Compute the global spatial runs test.

This function compute the global spatial runs test for spatial independence of a categorical spatial data set.

Usage

sp.runs.test(formula = NULL, data = NULL, fx = NULL,
listw = listw, alternative = "two.sided" ,
distr = "asymptotic", nsim = NULL,control = list())

Arguments

formula

a symbolic description of the factor (optional).

data

an (optional) data frame or a sf object containing the variable to testing for.

fx

a factor (optional).

listw

A neighbourhood list (type knn or nb) or a W matrix that indicates the order of the elements in each \(m_i-environment\) (for example of inverse distance). To calculate the number of runs in each \(m_i-environment\), an order must be established, for example from the nearest neighbour to the furthest one.

alternative

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

distr

A string. Distribution of the test "asymptotic" (default) or "bootstrap".

nsim

Number of permutations to obtain pseudo-value and confidence intervals (CI). Default value is NULL to don`t get CI of number of runs.

control

List of additional control arguments.

Value

A object of the htest and sprunstest class

data.name	a character string giving the names of the data.
method	the type of test applied ().
SR	total number of runs
dnr	empirical distribution of the number of runs
statistic	Value of the homogeneity runs statistic. Negative sign indicates global homogeneity
alternative	a character string describing the alternative hypothesis.
p.value	p-value of the SRQ
pseudo.value	the pseudo p-value of the SRQ test if nsim is not NULL
MeanNeig	Mean of the Maximum number of neighborhood
MaxNeig	Maximum number of neighborhood
listw	The list of neighborhood
nsim	number of boots (only for boots version)
SRGP	nsim simulated values of statistic.
SRLP	matrix with the number of runs for eacl localization.

Details

The order of the neighbourhoods (\(m_i-environments\)) is critical to obtain the test.
To obtain the number of runs observed in each \(m_i-environment\), each element must be associated with a set of neighbours ordered by proximity. Three kinds of lists can be included to identify \(m_i-environments\):

• knn: Objects of the class knn that consider the neighbours in order of proximity.
  

• nb: If the neighbours are obtained from an sf object, the code internally will call the function nb2nb_order it will order them in order of proximity of the centroids.
  

• matrix: If a object of matrix class based in the inverse of the distance in introduced as argument, the function nb2nb_order will also be called internally to transform the object the class matrix to a matrix of the class nb with ordered neighbours.
  

Two alternative sets of arguments can be included in this function to compute the spatial runs test:

Option 1	A factor (fx) and a list of neighborhood (listw) of the class knn.
Option 2	A sf object (data) and formula to specify the factor. A list of neighbourhood (listw)

Definition of spatial run

In this section define the concepts of spatial encoding and runs, and construct the main statistics necessary for testing spatial homogeneity of categorical variables. In order to develop a general theoretical setting, let us consider \(\{X_s\}_{s \in S}\) to be the categorical spatial process of interest with Q different categories, where S is a set of coordinates.

Spatial encoding: For a location \(s \in S\) denote by \(N_s = \{s_1,s_2 ...,s_{n_s}\}\) the set of neighbours according to the interaction scheme W, which are ordered from lesser to higher Euclidean distance with respect to location s.
The sequence as \(X_{s_i} , X_{s_i+1},...,, X_{s_i+r}\) its elements have the same value (or are identified by the same class) is called a spatial run at location s of length r.

Spatial run statistic

The total number of runs is defined as:
$$SR^Q=n+\sum_{s \in S}\sum_{j=1}^{n_s}I_j^s$$
where \(I_j^s = 1 \ if \ X_{s_j-1} \neq X_{s_j} \ and 0 \ otherwise\) for \(j=1,2,...,n_s\)
Following result by the Central Limit Theorem, the asymtotical distribution of \(SR^Q\) is:
$$SR^Q = N(\mu_{SR^Q},\sigma_{SR^Q})$$

In the one-tailed case, we must distinguish the lower-tailed test and the upper-tailed, which are associated with homogeneity and heterogeneity respectively. In the case of the lower-tailed test, the following hypotheses are used:

\(H_0:\{X_s\}_{s \in S}\) is i.i.d.
\(H_1\): The spatial distribution of the values of the categorical variable is more homogeneous than under the null hypothesis (according to the fixed association scheme). In the upper-tailed test, the following hypotheses are used:

\(H_0:\{X_s\}_{s \in S}\) is i.i.d.
\(H_1\): The spatial distribution of the values of the categorical variable is more heterogeneous than under the null hypothesis (according to the fixed association scheme).

These hypotheses provide a decision rule regarding the degree of homogeneity in the spatial distribution of the values of the spatial categorical random variable.

Control arguments

seedinit

Numerical value for the seed (only for boot version). Default value seedinit=123

References

• Ruiz, M., López, F., and Páez, A. (2021). A test for global and local homogeneity of categorical data based on spatial runs. Working paper.

