Package 'spqdep'

Testing for Spatial Dependence of Qualitative Data in Cross Section.

Description

spqdep offers the user a collection of functions to testing for spatial independence of categorical cross-section datasets.

Details

Some functionalities that have been included in spqdep package are:

1. Test Q

Testing for spatial dependence with the Q test based on the symbolic entropy. See references

2. Test QMap

Two statistics for the spatial analysis of qualitative data are obtained that are based on the symbolic entropy of the maps. See references

3. Spatial runs test

Testing for spatial dependence with the global test based on runs. See references

4. Local spatial runs test

Obtain the local test based on runs. See references

5. Scan test

Testing for spatial dependence with the Scan test. See references

6. Join-count tests

Testing for spatial dependence with the classical Join-count test. See references

7. Similarity test

Testing for spatial dependence with the Similarity test. See references

Datasets

(example provinces_spain, FastFood.sf) spqdep includes two different datasets: provinces_spain and FastFood.sf. These sets are used to illustrate the capabilities of different functions. Briefly, their main characteristics are the following

• The FastFood.sf A sf object with points of the localization of Fastfood restaurants in Toronto.

• The provinces_spain An object of the class sf with the geometry of spanish provinces and several data sets.

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Ruiz M, López FA and A Páez (2011). Comparison of Thematic Maps Using Symbolic Entropy. International Journal of Geographical Information Science, 26, 413-439.

• Ruiz, M., López, F., and Páez, A. (2010). Testing for spatial association of qualitative data using symbolic dynamics. Journal of Geographical Systems, 12(3), 281-309.0.

• Kulldorff M, Nagarwalla N. (1995). Spatial disease clusters: Detection and Inference. Statistics in Medicine. 14:799-810

• Jung I, Kulldorff M, Richard OJ (2010). A spatial scan statistic for multinomial data. Statistics in Medicine. 29(18), 1910-1918

• Páez, A, López-Hernández, FA, Ortega-García, JA, & Ruiz, M. (2016). Clustering and co-occurrence of cancer types: A comparison of techniques with an application to pediatric cancer in Murcia, Spain. Spatial Analysis in Health Geography, 69-90.

---

Boots.sf.

Description

A simple features object square regular lattice 16x16. from Fig. 3.3 in Upton and Fingleton (1985). In this figure, the cells coded black/white correspond to quadrats where the perennial shrub Atriplex hymenelytra is present/absent in a sample area in Death Valley, California.

Usage

data(Boots.sf)

Format

A simple features dataframe with 256 rows and 1 column:

BW

A factor with two levels: Black and White

Source

Boots, B. (2003) https://link.springer.com/content/pdf/10.1007/s10109-003-0110-3.pdf

References

• Boots, B. (2003). Developing local measures of spatial association for categorical data. Journal of Geographical Systems, 5(2), 139-160.

• Upton G., Fingleton B. (1985) Spatial data analysis by example. Volume 1: Point pattern and quantitative data. John Wiley & Sons, Chichester

Examples

data(Boots.sf)
 summary(Boots.sf)
 plot(Boots.sf)

---

A function to create symbols

Description

This function obtains the set of symbols to get the Q-statistic

Usage

cr_symb(k = k, m = m)

Arguments

k	number of categories
m	length of the m-surrounding

Details

...

Value

A list with two types of symbols. Permutation and Combinations-totals

p_symb	Matrix with symbols (permutations)
c_symb	Matrix with symbols (combinations)

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Ruiz M, López FA, A Páez. (2010). Testing for spatial association of qualitative data using symbolic dynamics. Journal of Geographical Systems. 12 (3) 281-309

See Also

m.surround

Examples

# Example 1: Obtain symbols for k=2 classes and m-surroundings of size 5
symb25 <- cr_symb(2,5)
symb25$p_symb # Permutations symbols
symb25$c_symb # Combinations-totals symbols

---

Generation of qualitative process with spatial structure

Description

The purpose of the function dgp.spq is to generate a random dataset with the dimensions and spatial structure decided by the user. This function may be useful in pure simulation experiments or with the aim of showing specific properties and characteristics of a spatial qualitative dataset.

Usage

dgp.spq(listw = listw, p = p,  rho = rho, control = list())

Arguments

listw	A listw object of the class nb, knn, listw o matrix created for example by nb2listw from spatialreg package; if nb2listw not given, set to the same spatial weights as the listw argument. It can also be a spatial weighting matrix of order (NxN) instead of a listw object. Default = NULL.
p	a vector with the percentage of elements of each categories. The lengths must be the number of categories. The sum of the elements of vector must be 1.
rho	the level of spatial dependence (values between -1 y 1)
control	List of additional control arguments. See control argument section.

Details

In order to obtain categorical random variables with controlled degrees of spatial dependence, we have designed a two- stage data generating process:

Firstly, we simulate autocorrelated data using the following model:

$Y = (I - \rho W)^{-1} \epsilon$

where $\epsilon = N(0,1)$ I is the $N \times N$ identity matrix, $\rho$ is a parameter of spatial dependence, and W is a connectivity matrix that determines the set of spatial relationships among points.

In the second step of the data generation process, the continuous spatially autocorrelated variable Y is used to define a discrete spatial process as follows. Let $b_{ij}$ be defined by:

$p(Y \leq b_{ij})= {i \over j} \ \ \ with \ \ \ i<j$

Let $A =\{a_1,a_2,...,a_k\}$ and define the discrete spatial process as:

$X_s=a_1 \ \ \ if \ \ \ Y_s \leq b_{1k}$

$X_s=a_i \ \ \ if \ \ \ b_{i-1k} < Y_s \leq b_{ik}$

$X_s=a_k \ \ \ if \ \ \ Y_s > b_{k-1k}$

Value

a factor of length N (listw is a matrix of order (NxN)) with levels the first capital letters: "A", "B", ....

The description of the DGP is available in Páez et al. 2010 (pag 291) and in details section.

Control arguments

seedinit

seed to generate the data sets

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Ruiz M, López FA, A Páez. (2010). Testing for spatial association of qualitative data using symbolic dynamics. Journal of Geographical Systems. 12 (3) 281-309

See Also

Q.test, Q.map.test, sp.runs.test, scan.test

Examples

#
N <- 100
cx <- runif(N)
cy <- runif(N)
coor <- cbind(cx,cy)
p <- c(1/6,3/6,2/6)
rho = 0.5
listw <- spdep::nb2listw(spdep::knn2nb(spdep::knearneigh(cbind(cx,cy), k = 4)))
xf <- dgp.spq(list = listw, p = p, rho = rho)

data(provinces_spain)
listw <- spdep::poly2nb(provinces_spain, queen = FALSE)
p <- c(1/6,3/6,2/6)
rho = 0.9
xf <- dgp.spq(p = p, listw = listw, rho = rho)
provinces_spain$xf <- xf
plot(provinces_spain["xf"])

---

Selection of fast food restaurants in Toronto

Description

A simple feature (sf) dataframe containing the locations of a selection of fast food restaurants in the city of Toronto, Canada (data are from 2008). The data are projected using EPSG: 26917 (WGS 84/UTM Zone 17N).

Usage

data(FastFood.sf)

Format

A simple features object with 614 rows and 3 variables:

ID

Unique identifier of record.

Class

Factor with 3 types of fast food restaurants: [P]izza, [S]andwich, and [H]amburger

geometry

Geometry of simple features.

Source

Ruiz et al. (2010) https://link.springer.com/article/10.1007/s10109-009-0100-1

References

• Ruiz M, López FA, A Páez. (2010). Testing for spatial association of qualitative data using symbolic dynamics. Journal of Geographical Systems. 12 (3) 281-309

---

A function to compute joint-count test for binomial and multinomial (asymptotic and permutation distributions).

