spdep (version 1.3-13)

spatialdelta: Weighted Multivariate Spatial Autocorrelation Measures

Description

The kernel-based weighted multivariate spatial autocorrelation measure delta proposed in Bavaud (2024) is implemented, together with support functions to create spatial weights matrices from symmetric binary adjacency matrices; summary and display methods are provided.

Usage

spatialdelta(dissimilarity_matrix, adjusted_spatial_weights,
 regional_weights=NULL, alternative = "greater")
# S3 method for spatialdelta
summary(object, ...)
# S3 method for summary.spatialdelta
print(x, digits=getOption("digits"), ...)
linearised_diffusive_weights(adjacency_matrix, regional_weights, t_choice = 2)
metropolis_hastings_weights(adjacency_matrix, regional_weights)
iterative_proportional_fitting_weights(adjacency_matrix, regional_weights,
 g=0.001, iter=1000, tol=1e-10, tol.margins=1e-10, print=FALSE)
graph_distance_weights(adjacency_matrix, regional_weights, c=NULL)
# S3 method for adjusted_spatial_weights
as.matrix(x, ...)
plot_spatialcoords(x, ...)
# S3 method for default
plot_spatialcoords(x, ...)
# S3 method for spatialdelta
plot_spatialcoords(x, cols = c(1L, 2L),
 mult = c(1, 1), power = 1L, fmult = NULL, names = attr(x, "rnames"), bg = 1,
 pos = 3, cex = 0.6, largest = NULL, ...)
plot_moran(x, y, ...)
# S3 method for default
plot_moran(x, y, ...)
# S3 method for spatialdelta
plot_moran(x, y, fmult = NULL, names = attr(x, "rnames"),
 bg = 1, pos = 3, cex = 0.6, largest = NULL, ...)
plot_spatialscree(x, ...)
# S3 method for default
plot_spatialscree(x, ...)
# S3 method for spatialdelta
plot_spatialscree(x, lwd=2, ...)
factorial_coordinates(x)
# S3 method for default
factorial_coordinates(x)
# S3 method for spatialdelta
factorial_coordinates(x)
plot_factorialcoords(x, ...)
# S3 method for default
plot_factorialcoords(x, ...)
# S3 method for spatialdelta
plot_factorialcoords(x, cols = c(1L, 2L),
 mult = c(1, 1), fmult = NULL, names = attr(x, "rnames"), bg = 1, pos = 3,
 cex = 0.6, largest = NULL, ...)
plot_factorialscree(x, ...)
# S3 method for default
plot_factorialscree(x, ...)
# S3 method for spatialdelta
plot_factorialscree(x, ...)
localdelta(x, ...)
# S3 method for default
localdelta(x, ...)
# S3 method for spatialdelta
localdelta(x, names = attr(x, "rnames"), ...)
cornish_fisher(x, ...)
# S3 method for default
cornish_fisher(x, ...)
# S3 method for spatialdelta
cornish_fisher(x, ...)

Value

spatialdelta returns an object with classes "spatialdelta" and "htest". Its print method treats it as an "htest" object. Other methods use additional attributes:

Kx

Multivariate kernel matrix eq. 16

Kw

Spatial weights kernel matrix eq. 21

regional_weights

Input observation weights

adjusted_spatial_weights

Input spatial adjusted_spatial_weightseights

B

symmetric matrix of scalar products eq. 15

VI

Spatial component of delta variance

Vx

Multivariate component of delta variance

Vd0

Variance of delta calculated in another way eq. 38

alphamu

Spatial component of delta skewness

alphalambda

Multivariate component of delta skewness

gammamu

Spatial component of delta excess kurtosis

gammalambda

Multivariate component of delta excess kurtosis

rnames

Region names passed through from row/column names of adjusted_spatial_weights, from the "region.id" attribute of the "nb" object used to create the adjusted_spatial_weights matrix from the "adjacency_matrix" object

linearised_diffusive_weights, metropolis_hastings_weights, iterative_proportional_fitting_weights and graph_distance_weights return square spatial weights matrices (the latter two return matrices created by metropolis_hastings_weights if igraph or mipfp are not available); factorial_coordinates returns a square factorial matrix; localdelta returns a vector of local deltas eq. 30.