See also

local.sp.runs.test, dgp.spq, Q.test,

Author

Fernando López	fernando.lopez@upct.es
Román Mínguez	roman.minguez@uclm.es
Antonio Páez	paezha@gmail.com
Manuel Ruiz	manuel.ruiz@upct.es

Examples

# Case 1: SRQ test based on factor and knn
# \donttest{
n <- 100
cx <- runif(n)
cy <- runif(n)
x <- cbind(cx,cy)
listw <- spdep::knearneigh(cbind(cx,cy), k=3)
p <- c(1/6,3/6,2/6)
rho <- 0.5
fx <- dgp.spq(listw = listw, p = p, rho = rho)
srq <- sp.runs.test(fx = fx, listw = listw)
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Asymptotic
#> 	version
#> 
#> data:  mxf
#> sp.runs test = -1.3614, p-value = 0.1734
#> alternative hypothesis: two.sided
#> sample estimates:
#>          Total runs     Mean total runs Variance total runs 
#>            270.0000            285.5152            129.8759 
#> 
plot(srq)


# Boots Version
control <- list(seedinit = 1255)
srq <- sp.runs.test(fx = fx, listw = listw, distr = "bootstrap" , nsim = 299, control = control)
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Boots version
#> 
#> data:  mxf
#> sp.runs test = -1.3614, p-value = 0.1533
#> alternative hypothesis: two.sided
#> sample estimates:
#>       Observed Total runs     Mean total runs boots Variance total runs boots 
#>                  270.0000                  284.8194                  126.7391 
#> 
plot(srq)


# Case 2: SRQ test with formula, a sf object (points) and knn
data("FastFood.sf")
x <- sf::st_coordinates(sf::st_centroid(FastFood.sf))
#> Warning: st_centroid assumes attributes are constant over geometries of x
#> Warning: bounding box has potentially an invalid value range for longlat data
#> Warning: st_centroid does not give correct centroids for longitude/latitude data
listw <- spdep::knearneigh(x, k=4)
formula <- ~ Type
srq <- sp.runs.test(formula = formula, data = FastFood.sf, listw = listw)
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Asymptotic
#> 	version
#> 
#> data:  Type
#> sp.runs test = 3.899, p-value = 9.657e-05
#> alternative hypothesis: two.sided
#> sample estimates:
#>          Total runs     Mean total runs Variance total runs 
#>            3408.000            3216.242            2418.743 
#> 
plot(srq)

# Version boots
srq <- sp.runs.test(formula = formula, data = FastFood.sf, listw = listw,
distr = "bootstrap", nsim = 199)
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Boots version
#> 
#> data:  Type
#> sp.runs test = 3.899, p-value < 2.2e-16
#> alternative hypothesis: two.sided
#> sample estimates:
#>       Observed Total runs     Mean total runs boots Variance total runs boots 
#>                  3408.000                  3213.030                  2597.191 
#> 
plot(srq)


# Case 3: SRQ test (permutation) using formula with a sf object (polygons) and nb
library(sf)
fname <- system.file("shape/nc.shp", package="sf")
nc <- sf::st_read(fname)
#> Reading layer `nc' from data source 
#>   `/home/runner/work/_temp/Library/sf/shape/nc.shp' using driver `ESRI Shapefile'
#> Simple feature collection with 100 features and 14 fields
#> Geometry type: MULTIPOLYGON
#> Dimension:     XY
#> Bounding box:  xmin: -84.32385 ymin: 33.88199 xmax: -75.45698 ymax: 36.58965
#> Geodetic CRS:  NAD27
listw <- spdep::poly2nb(as(nc,"Spatial"), queen = FALSE)
#> although coordinates are longitude/latitude, st_intersects assumes that they are planar
p <- c(1/6,3/6,2/6)
rho = 0.5
co <- sf::st_coordinates(sf::st_centroid(nc))
#> Warning: st_centroid assumes attributes are constant over geometries of x
#> Warning: st_centroid does not give correct centroids for longitude/latitude data
nc$fx <- dgp.spq(listw = listw, p = p, rho = rho)
plot(nc["fx"])

formula <- ~ fx
srq <- sp.runs.test(formula = formula, data = nc, listw = listw,
distr = "bootstrap", nsim = 399)
#> Warning: st_centroid assumes attributes are constant over geometries of x
#> Warning: st_centroid does not give correct centroids for longitude/latitude data
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Boots version
#> 
#> data:  fx
#> sp.runs test = -1.6349, p-value = 0.0925
#> alternative hypothesis: two.sided
#> sample estimates:
#>       Observed Total runs     Mean total runs boots Variance total runs boots 
#>                  366.0000                  384.9248                  140.7984 
#> 
plot(srq)