Description

A function to compute joincount tests for spatial qualitative data. This function is a wrapper of joincount.multi and joincount.test in spdep package.

Usage

jc.test(formula = NULL,
               data = NULL,
               fx = NULL,
               listw = NULL,
               na.action,
               zero.policy = NULL,
               distr = "asymptotic",
               alternative = "greater",
               control =list())

Arguments

formula	a symbolic description of the factor(s).
data	an (optional) data frame or a sf object with points/multipolygons geometry containing the variable(s) to be tested.
fx	a factor or a matrix of factors in columns
listw	A listw object created for example by nb2listw from spdep package; if nb2listw not given, the spatial weights are built using the object given in listw argument (usually an sf object). Default = NULL.
na.action	A function (default options("na.action")), can also be na.omit or na.exclude. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations.
zero.policy	Similar to the corresponding parameter of lagsarlm function in spatialreg package. If TRUE assign zero to the lagged value of zones without neighbours. Default = NULL.
distr	character. Distribution type "asymptotic" (default) or "mc".
alternative	character string specifying the alternative hypothesis, must be one of "greater" (default), or "less".
control	list of additional arguments.

Value

An spjctest object. This type of object is a list of htest objects. The length of the list is the number of factor variables included in the formula or the number of columns in fx. Each element of the list can be a jclist object, for binomial factors, or a jcmulti object for multinomial factors. See joincount.test or joincount.multi for additional details.

Control arguments

nsim	number of permutations used in the Monte Carlo distribution. Default nsim = 999.
seedinit	seed to select the initial element during the simulations. Default seedinit = 1111.
adjust.n	default *TRUE*, if *FALSE* the number of observations is not adjusted for no-neighbour observations, if *TRUE*, the number of observations is adjusted consistently. See joincount.multi for aditional information.
spChk	should the data vector names be checked against the spatial objects for identity integrity, *TRUE*, or *FALSE*, default *NULL* to use *get.spChkOption()*. See joincount.multi for aditional information.
sampling	default *nonfree*, may be *free*. See joincount.test for additional information.
queen	default *TRUE*. Defines the neighborhood criteria for *sf* objects.
style	defines the style for listw. Default = *B* (binary).
knn	chooses the number of neighboors when this criteria is used. Default knn = 5.

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, pp. 19-20.

• Upton, G., Fingleton, B. 1985 Spatial data analysis by example: point pattern and qualitative data, Wiley, pp. 158–170.

See Also

print.summary.spqtest, joincount.test, joincount.multi

Examples

## Case 1
## Multinomial + Binomial using a sf multipolygon

data("provinces_spain")
sf::sf_use_s2(FALSE)
provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman")
provinces_spain$Older <- cut(provinces_spain$Older, breaks = c(-Inf,19,22.5,Inf))
levels(provinces_spain$Older) = c("low","middle","high")
f1 <- ~ Older + Male2Female
jc1 <- jc.test(formula = f1,
               data = provinces_spain,
               distr = "mc",
               alternative = "greater",
               zero.policy = TRUE)
summary(jc1)

provinces_spain$Coast <- factor(provinces_spain$Coast)
levels(provinces_spain$Coast) = c("no","yes")
f2 <- ~ Male2Female + Coast
jc2 <- jc.test(formula = f2,
               data = provinces_spain,
               distr = "mc",
               zero.policy = TRUE)
summary(jc2)


# Case 2:
## Multinomial using a sf multipoint
data("FastFood.sf")
sf::sf_use_s2(FALSE)
f1 <- ~ Type
jc3 <- jc.test(formula = f1,
               data = FastFood.sf,
               distr = "asymptotic",
               control = list(knn = 6))
 summary(jc3)

# Examples function joincount.test
data(oldcol, package = "spdep")
HICRIME <- cut(COL.OLD$CRIME, breaks = c(0,35,80), labels = c("low","high"))
names(HICRIME) <- rownames(COL.OLD)
jc4 <- jc.test(fx = HICRIME,
               listw = spdep::nb2listw(COL.nb,
               style="B"))
summary(jc4)
 jc5 <- jc.test(fx = HICRIME,
                listw = spdep::nb2listw(COL.nb, style="B"),
                distr = "mc")
 summary(jc5)
 HICRIME <- cut(COL.OLD$CRIME, breaks = c(0, 35, 80),
                labels = c("low","high"))
 names(HICRIME) <- rownames(COL.OLD)
 jc6 <- jc.test(fx = HICRIME,
                listw = spdep::nb2listw(COL.nb,
                                 style = "B"))
summary(jc6)

---

A function to calculate the local Join Count tests.

Description

This function calculates the local Join Count tests.This function is a wrapper of local_joincount in rgeoda package.

Usage

local.jc.test(formula = NULL, data = NULL, fx = NULL, case = NULL, coor = NULL,
 listw = listw, nsim = 999, control = list())

Arguments

formula	An (optional) formula with the factor included in data
data	An (optional) data frame or a sf object containing the variable to testing for.
fx	An (optional) factor of observations with the same length as the neighbours list in listw
case	level of the factor used to obtain the local Join Count test. See details.
coor	(optional) a 2xN vector with spatial coordinates. Used when *data* is not a spatial object.
listw	A neighbourhood list (an object type knn, listw or nb).
nsim	The number of permutation to obtain the local Join Count tests. Default value 999.
control	Optional argument. See Control argument section.

Details

This test develop by Anselin and Li (2019) can be apply only for binary variables (usually named B and W), and count the number of joins of type B-B where the proportion of observations with B is lower than half. The interest lies in identifying co-occurrences of uncommon events, i.e., situations where observations that take on the value of B constitute much less than half of the sample.
The statistic is defined as:

$BB_i = x_i \sum_j{w_{ij}x_j}$

only meaningful for those observations where $x_i=1$, since for $x_i=0$ the result will always equal zero.

The object listw can be the class:

• 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.

Value

The output is an object of the class localjc

local.SRQ A matrix with

nn	number of neighbourhood in the localization 'i'.
ljc	local Join Count statistics.
pseudo.value	p-value of local jc tests.

Control arguments

seedinit	Numerical value for the seed in boot version. Default value seedinit = 123

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Anselin, L., and Li, X. (2019). Operational local join Count statistics for cluster detection. Journal of Geographical Systems, 21(2), 189-210.

See Also

local.sp.runs.test, dgp.spq

Examples

# Case 1: Local spatial runs test based on knn
library(lwgeom)
N <- 100
cx <- runif(N)
cy <- runif(N)
coor <- cbind(cx,cy)
listw <- spdep::knearneigh(coor, k = 4)
p <- c(1/2,1/2)
rho <- 0.5
fx <- dgp.spq(p = p, listw = listw, rho = rho)
ljc <- local.jc.test(fx = fx, coor = coor, case= "A", listw = listw)
print(ljc)
plot(ljc, coor = coor, sig = 0.1)


# Case 2:Fastfood example. sf (points)
library(lwgeom)
data("FastFood.sf")
sf::sf_use_s2(FALSE)
x <- sf::st_coordinates(sf::st_centroid(FastFood.sf))
listw <- spdep::knearneigh(x, k = 6)
formula <- ~ Type
ljc <- local.jc.test(formula = formula, data = FastFood.sf, case = "H",listw = listw)
print(ljc)
plot(ljc, sf = FastFood.sf, sig = 0.05)

# Case 3: With a sf object (poligons)
library(lwgeom)
fname <- system.file("shape/nc.shp", package="sf")
nc <- sf::st_read(fname)
listw <- spdep::poly2nb(as(nc,"Spatial"), queen = FALSE)
p <- c(1/3,2/3)
rho = 0.5
nc$fx <- dgp.spq(p = p, listw = listw, rho = rho)
plot(nc["fx"])
formula <- ~ fx
ljc <- local.jc.test(formula = formula, data = nc, case = "A", listw = listw)
ljc
plot(ljc, sf = nc)