Arguments

dissimilarity_matrix

dissimilarity_matrix is a square matrix of dissimilarities between (possibly multivariate) observations

adjusted_spatial_weights

adjusted_spatial_weights is a square matrix of spatial weights as returned by construction functions linearised_diffusive_weights, metropolis_hastings_weights, iterative_proportional_fitting_weights, graph_distance_weights or similar; only the named construction functions pass regional_weights through to spatialdelta, so for matrices constructed in other ways, this argument is required

regional_weights

default NULL, regional_weights are weights reflecting the contribution of each observation to the (possibly multivariate) data set, they may be uniform, but none can be zero, and they must sum to unity; if adjusted_spatial_weights is an "adjusted_spatial_weights" object created by linearised_diffusive_weights, metropolis_hastings_weights, iterative_proportional_fitting_weights or graph_distance_weights, regional_weights is passed as an attribute

alternative

a character string specifying the alternative hypothesis, must be one of "greater" (default), "less" or "two.sided"

object, x

object, x are objects returned by spatialdelta of class "spatialdelta", or of class "adjusted_spatial_weights"

digits

default getOption("digits"), or a non-null value specifying the minimum number of significant digits to be printed in values

adjacency_matrix

adjacency_matrix is a symmetric binary adjacency matrix

t_choice

default 2, the inverse of the largest eigenvalue of the adjusted Laplacian adjacency matrix (t2, Bavaud 2024, page 583), otherwise 1 (t1, page 579)

g

default 0.001, (Bavaud 2024, page 579, 589), a small quantity to lift binary adjacency matrix above zero

iter, tol, tol.margins, print

arguments passed to Ipfp

c

default NULL, a coefficient greater than zero and less than or equal to -1/min(B) where B is the matrix of scalar products associated with the distance matrix; if not given, -1/min(B) is used

y

a numeric vector from which to create a Moran plot

cols

integer vector of length 2, default c(1L, 2L), giving the two columns of the regional or factorial coordinates to plot

mult

numeric vector of length 2, default c(1, 1), to scale or reverse axes as signs are not given to be the same for different eigenproblem implementations

power

integer vector of length 1, default 1; if higher, use a powered kernel

fmult

default NULL for automatic scaling of the circle symbols expressing the observation weights f in the plot as (0.02*diff(range(X)))/diff(range(regional_weights)) where X is the first vector of coordinates; may be set manually to a numeric value

names

default attr(x, "rnames"), the coordinates will be annotated with these strings passed through from the "region.id" attribute of the "nb" object used to create the adjusted_spatial_weights matrix from the "adjacency_matrix" object as row and column names

bg, pos, cex, lwd

arguments passed to graphics functions or methods

largest

default NULL, may be integer > 0 and <= the region count, permitting names to be plotted only for the largest circle symbols

...

other arguments passed to other functions or methods

Author

Roger Bivand,

Details

In general, the user should choose one of the functions for constructing spatial weights from the symmetric binary, or similar to symmetric - for example row-standardised - adjacency matrix. The cornish_fisher method updates the standard deviate and p-value if the pair of skewness and excess kurtosis estimates are within the required domain.

References

Bavaud, F. (2024), Measuring and Testing Multivariate Spatial Autocorrelation in a Weighted Setting: A Kernel Approach. Geographical Analysis, 56: 573-599. tools:::Rd_expr_doi("10.1111/gean.12390")

