a connectivity histogram for the number of neighbors.
plot(W)
Convert to other objects
Spatial weights objects can be converted to a Matrix using the Matrix constructor:
Matrix(W)85×85 Matrix{Float64}:
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.25 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.333333 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.333333 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
⋮ ⋱ ⋮
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.333333 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.166667 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.166667 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.166667 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0and to a sparse matrix with the sparse function:
sparse(W)85×85 SparseArrays.SparseMatrixCSC{Float64, Int64} with 420 stored entries:
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