# Case 4: With isolated areas
library(lwgeom)
data(provinces_spain)
sf::sf_use_s2(FALSE)
listw <- spdep::poly2nb(as(provinces_spain,"Spatial"), queen = FALSE)
provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman")
formula <- ~ Male2Female
ljc <- local.jc.test(formula = formula, data = provinces_spain, listw = listw)
print(ljc)
plot(ljc, sf = provinces_spain, sig = 0.1)

# Case 5: Regular lattice
data(Boots.sf)
listw <- spdep::poly2nb(as(Boots.sf,"Spatial"), queen = TRUE)
formula <- ~ BW
ljc <- local.jc.test(formula = formula, data = Boots.sf, case="B", listw = listw)
print(ljc)
plot(ljc, sf = Boots.sf, sig = 0.05)
lsr <- local.sp.runs.test(formula = formula, data = Boots.sf,
distr = "bootstrap", nsim= 99, listw = listw)
print(lsr)
plot(lsr, sf = Boots.sf, sig = 0.1)
scan <- scan.test(formula = formula, data = Boots.sf, nsim = 299,
dist = "bernoulli", nv = 60 , case = "B", windows = "elliptic")
plot(scan, sf = Boots.sf)

---

A function to calculate the local spatial runs tests.

Description

This function calculates the local spatial runs tests for all localizations.

Usage

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

Arguments

formula	An (optional) formula with the factor included in data
data	An (optional) data frame or a sf object containing the variable to testing for.
fx	An (optional) factor of observations with the same length as the neighbors list in listw
distr	a character string specifying the distribution "asymptotic" (default) or "bootstrap"
listw	A neighbourhood list (an object 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".
nsim	Default value is NULL to obtain the asymptotic version of the local test. For the bootstrap version nsim is the number of permutations to obtain the pseudo-value.
control	Optional argument. See Control Argument section.

Details

The object listw can be the class:

• 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.

Value

The output is an object of the class localsrq

local.SRQ A matrix with

runs.i	number of runs in the localization 'i'.
E.i	expectation of local runs statistic in the localization 'i'.
Sd.i	standard deviate of local runs statistic in the localization 'i'.
z.value	standard value of local runs statistic (only for asymptotic version).
p.value	p-value of local local runs statistic (only for asymptotic version).
zseudo.value	standard value of local runs statistic (only for boots version).
pseudo.value	p-value of local runs statistic (only for boots version).

MeanNeig Mean of run.i
MaxNeig Maximum of run.i
listw the object listw
alternative a character string describing the alternative hypothesis

Control arguments

seedinit	Numerical value for the seed in boot version. Default value seedinit = 123

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

@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

sp.runs.test, dgp.spq

Examples

# Case 1: Local spatial runs test based on knn
library(lwgeom)
N <- 100
cx <- runif(N)
cy <- runif(N)
x <- cbind(cx,cy)
listw <- spdep::knearneigh(cbind(cx,cy), k = 10)
p <- c(1/6,3/6,2/6)
rho <- 0.5
fx <- dgp.spq(p = p, listw = listw, rho = rho)

# Asymtotic version
lsrq <- local.sp.runs.test(fx = fx, listw = listw, alternative = "less")
print(lsrq)
plot(lsrq, sig = 0.05)
# Asymtotic version
lsrq <- local.sp.runs.test(fx = fx, listw = listw, alternative = "two.sided",
                           distr ="bootstrap", nsim = 399)
print(lsrq)
plot(lsrq, sig = 0.1)

# Case 2:Fastfood example. sf (points)
library(lwgeom)
data("FastFood.sf")
sf::sf_use_s2(FALSE)
x <- sf::st_coordinates(sf::st_centroid(FastFood.sf))
listw <- spdep::knearneigh(x, k = 10)
formula <- ~ Type
lsrq <- local.sp.runs.test(formula = formula, data = FastFood.sf, listw = listw)
print(lsrq)
plot(lsrq, sf = FastFood.sf, sig = 0.05)


# Case 3: With a sf object (poligons)
library(lwgeom)
fname <- system.file("shape/nc.shp", package="sf")
nc <- sf::st_read(fname)
listw <- spdep::poly2nb(as(nc,"Spatial"), queen = FALSE)
p <- c(1/6,3/6,2/6)
rho = 0.5
nc$fx <- dgp.spq(p = p, listw = listw, rho = rho)
plot(nc["fx"])
formula <- ~ fx
lsrq <- local.sp.runs.test(formula = formula, data = nc, listw = listw)
print(lsrq)
plot(lsrq, sf = nc)
# Version boot
lsrq <- local.sp.runs.test(formula = formula, data = nc, listw = listw,
                           distr ="bootstrap", nsim = 399)
print(lsrq)
plot(lsrq, sf = nc)

# Case 4: With isolated areas
library(lwgeom)
data(provinces_spain)
listw <- spdep::poly2nb(as(provinces_spain,"Spatial"), queen = FALSE)
provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman")
plot(provinces_spain["Male2Female"])
formula <- ~ Male2Female
lsrq <- local.sp.runs.test(formula = formula, data = provinces_spain, listw = listw)
print(lsrq)
plot(lsrq, sf = provinces_spain, sig = 0.1)

# Boots Version
lsrq <- local.sp.runs.test(formula = formula, data = provinces_spain, listw = listw,
                           distr ="bootstrap", nsim = 199)
print(lsrq)
plot(lsrq, sf = provinces_spain, sig = 0.10)

# Case 5: SRQ test based on a distance matrix (inverse distance)

library(lwgeom)
N <- 100
cx <- runif(N)
cy <- runif(N)
coor <- as.data.frame(cbind(cx,cy))
coor <- sf::st_as_sf(coor,coords = c("cx","cy"))
n = dim(coor)[1]
dis <- 1/matrix(as.numeric(sf::st_distance(coor,coor)), 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(p = p, listw = dis, rho = rho)
lsrq <- local.sp.runs.test(fx = fx, listw = dis)
print(lsrq)
plot(lsrq, coor = cbind(cx,cy), sig = 0.05)
lsrq <- local.sp.runs.test(fx = fx, listw = dis, data = )
print(lsrq)
plot(lsrq, sf = coor)
# Version boots
lsrq <- local.sp.runs.test(fx = fx, listw = dis, data = coor,
                           distr ="bootstrap", nsim = 299)
print(lsrq)
plot(lsrq, sf = coor)

# SRQ test based on inverse distance
library(lwgeom)
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)
diag(dis) <- 0
dis <- (dis < quantile(dis,.01))*dis
formula <- ~ Type
lsrq <- local.sp.runs.test(formula = formula, data = FastFood.sf, listw = dis)
print(lsrq)
plot(lsrq, sf = FastFood.sf)

---

A function to generate m-surroundings

Description

This function obtains the m-surroundings by selecting the *m-1* nearest neighbors of each observation, allowing for a degree of overlap of *r*.

Usage

m.surround(x, m, r = 1, distance = "Euclidean", control = list())

Arguments

x	input sf object with points/multipolygons geometry or matrix of spatial coordinates
m	dimension of m-surrounding
r	maximum overlapping between any two m-surroundings.
distance	character. For Cartesian coordinates only: one of Euclidean, Hausdorff or Frechet; for geodetic coordinates, Great Circle distances are computed. (see sf::st_distance()). Default = "Euclidean".
control	List of additional control arguments.

Value

A list of class list and m_surr containing the following components:

ms	a matrix. Each row is a m-surrounding.
R	total number of observations.
rowexcl	index of rows excluded.
mdtms	distances between the observations of each m-surrounding.
N	total number of symbolized observations.
m	length of the m-surroundings.
r	overlapping degree.
initobs	element to start the generation of m-surroundings.
distance	type of distance.
m	length of the m-surroundings.
x	the input "x" as sf-object.