See Also

Ipfp, distances

Examples

toy_f <- c(0.4, 0.1, 0.3, 0.2)
toy_xa <- c(4, 9, 1, 1)
toy_nba <- structure(list(c(2L, 3L, 4L), c(1L), c(1L, 4L), c(1L, 3L)),
 class="nb", region.id=as.character(1:4))
toy_gla <- list(c(0.25, 0.5, 0.25), c(1), c(2/3, 1/3), c(0.5, 0.5))
toy_wa <- nb2mat(toy_nba, glist=toy_gla, style="W")
spatialdelta(dissimilarity_matrix=as.matrix(dist(toy_xa))^2,
 adjusted_spatial_weights=toy_wa, regional_weights=toy_f, alternative="two.sided")
toy_xb <- toy_xa
toy_glb <- list(c(0.5, 0.125, 0.25, 0.125), c(0.5, 0.5), c(1/3, 0.5, 1/6), c(0.25, 0.25, 0.5))
toy_nbb <- include.self(toy_nba)
toy_wb <- nb2mat(toy_nbb, glist=toy_glb, style="W")
spatialdelta(dissimilarity_matrix=as.matrix(dist(toy_xb))^2,
 adjusted_spatial_weights=toy_wb, regional_weights=toy_f, alternative="two.sided")
toy_xc <- c(3, 3, 1, 1)
toy_nbc <- structure(list(c(3L, 4L), c(3L, 4L), c(1L, 2L), c(1L, 2L)),
 class="nb", region.id=as.character(1:4))
toy_glc <- list(c(0.6, 0.4), c(0.6, 0.4), c(0.8, 0.2), c(0.8, 0.2))
toy_wc <- nb2mat(toy_nbc, glist=toy_glc, style="W")
spatialdelta(dissimilarity_matrix=as.matrix(dist(toy_xc))^2,
 adjusted_spatial_weights=toy_wc, regional_weights=toy_f, alternative="two.sided")
# \donttest{
if (require(Guerry, quietly=TRUE)) {
data(gfrance85)
gfrance85 <- st_as_sf(gfrance85)
moral <- scale(st_drop_geometry(gfrance85)[, 7:12])
Dmoral <- as.matrix(dist(moral))^2
fG <- gfrance85$Pop1831/sum(gfrance85$Pop1831)
gnb <- poly2nb(gfrance85, row.names=gfrance85$Department)
frG <- nb2mat(gnb, style="B")
ldwG <- linearised_diffusive_weights(adjacency_matrix=frG, regional_weights=fG)
mhwG <- metropolis_hastings_weights(adjacency_matrix=frG, regional_weights=fG)
ifwG <- iterative_proportional_fitting_weights(adjacency_matrix=frG, regional_weights=fG)
gdwG <- graph_distance_weights(adjacency_matrix=frG, regional_weights=fG)
(moral_ldw <- spatialdelta(dissimilarity_matrix=Dmoral, adjusted_spatial_weights=ldwG))
(cornish_fisher(moral_ldw))
moral_mhw <- spatialdelta(dissimilarity_matrix=Dmoral, adjusted_spatial_weights=mhwG)
moral_ifw <- spatialdelta(dissimilarity_matrix=Dmoral, adjusted_spatial_weights=ifwG)
moral_gdw <- spatialdelta(dissimilarity_matrix=Dmoral, adjusted_spatial_weights=gdwG)
opar <- par(mfrow=c(2, 2))
plot_spatialcoords(moral_ldw, cols=c(2,1), mult=c(-1, -1), main="Linearised Diffusive", asp=1)
plot_spatialcoords(moral_mhw, cols=c(2,1), mult=c(1, 1), main="Metropolis-Hastings", asp=1)
plot_spatialcoords(moral_ifw, cols=c(2,1), mult=c(1, 1), main="Iterative fitting", asp=1)
plot_spatialcoords(moral_gdw, cols=c(2,1), mult=c(1, 1), main="Graph distance", asp=1)
par(opar)
opar <- par(mfrow=c(2, 2))
plot_spatialscree(moral_ldw, main="linearised diffusive")
plot_spatialscree(moral_mhw, main="Metropolis-Hastings")
plot_spatialscree(moral_ifw, main="iterative fitting")
plot_spatialscree(moral_gdw, main="graph distance")
par(opar)
Crime_pers <- scale(gfrance85$Crime_pers)
funif <- rep(1/length(Crime_pers), length(Crime_pers))
lw <- nb2listw(gnb, style="W")
Crime_pers_delta <- spatialdelta(dissimilarity_matrix=as.matrix(dist(Crime_pers))^2,
 adjusted_spatial_weights=listw2mat(lw), regional_weights=funif)
Crime_pers_delta$estimate["delta"]
locdelta <- localdelta(Crime_pers_delta)
moran.test(Crime_pers, lw, randomisation=FALSE)$estimate["Moran I statistic"]
locmoran <- localmoran(c(Crime_pers), lw)[,1]
all.equal(locmoran, locdelta) 
all.equal(unname(moral_ldw$estimate["delta"]), # Eq. 30 p. 588
 sum(fG * localdelta(moral_ldw)))
all.equal(unname(moral_mhw$estimate["delta"]), # Eq. 30 p. 588
 sum(fG * localdelta(moral_mhw)))
all.equal(unname(moral_ifw$estimate["delta"]), # Eq. 30 p. 588
 sum(fG * localdelta(moral_ifw)))
all.equal(unname(moral_gdw$estimate["delta"]), # Eq. 30 p. 588
 sum(fG * localdelta(moral_gdw)))
}
# }