# Case 4: SRQ test (Asymptotic) using formula with a sf object (polygons) and nb
data(provinces_spain)
sf::sf_use_s2(FALSE)
listw <- spdep::poly2nb(provinces_spain, queen = FALSE)
#> although coordinates are longitude/latitude, st_intersects assumes that they are planar
provinces_spain$Coast <- factor(provinces_spain$Coast)
levels(provinces_spain$Coast) = c("no","yes")
plot(provinces_spain["Coast"])

formula <- ~ Coast
srq <- sp.runs.test(formula = formula, data = provinces_spain, listw = listw)
#> Warning: st_centroid assumes attributes are constant over geometries of x
#> Warning: st_centroid does not give correct centroids for longitude/latitude data
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Asymptotic
#> 	version
#> 
#> data:  Coast
#> sp.runs test = -2.7795, p-value = 0.005444
#> alternative hypothesis: two.sided
#> sample estimates:
#>          Total runs     Mean total runs Variance total runs 
#>           133.00000           160.36571            96.93415 
#> 
plot(srq)


# Boots version
srq <- sp.runs.test(formula = formula, data = provinces_spain, listw = listw,
distr = "bootstrap", nsim = 299)
#> Warning: st_centroid assumes attributes are constant over geometries of x
#> Warning: st_centroid does not give correct centroids for longitude/latitude data
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Boots version
#> 
#> data:  Coast
#> sp.runs test = -2.7795, p-value = 0.003333
#> alternative hypothesis: two.sided
#> sample estimates:
#>       Observed Total runs     Mean total runs boots Variance total runs boots 
#>                 133.00000                 157.27425                  92.83058 
#> 
plot(srq)


# Case 5: SRQ test based on a distance matrix (inverse distance)
N <- 100
cx <- runif(N)
cy <- runif(N)
data <- as.data.frame(cbind(cx,cy))
data <- sf::st_as_sf(data,coords = c("cx","cy"))
n = dim(data)[1]
dis <- 1/matrix(as.numeric(sf::st_distance(data,data)),ncol=n,nrow=n)
diag(dis) <- 0
dis <- (dis < quantile(dis,.10))*dis
p <- c(1/6,3/6,2/6)
rho <- 0.5
fx <- dgp.spq(listw = dis , p = p, rho = rho)
srq <- sp.runs.test(fx = fx, listw = dis)
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Asymptotic
#> 	version
#> 
#> data:  mxf
#> sp.runs test = -1.3821, p-value = 0.1669
#> alternative hypothesis: two.sided
#> sample estimates:
#>          Total runs     Mean total runs Variance total runs 
#>            605.0000            656.5455           1390.9461 
#> 
plot(srq)


srq <- sp.runs.test(fx = fx, listw = dis, data = data)
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Asymptotic
#> 	version
#> 
#> data:  mxf
#> sp.runs test = -1.3821, p-value = 0.1669
#> alternative hypothesis: two.sided
#> sample estimates:
#>          Total runs     Mean total runs Variance total runs 
#>            605.0000            656.5455           1390.9461 
#> 
plot(srq)


# Boots version
srq <- sp.runs.test(fx = fx, listw = dis, data = data, distr = "bootstrap", nsim = 299)
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Boots version
#> 
#> data:  mxf
#> sp.runs test = -1.3821, p-value = 0.21
#> alternative hypothesis: two.sided
#> sample estimates:
#>       Observed Total runs     Mean total runs boots Variance total runs boots 
#>                  605.0000                  635.7793                 1406.2800 
#> 
plot(srq)


# Case 6: SRQ test based on a distance matrix (inverse distance)
data("FastFood.sf")
sf::sf_use_s2(FALSE)
n = dim(FastFood.sf)[1]
dis <- 1000000/matrix(as.numeric(sf::st_distance(FastFood.sf,FastFood.sf)), ncol = n, nrow = n)
#> Warning: bounding box has potentially an invalid value range for longlat data
#> Warning: bounding box has potentially an invalid value range for longlat data
#> Warning: bounding box has potentially an invalid value range for longlat data
diag(dis) <- 0
dis <- (dis < quantile(dis,.005))*dis
p <- c(1/6,3/6,2/6)
rho = 0.5
co <- sf::st_coordinates(sf::st_centroid(FastFood.sf))
#> Warning: st_centroid assumes attributes are constant over geometries of x
#> Warning: bounding box has potentially an invalid value range for longlat data
#> Warning: st_centroid does not give correct centroids for longitude/latitude data
FastFood.sf$fx <- dgp.spq(p = p, listw = dis, rho = rho)
plot(FastFood.sf["fx"])
#> Warning: bounding box has potentially an invalid value range for longlat data
#> Warning: bounding box has potentially an invalid value range for longlat data

formula <- ~ fx

# Boots version
srq <- sp.runs.test(formula = formula, data = FastFood.sf, listw = dis,
distr = "bootstrap", nsim = 299)
#> Warning: st_centroid assumes attributes are constant over geometries of x
#> Warning: bounding box has potentially an invalid value range for longlat data
#> Warning: st_centroid does not give correct centroids for longitude/latitude data
print(srq)
#> 
#> 	Runs test of spatial dependence for qualitative data. Boots version
#> 
#> data:  fx
#> sp.runs test = -5.6155, p-value = 0.006667
#> alternative hypothesis: two.sided
#> sample estimates:
#>       Observed Total runs     Mean total runs boots Variance total runs boots 
#>                  2399.000                  2541.642                  2778.801 
#> 
plot(srq)

# }