Control arguments

• initobs: Initial observation to begin the m-surrounding process. Default = 1.

• dtmaxabs: Threshold of distance (in absolute value) to prune the m-surroundings. Any m-surrounding exceeding the threshold is excluded. If dtmaxabs = 0 there is no exclusion of m-surroundings. Default = 0.

• dtmaxpc: Threshold of distance (as a percentage of the maximum distance between observations) to prune the m-surroundings. Any m-surrounding exceeding the threshold is excluded. Example if dtmaxpc = 0.1 the m-surrounding exceeding the 10 If dtmaxpc = 0 there is no exclusion of m-surroundings. Default = 0.

• dtmaxknn: Eliminate m-surroundings where some of the elements are not among the closest knn (k-nearest-neighbors). Example, if dtmaxknn = 4 exclude m-surroundings where some of the elements are not between the 4 closest. Default dtmaxknn = 0 (no exclusion)

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Ruiz M, López FA, A Páez. (2010). Testing for spatial association of qualitative data using symbolic dynamics. Journal of Geographical Systems. 12 (3) 281-309

Examples

# Example 1: Obtain m-surroundings with degree of overlapping r
N <- 100
cx <- runif(N)
cy <- runif(N)
x <- cbind(cx,cy)
m <- 3
r <- 1
msurr_points <- m.surround(x = x, m = m, r = r)
plot(msurr_points, type = 1)
plot(msurr_points, type = 2)
summary(msurr_points)
msurr_points <- m.surround(x = x, m = m, r = r,
                control = list(dtmaxpc = 0.1))
plot(msurr_points, type = 1)
plot(msurr_points, type = 2)
summary(msurr_points)
msurr_points <- m.surround(x = x, m = m, r = r,
                   control = list(dtmaxknn = 20))
plot(msurr_points, type = 1)
plot(msurr_points, type = 2)
summary(msurr_points)

# Example 2:

data("FastFood.sf")
m <- 3
r <- 1
msurr_points <- m.surround(x = FastFood.sf, m = m, r = r,
                           distance = "Euclidean",
                           control = list(dtmaxpc = .001))
plot(msurr_points, type = 1)
plot(msurr_points, type = 2)
print(msurr_points)
summary(msurr_points)
msurr_points <- m.surround(x = FastFood.sf, m = m, r = r,
                           distance = "Euclidean",
                           control = list(dtmaxknn = 20))
plot(msurr_points, type = 1)
plot(msurr_points, type = 2)
summary(msurr_points)


# Example 3: With isolated areas
data(provinces_spain)
sf::sf_use_s2(FALSE)
plot(sf::st_geometry(provinces_spain))
m <- 3
r <- 1
msurr_points <- m.surround(x = provinces_spain, m = m, r = r,
                           distance = "Euclidean",
                           control = list(dtmaxknn = 8))
plot(msurr_points, type = 1)
plot(msurr_points, type = 2)

# Example 4: Examples with multipolygons
fname <- system.file("shape/nc.shp", package="sf")
nc <- sf::st_read(fname)
plot(sf::st_geometry(nc))
m <- 3
r <- 1
msurr_polygonsf <- m.surround(x = nc, m = m, r = r,
                   distance = "Great Circle",
                   control=list(dtmaxpc = 0.20))
plot(msurr_polygonsf, type = 1)
plot(msurr_polygonsf, type = 2)

# Example 5: With regular lattice
sfc = sf::st_sfc(sf::st_polygon(list(rbind(c(0,0), c(1,0), c(1,1), c(0,1), c(0,0)))))
hexs <- sf::st_make_grid(sfc, cellsize = 0.1, square = FALSE)
hexs.sf <- sf::st_sf(hexs)
listw  <- spdep::poly2nb(as(hexs.sf, "Spatial"), queen = FALSE)
m <- 3
r <- 1
msurr_polygonsf <- m.surround(x = hexs.sf, m = m, r = r)
plot(msurr_polygonsf, type = 1)
plot(msurr_polygonsf, type = 2)
summary(msurr_polygonsf)

---

Methods for class localjc

Description

The plot() function allows the user to plot significant observations. The print() function is used to print the number of runs in each localization. Additional information of expected values and standard deviation, z-value ans p-value is prited for each observation.

Usage

## S3 method for class 'localjc'
print(x, ...)

## S3 method for class 'localjc'
plot(x, ..., sf = NULL, coor = NULL, sig = 0.05)

Arguments

x	a localjc object created by Q.test.
...	further arguments passed to or from other methods.
sf	optional argument for plot() method to include a sf object (default = NULL)
coor	optional argument for plot() method to include coordinates of points (default = NULL)
sig	significant level for each observation in plot() method. Default sig = 0.05

Value

No return value, called for side effects

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

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.

Examples

# Example 1: Local spatial runs test based on knn
library(lwgeom)
N <- 100
cx <- runif(N)
cy <- runif(N)
x <- cbind(cx,cy)
listw <- spdep::knearneigh(cbind(cx,cy), k = 10)
p <- c(1/6,3/6,2/6)
rho <- 0.5
fx <- dgp.spq(p = p, listw = listw, rho = rho)

# Asymtotic version
lsrq <- local.sp.runs.test(fx = fx, listw = listw, alternative = "less")
print(lsrq)
plot(lsrq, sig = 0.05)

---

Methods for class localsrq

Description

The plot() function allows the user to plot significant observations. The print() function is used to print the number of runs in each localization. Additional information of expected values and standard deviation, z-value ans p-value is prited for each observation.

Usage

## S3 method for class 'localsrq'
print(x, ...)

## S3 method for class 'localsrq'
plot(x, ..., sf = NULL, coor = NULL, sig = 0.05)

Arguments

x	a localsrq object created by Q.test.
...	further arguments passed to or from other methods.
sf	optional argument for plot() method to include a sf object (default = NULL)
coor	optional argument for plot() method to include coordinates of points (default = NULL)
sig	significant level for each observation in plot() method. Default sig = 0.05

Value

No return value, called for side effects

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

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.

Examples

# Example 1: Local spatial runs test based on knn
library(lwgeom)
N <- 100
cx <- runif(N)
cy <- runif(N)
x <- cbind(cx,cy)
listw <- spdep::knearneigh(cbind(cx,cy), k = 10)
p <- c(1/6,3/6,2/6)
rho <- 0.5
fx <- dgp.spq(p = p, listw = listw, rho = rho)

# Asymtotic version
lsrq <- local.sp.runs.test(fx = fx, listw = listw, alternative = "less")
print(lsrq)
plot(lsrq, sig = 0.05)

---

Method for class m_surr

Description

A function to plots the m-surrounds give an object of the class m_surr obtain with the code m.surround.
The plot() function allows the user view the configuration of the m-surroundings.
The argument type select the type o visualization.
The print() print the matrix of the m-surrounding.
. The summary give information about the characteristics of the m-surroundings.
.

Usage

## S3 method for class 'm_surr'
summary(object, ...)

## S3 method for class 'm_surr'
plot(x, ..., type = 1)

## S3 method for class 'm_surr'
print(x, ...)

Arguments

object	object of class m_surr. 2 plot W matrix with network
...	further arguments passed to or from other methods.
x	object of class m_surr
type	numeric. 1 (default) to get the plot with igraph.

Value

No return value, called for side effects

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

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.

Examples

# Example 1: Obtain m-surroundings with degree of overlapping r
N <- 100
cx <- runif(N)
cy <- runif(N)
x <- cbind(cx,cy)
m = 4
r = 2
msurr_points <- m.surround(x = x, m = m, r = r,control = list(dtmaxabs = 0.5))
plot(msurr_points, type = 1)
plot(msurr_points, type = 2)
print(msurr_points)

# Example 2:
data("FastFood.sf")
m = 6
r = 1
msurr_points <-  m.surround(x = FastFood.sf, m = m, r = r, distance = "Euclidean",
                            control = list(dtmaxpc = .01))
plot(msurr_points, type = 1)
plot(msurr_points, type = 2)
print(msurr_points)

---

Method for class qmap

Description

A function to plot the difference in frequencies of symbols of each map. The plot() function to obtain the plot. The argument ci select the confidence level.

Arguments

x	object of class qmap
ci	confidence level for the difference of probabilities of symbols in 'plot' method. Default ci = 0.95
...	further arguments passed to or from other methods.

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Ruiz M, López FA and A Páez (2011). Comparison of Thematic Maps Using Symbolic Entropy. International Journal of Geographical Information Science, 26, 413-439.

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

Examples

# Example 1:
N <- 100
cx <- runif(N)
cy <- runif(N)
coor <- cbind(cx,cy)
p <- c(1/6,3/6,2/6)
rho = 0.5
listw <- spdep::nb2listw(spdep::knn2nb(spdep::knearneigh(cbind(cx,cy), k = 4)))
fx <- dgp.spq(list = listw, p = p, rho = rho)
q.test <- Q.test(fx = fx, coor = coor, m = 3, r = 1)
plot(q.test)

---

Methods for class scantest

Description

The plot() function allows the user plot the significant cluster(s).
summary list information about the most likelihood cluster.

Usage

## S3 method for class 'scantest'
plot(x, ..., sf = NULL, coor = NULL)

## S3 method for class 'scantest'
summary(object, ...)

Arguments

x	is a scantest object created by scan.test for plot() method.
...	further arguments passed to or from other methods.
sf	optional argument for plot() method to include a sf object (default = NULL)
coor	optional argument for plot() method to include coordinates of points (default = NULL)
object	a scantest object created by scan.test for summary() method.

Value

No return value, called for side effects

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

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.

---

Methods for class spqtest

Description

The plot() function allows the user obtain the plot of relative frequency of each symbol (standard and equivalent) with the confidence interval. The print() function is used to get a list with the values of Q test for standard symbolization based on permutations and for equivalent symbolization based on combinations. summary() print a table with the output of the Q test.

Usage

## S3 method for class 'spqtest'
plot(x, ..., ci = 0.95)

Arguments

x	a spqtest object created by Q.test.
...	further arguments passed to or from other methods
ci	confidence level for the intervals in plot method. Default ci = 0.95

Value

This functions does not return any value

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

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.

---

A function to order the elements of the m_i-subrrounds

Description

An auxiliary function. In the case of obtaining the list of neighbors of class nb or poly2nb, it is necessary to reorder the elements based on distance and/or angle.

Usage

nb2nb_order(listw = listw, sf = NULL)

Arguments

listw	an object of the nb class.
sf	the sf object used to get the listw .

Details

Sort the elements of a list nb. First by distance and

Value

An object of the nb class with the elements in order.

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

@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

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

Examples

# With a sf object (irregular lattice)
library(sf)
fname <- system.file("shape/nc.shp", package="sf")
nc <- sf::st_read(fname)
listw <- spdep::poly2nb(as(nc,"Spatial"), queen = FALSE)
listw.order <- nb2nb_order(listw = listw, sf = nc)

# With a sf object (regular lattice: hexagons)
sfc = sf::st_sfc(sf::st_polygon(list(rbind(c(0,0), c(1,0), c(1,1), c(0,1), c(0,0)))))
hexs <- sf::st_make_grid(sfc, cellsize = 0.1, square = FALSE)
hexs.sf <- sf::st_sf(hexs)
listw  <- spdep::poly2nb(as(hexs.sf, "Spatial"), queen = FALSE)
listw.order <- nb2nb_order(listw = listw, sf = hexs.sf)

---

Extract of 1880 US Census for Newark, New Jersey.

Description

A simple features object with geocoded information about respondents in the 1880 US Census with selected demographic information coded as dummy variables. The data are projected using EPSG: 32618 (WGS 84/UTM Zone 18N). The coordinates have been and jiggled to create unique coordinates for each observation.

Usage

data(Newark.sf)

Format

A simple features dataframe with 21,520 rows and 8 columns:

ID

Unique identifier of record.

YANKEE

Dummy variable for ethnicity of respondent: 1 if Yankee, 0 otherwise.

IRISH

Dummy variable for ethnicity of respondent: 1 if Irish, 0 otherwise.

GERMAN

Dummy variable for ethnicity of respondent: 1 if German, 0 otherwise.

under30

Dummy variable for age of respondent: 1 if younger than 30 years old, 0 otherwise.

mar

Dummy variable for marital status of respondent: 1 if married, 0 otherwise.

usborn

Dummy variable for place of birth of respondent: 1 if born in the US, 0 otherwise.

geometry

geometry of the simple features object

Source

Páez et al. (2012) doi:10.1080/00045608.2011.620502

References

• Paez, A., Ruiz, M., Lopez, F. & Logan, J. (2012). Measuring Ethnic Clustering and Exposure with the Q Statistic: An Exploratory Analysis of Irish, Germans, and Yankees in 1880 Newark.. Annals of the Association of American Geographers.

Examples

data(Newark.sf)
 summary(Newark.sf)

---

Plot the empirical distribution of runs

Description

Plot the empirical distribution of runs

Usage

## S3 method for class 'sprunstest'
plot(x, ...)

Arguments

x	A object of class sprunstest.
...	further arguments passed to or from other methods.

Details

Plot the histogram with the empirical distribution of the runs

Value

No return value, called for side effects

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

See Also

sp.runs.test.

Examples

# Example 1: Fastfood example. sf (points)
data("FastFood.sf")
x <- sf::st_coordinates(sf::st_centroid(FastFood.sf))
listw <- spdep::knearneigh(x, k = 2)
formula <- ~ Type
srq <- sp.runs.test(formula = formula, data = FastFood.sf, listw = listw, nsim = 299)
plot(srq)

# Example 2: Spain example (poligons with 0 neinghbourhood)
data("provinces_spain")
sf::sf_use_s2(FALSE)
listw <- spdep::poly2nb(as(provinces_spain,"Spatial"), queen = FALSE)
provinces_spain$Older <- cut(provinces_spain$Older, breaks = c(-Inf,19,22.5,Inf))
levels(provinces_spain$Older) = c("low","middle","high")
formula <- ~ Older
srq <- sp.runs.test(formula = formula, data = provinces_spain, listw = listw, nsim = 299)
plot(srq)
provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman")
formula <- ~ Male2Female
srq <- sp.runs.test(formula = formula, data = provinces_spain, listw = listw, nsim = 299)
plot(srq)

---

Print method for objects of class summary.spjctest.

Description

Print method for objects of class summary.spjctest.

Usage

## S3 method for class 'summary.spjctest'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

x	object of class summary.spjctest.
digits	number of digits to show in printed tables. Default: max(3L, getOption("digits") - 3L).
...	further arguments passed to or from other methods.

Value

No return value, called for side effects

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

See Also

summary.spqtest.

---

Print method for objects of class summary.spqtest.

Description

Print method for objects of class summary.spqtest.

Usage

## S3 method for class 'summary.spqtest'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

x	object of class summary.spqtest.
digits	number of digits to show in printed tables. Default: max(3L, getOption("digits") - 3L).
...	further arguments passed to or from other methods.

Value

No return value, called for side effects

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

See Also

summary.spqtest.

---

Provinces in Spain.

Description

A simple features object with the provinces in Spain and selected demographic and economic information.

Usage

data(provinces_spain)

Format

A simple features dataframe with 50 rows and 15 columns:

province

Names of provinces in Spain as factor

CCAA

Names of Autonomous Communities in Spain as factor

ID_INE

National Institute of Statistics unique identifier of the provinces

Population

Population in the province in 2020

Density

Population density in the province in persons/km^2

Older

Percentage of population 65 and older in the provice in 2020

Median_Age

Median age of population in the province in 2020

Male2Female

Ratio of male to female population in the province in 2020

GDPpc

GDP per capita in the province in 2016

Transit

Dummy variable for mass transit system in province; 1: YES

Area

Area of the province

Altitude

Altitude of the province

Coast

A dummy variable that indicates whether the province is in the coast; 1: YES

Meteo_Station

Identifier of meteorological station representative of the province used to retrieve climatic variables

geometry

geometry of the simple features object

Source

Instituto Nacional de Estadistica http://www.ine.es/

Climatic data: Agencia Estatal de Meteorologia http://www.aemet.es/

Páez et al. (2020)

References

• Paez, A., Lopez, F.A., Menezes, T., Cavalcanti, R., & Pitta, M. (2020). A Spatio‐Temporal Analysis of the Environmental Correlates of COVID‐19 Incidence in Spain. Geographical Analysis. 53(3) 397-421

Examples

data(provinces_spain)
 summary(provinces_spain)

---

Compute the QE and QI tests of Equivalence and Independence between maps

Description

This function compute the QE and QI tests for maps comparison based on symbolic entropy.

Usage

Q.map.test(formula = formula, data = data, coor = NULL, m = m, r = 1,
type = "combinations", control = list())

Arguments

formula	a symbolic description of the two factors.
data	(optional) a data frame or a sf object containing the variables to testing for.
coor	(optional) a 2xN vector with coordinates.
m	length of m-surrounding.
r	maximum overlapping between any two m-surroundings (default = 1).
type	Type of symbols: "permutations" or "combinations". Default "combinations"
control	Optional argument. See Control Argument section.

Details

If data is not a sf object the coor argument with the coordinates of each observation must be included.

Value

A list with two objects of the class htest. The first one is the QE test of Equivalence between maps and the second one is the QI test of independence between maps. the elements of each test are:

method	a character string giving description of the method.
data.name	a character string giving the name(s) of the data.
statistic	the value of the statistic QE or/and QI.
alternative	a character string describing the alternative hypothesis.
p.value	p-value for QE or QI.
parameter	free degree of the statistic for QE or QI.
symb	A matrix with the symbols.
mh	m-surrounding of th map.
Tm	number of maps (ONLY 2).
sample.size	number of symbolized observations.
nsk	a matrix Tm x symbols with the frequency of the number of symbols of each map.

Control arguments

Several parameters to construct the m-surrounding

dtmaxabs

Delete degenerate surrounding based on the absolute distance between observations.

dtmaxpc

A value between 0 and 1. Delete degenerate surrounding based on the distance. Delete m-surrounding when the maximum distance between observation is upper than k percentage of maximum distance between anywhere observation.

dtmaxknn

A integer value 'k'. Delete degenerate surrounding based on the near neighborhood criteria. Delete m-surrounding is a element of the m-surrounding is not include in the set of k near neighborhood of the first element

seedinit

seed to select the initial element to star the algorithm to compute the m-surroundings.

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Ruiz M, López FA and A Páez (2011). Comparison of Thematic Maps Using Symbolic Entropy. International Journal of Geographical Information Science, 26, 413-439.

• Ruiz, M., López, FA, and Páez, A. (2010). Testing for spatial association of qualitative data using symbolic dynamics. Journal of Geographical Systems, 12(3), 281-309.0.

See Also

dgp.spq, m.surround, Q.test

Examples

# Case 1:
N <- 200
cx <- runif(N)
cy <- runif(N)
x <- cbind(cx,cy)
listw <- spdep::nb2listw(spdep::knn2nb(
           spdep::knearneigh(cbind(cx,cy), k = 4)))
p <- c(1/6, 3/6, 2/6)
rho = 0.5
QY1 <- dgp.spq(p = p, listw = listw, rho = rho)
rho = 0.8
QY2 <- dgp.spq(p = p, listw = listw, rho = rho)
dt = data.frame(QY1,QY2)
m = 3
r = 1
formula <- ~ QY1 + QY2
control <- list(dtmaxknn = 10)
qmap <- Q.map.test(formula = formula, data = dt, coor = x, m = m, r = r,
                   type ="combinations", control = control)
print(qmap)
plot(qmap)
plot(qmap, ci=.6)
plot(qmap[[1]]$mh)
summary(qmap[[1]]$mh)

control <- list(dtmaxknn = 20)
qmap <- Q.map.test(formula = formula, data = dt, coor = x, m = m, r = r,
                   type ="permutations", control = control)
print(qmap)
plot(qmap)
plot(qmap[[1]]$mh)
qmap <- Q.map.test(formula = formula, data = dt, coor = x, m = m, r = r,
                   type ="combinations")
print(qmap)
plot(qmap)
control <- list(dtmaxknn = 10)
qmap <- Q.map.test(formula = formula, data = dt, coor = x, m = m, r = r,
                   type ="combinations", control = control)
print(qmap)
plot(qmap)

# Case 2:
data(provinces_spain)
sf::sf_use_s2(FALSE)
m = 3
r = 1

provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman")
provinces_spain$Coast <- factor(provinces_spain$Coast)
levels(provinces_spain$Coast) = c("no","yes")
formula <- ~ Coast + Male2Female
qmap <- Q.map.test(formula = formula, data = provinces_spain, m = m, r = r,
                   type ="combinations")
print(qmap)
plot(qmap)
plot(qmap[[1]]$mh)

control <- list(dtmaxknn = 6)
qmap <- Q.map.test(formula = formula, data = provinces_spain, m = m, r = r,
                   type ="combinations", control = control)
print(qmap)
plot(qmap[[1]]$mh)
summary(qmap[[1]]$mh)

---

A function to compute Q test for spatial qualitative data

Description

A function to compute Q test for spatial qualitative data.

Usage

Q.test(formula = NULL, data = NULL, na.action,
fx = NULL, coor = NULL, m = 3, r = 1, distr = "asymptotic",
control = list())

Arguments

formula	a symbolic description of the factor(s).
data	an (optional) data frame or a sf object with points/multipolygons geometry containing the variable(s) to be tested.
na.action	action with NA values
fx	a factor or a matrix of factors in columns
coor	(optional) a 2xN vector with spatial coordinates. Used when *data* is not a spatial object
m	length of m-surrounding (default = 3).
r	only for asimtotic distribution. Maximum overlapping between any two m-surroundings (default = 1).
distr	character. Distribution type "asymptotic" (default) or "mc".
control	Optional argument. See Control Argument section.

Details

The Q-test is a simple, consistent, and powerful statistic for qualitative spatial independence that we develop using concepts from symbolic dynamics and symbolic entropy. The Q test can be used to detect, given a spatial distribution of events, patterns of spatial association of qualitative variables in a wide variety of settings.
The Q(m) statistic was introduced by Ruiz et al. (2010) as a tool to explore geographical co-location/co-occurrence of qualitative data. Consider a spatial variable X which is the result of a qualitative process with a set number of categorical outcomes $a_j$ (j=1,...,k). The spatial variable is observed at a set of fixed locations indexed by their coordinates $s_i$ (i=1,..., N), so that at each location si where an event is observed, $X_i$ takes one of the possible values $a_j$.

Since the observations are georeferenced, a spatial embedding protocol can be devised to assess the spatial property of co-location. Let us define, for an observation at a specified location, say $s_0$, a surrounding of size m, called an m-surrounding.
The m-surrounding is the set of m-1 nearest neighbours from the perspective of location $s_0$. In the case of distance ties, a secondary criterion can be invoked based on direction.
Once that an embedding protocol is adopted and the elements of the m-surrounding for location $s_0$ have been determined, a string can be obtained that collects the elements of the local neighborhood (the m-1 nearest neighbors) of the observation at $s_0$. The m-surrounding can then be represented in the following way:

$X_m(s_0)=(X_{s_0},X_{s_1},...X_{s_{m-1}})$

Since each observation Xs takes one of k possible values, and there are m observations in the m-surrounding, there are exactly k possible unique ways in which those values can co-locate. This is the number of permutations with replacement.
For instance, if k=2 (e.g. the possible outcomes are a1=0 and a2=1) and m=3, the following eight unique patterns of co-location are possible (the number of symbols is $n_{\sigma}$=8): 0,0,0, 1,0,0, 0,1,0, 0,0,1, 1,1,0, 1,0,1, 0,1,1, and 1,1,1. Each unique co-locationtype can be denoted in a convenient way by means of a symbol $\sigma_i$ $(i=1, 2,...,k^m)$. It follows that each site can be uniquely associated with a specific symbol, in a process termed symbolization. In this way, we say that a location s is of type $\sigma_i$ if and only if $X_m(s)=\sigma_i$.
Equivalent symbols (see Páez, et al. 2012) can be obtained by counting the number of occurrences of each category within an m-surrounding. This surrenders some topological information (ordering within the m-surrounding is lost) in favor of a more compact set of symbols, since the number of combinations with replacement.

Definition of Q(m) statistic

Let $\{X_s\}_{s \in R}$ be a discrete spatial process and m be a fixed embedding dimension. The statistic Q testing the null hypothesis:

$H_0:\{X_s\}_{s \in R}$ is spatially independent, against any other alternative.

For a fixed $m \geq 2$, the relative frequency of symbols can be used to define the symbolic entropy of the spatial process as the Shanon entropy of the distinct symbols:

$h(m) = - \sum_j p_{\sigma_j}ln(p_{\sigma_j})$

where

$p_{\sigma_j}={ n_{\sigma_j} \over R}$

with $n_{\sigma_j}$ is simply the number of times that the symbol $\sigma_j$ is observed and R the number of symbolized locations. The entropy function is bounded between $0 < h (m) \leq \eta$.

The Q statistic is essentially a likelihood ratio test between the symbolic entropy of the observed pattern and the entropy of the system under the null hypothesis of a random spatial sequence:

$Q(m)=2R(\eta-h(m))$

with $\eta = ln(k^m)$. The statistic is asymptotically $\chi^2$ distributed with degrees of freedom equal to the number of symbols minus one.

Value

An list of two object of the class htest. Each element of the list return the:

data.name	a character string giving the names of the data.
statistic	Value of the Q test
N	total number of observations.
R	total number of symbolized observations.
m	length m-surrounding.
r	degree of overlapping.
df	degree of freedom.
distr	type of distribution used to get the significance of the Q test.
type	type of symbols.

Control arguments

distance

character to select the type of distance. Default = "Euclidean" for Cartesian coordinates only: one of Euclidean, Hausdorff or Frechet; for geodetic coordinates, great circle distances are computed (see sf::st_distance())

dtmaxabs

Delete degenerate surrounding based on the absolute distance between observations.

dtmaxpc

A value between 0 and 1. Delete degenerate surrounding based on the distance. Delete m-surrounding when the maximum distance between observation is upper than k percentage of maximum distance between anywhere observation.

dtmaxknn

A integer value 'k'. Delete degenerate surrounding based on the near neighbourhood criteria. Delete m-surrounding is a element of the m-surrounding is not include in the set of k near neighbourhood of the first element

nsim

number of simulations for get the Monte Carlo distribution. Default = 999

seedinit

seed to select the initial element to star the algorithm to get compute the m-surroundings or to start the simulation

Standard-Permutation vs Equivalent-Combination Symbols

The symbolization protocol proposed by Ruiz et al. (2010) - call these Standard-Permutation Symbols — contains a large amount of topological information regarding the units of analysis, including proximity and direction. In this sense, the protocol is fairly general. On the other hand, it is easy to see that the combinatorial possibilities can very quickly become unmanageable. For a process with k = 3 outcomes and m = 5, the number of symbols becomes $3^5 = 243$; for k = 6 and m = 4 it is $6^4 = 1,296$. Depending on the number of observations N, the explosion in the number of symbols can very rapidly consume degrees of freedom for hypothesis testing, because as a rule of thumb it is recommended that the number of symbolized locations be at least five times the number of symbols used (e.g., $R \geq 5k^m$), and R will usually be a fraction of N.

As an alternative, we propose a symbolization protocol that sacrifices some amount of topological detail for conciseness. The alternative is based on the standard scheme; however, instead of retaining proximity and direction relationships, it maintains only the total number of occurrences of each outcome in an m-surrounding. We call these Equivalent-Combination Symbols. Because order in the sequence is not considered in this protocol, instead of a permutation with repetition, the number of symbols reflects a combination with repetition.

Selection of m-surrounding with Controlled Degree of Overlapping (r)

To select S locations for the analysis, coordinates are selected such that for any two coordinates $s_i$ , $s_j$ the number of overlapping nearest neighbours of $s_i$ and $s_j$ are at most r. The set S, which is a subset of all the observations N, is defined recursively as follows. First choose a location $s_0$ at random and fix an integer r with $0 \leq r < m$. The integer r is the degree of overlap, the maximum number of observations that contiguous m-surroundings are allowed to have in common.
Let $\{s_1^0, s_2^0,...,s_{m-1}^0 \}$ be the set of nearest neighbours to $s_0$, where the $s_i^0$ are ordered by distance to $s_0$, or angle in the case of ties. Let us call $s_1 = s_{m-r-1}^0$ and define $A_0 = \{s_0,s_0^1,...,s^0_{m-r-2} \}$ . Take the set of nearest neighbours to $s_1$, namely, $\{ s_1^1, s_2^1,...,s^1_{m-1} \}$ in the set of locations $S \setminus A_0$ and define $s_2=s^1_{m-r-1}$. Nor for i>1 we define $s_i = s^{i-1}_{m-r-1}$ where $s^{i-1}_{m-r-1}$ is in the ser of nearest neighbors to $s_{i-1}$, $\{ s_1^{i-1},s_2^{i-1},...,s_{m-1}^{i-1} \}$ ,of the set $S \setminus \{ \cup_{j=0}^{i-1} A_j \}$. Continue this process while there are locations to symbolize.

Selection of m-surroundings for bootstrap distribution

The bootstrapped-based testing can provide an advantage since overlapping between m-surroundings is not a consideration, and the full sample can be used.

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Ruiz M, López FA, A Páez. (2010). Testing for spatial association of qualitative data using symbolic dynamics. Journal of Geographical Systems. 12 (3) 281-309

• López, FA, and A Páez. (2012). Distribution-free inference for Q(m) based on permutational bootstrapping: an application to the spatial co-location pattern of firms in Madrid Estadística Española, 177, 135-156.

Examples

# Case 1: With coordinates
N <- 200
cx <- runif(N)
cy <- runif(N)
coor <- cbind(cx,cy)
p <- c(1/6,3/6,2/6)
rho = 0.3
listw <- spdep::nb2listw(spdep::knn2nb(spdep::knearneigh(cbind(cx,cy), k = 4)))
fx <- dgp.spq(list = listw, p = p, rho = rho)
q.test <- Q.test(fx = fx, coor = coor, m = 3, r = 1)
summary(q.test)
plot(q.test)
print(q.test)

q.test.mc <- Q.test(fx = fx, coor = coor, m = 3, distr = "mc", control = list(nsim = 999))
summary(q.test.mc)
plot(q.test.mc)
print(q.test.mc)


# Case 2: With a sf object
data("FastFood.sf")
f1 <- ~ Type
q.test <- Q.test(formula = f1, data = FastFood.sf, m = c(3, 4),
r = c(1, 2, 3), control = list(distance ="Euclidean"))
summary(q.test)
plot(q.test)
print(q.test)

# Case 3: With a sf object with isolated areas
data("provinces_spain")
sf::sf_use_s2(FALSE)
provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman")
provinces_spain$Older <- cut(provinces_spain$Older, breaks = c(-Inf,19,22.5,Inf))
levels(provinces_spain$Older) = c("low","middle","high")
f1 <- ~ Older + Male2Female
q.test <- Q.test(formula = f1,
data = provinces_spain, m = 3, r = 1, control = list(seedinit = 1111))
summary(q.test)
print(q.test)
plot(q.test)
q.test.mc <- Q.test(formula = f1, data = provinces_spain, m = 4, r = 3, distr = "mc",
control = list(seedinit = 1111))
summary(q.test.mc)
print(q.test.mc)
plot(q.test.mc)

# Case 4: Examples with multipolygons
library(sf)
fname <- system.file("shape/nc.shp", package="sf")
nc <- sf::st_read(fname)
qb79 <- quantile(nc$BIR79)
nc$QBIR79 <- (nc$BIR79 > qb79[2]) + (nc$BIR79 > qb79[3]) +
(nc$BIR79 >= qb79[4]) + 1
nc$QBIR79 <- as.factor(nc$QBIR79)
plot(nc["QBIR79"], pal = c("#FFFEDE","#FFDFA2", "#FFA93F", "#D5610D"),
     main = "BIR79 (Quartiles)")
sid79 <- quantile(nc$SID79)
nc$QSID79 <- (nc$SID79 > sid79[2]) + (nc$SID79 > sid79[3]) +
(nc$SID79 >= sid79[4]) + 1
nc$QSID79 <- as.factor(nc$QSID79)
plot(nc["QSID79"], pal = c("#FFFEDE","#FFDFA2", "#FFA93F", "#D5610D"),
     main = "SID79 (Quartiles)")
f1 <- ~ QSID79 + QBIR79
lq1nc <- Q.test(formula = f1, data = nc, m = 5, r = 2,
control = list(seedinit = 1111, dtmaxpc = 0.5, distance = "Euclidean") )
print(lq1nc)

lq2nc <- Q.test(formula = f1, data = nc, m = 5, r = 2,
control = list(dtmaxpc = 0.2) )
print(lq2nc)

lq3nc <- Q.test(formula = f1, data = nc, m = 5, r = 2,
control = list(dtmaxknn = 5) )
print(lq3nc)

# Case 5: Examples with points and matrix of variables
fx <- matrix(c(nc$QBIR79, nc$QSID79), ncol = 2, byrow = TRUE)
mctr <- suppressWarnings(sf::st_centroid(st_geometry(nc)))
mcoor <- st_coordinates(mctr)[,c("X","Y")]
q.test <- Q.test(fx = fx, coor = mcoor, m = 5, r = 2,
                 control = list(seedinit = 1111, dtmaxpc = 0.5))
print(q.test)
plot(q.test)

---

Compute the scan test

Description

This function compute the spatial scan test for Bernoulli and Multinomial categorical spatial process, and detect spatial clusters

Usage

scan.test(formula = NULL, data = NULL, fx = NULL, coor = NULL, case = NULL,
nv = NULL, nsim = NULL, distr = NULL, windows = "circular", listw = NULL,
alternative = "High", minsize = 1, 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).
coor	(optional) coordinates of observations.
case	Only for Bernoulli distribution. A element of factor, there are cases and non-cases for testing for cases versus non-cases
nv	Maximum windows size, default nv = N/2. The algorithm scan for clusters of geographic size between 1 and the upper limit (nv) defined by the user.
nsim	Number of permutations.
distr	distribution of the spatial process: "bernoulli" for two levels or "multinomial" for three or more levels.
windows	a string to select the type of cluster "circular" (default) of "elliptic".
listw	only for flexible windows. A neighbours list (an object of the class listw, nb or knn frop spdep) or an adjacency matrix.
alternative	Only for Bernoulli spatial process. A character string specifying the type of cluster, must be one of "High" (default), "Both" or "Low".
minsize	Minimum number of observations inside of Most Likely Cluster and secondary clusters.
control	List of additional control arguments.

Details

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

• Option 1: A factor (fx) and coordinates (coor).

• Option 2: A sf object (data) and the formula to specify the factor. The function consider the coordinates of the centroids of the elements of th sf object.

The spatial scan statistics are widely used in epidemiology, criminology or ecology. Their purpose is to analyse the spatial distribution of points or geographical regions by testing the hypothesis of spatial randomness distribution on the basis of different distributions (e.g. Bernoulli, Poisson or Normal distributions). The scan.test function obtain the scan statistic for two relevant distributions related with categorical variables: the Bernoulli and Multinomial distribution.
The spatial scan statistic is based on the likelihood ratio test statistic and is formulated as follows:

$\Delta = { \max_{z \in Z,H_A} L(\theta|z) \over \max_{z \in Z,H_0} L(\theta|z)}$

where Z represents the collection of scanning windows constructed on the study region, $H_A$ is an alternative hypothesis, $H_0$ is a null hypothesis, and $L(\theta|z)$ is the likelihood function with parameter $\theta$ given window Z
. The null hypothesis says that there is no spatial clustering on the study region, and the alternative hypothesis is that there is a certain area with high (or low) rates of outcome variables. The null and alternative hypotheses and the likelihood function may be expressed in different ways depending on the probability model under consideration.
To test independence in a spatial process, under the null, the type of windows is irrelevant but under the alternative the elliptic windows can to identify with more precision the cluster.

For big data sets (N >>) the windows = "elliptic" can be so slowly

Bernoulli version

When we have dichotomous outcome variables, such as cases and noncases of certain diseases, the Bernoulli model is used. The null hypothesis is written as

$H_0 : p = q \ \ for \ \ all \ \ Z$

and the alternative hypothesis is

$H_A : p \neq q \ \ for \ \ some \ \ Z$

where p and q are the outcome probabilities (e.g., the probability of being a case) inside and outside scanning window Z, respectively. Given window Z, the test statistic is:

where cz and nz are the numbers of cases and observations (cases and noncases) within z, respectively, and C and N are the total numbers of cases and observations in the whole study region, respectively.

$\Delta =$

Multinomial version of the scan test

The multinomial version of the spatial scan statistic is useful to investigate clustering when a discrete spatial variable can take one and only one of k possible outcomes that lack intrinsic order information. If the region defined by the moving window is denoted by Z, the null hypothesis for the statistic can be stated as follows:

$H_0: p_1 = q_1;p_2 = q_2;...;p_k = q_k$

where $p_j$ is the probability of being of event type j inside the window Z, and $q_j$ is the probability of being of event type j outside the window. The alternative hypothesis is that for at least one type event the probability of being of that type is different inside and outside of the window.

The statistic is built as a likelihood ratio, and takes the following form after transformation using the natural logarithm:

$\Delta = \max_Z \{\sum_j \{ S_j^Z log({ S_j^Z \over S^Z }) + (S_j-S_j^Z) log({ {S_j-S_j^Z} \over {S-S^Z} })\}\}-\sum_j S_j log({ S_j \over S })$

where S is the total number of events in the study area and $S_j$ is the total number of events of type j. The superscript Z denotes the same but for the sub-region defined by the moving window.
The theoretical distribution of the statistic under the null hypothesis is not known, and therefore significance is evaluated numerically by simulating neutral landscapes (obtained using a random spatial process) and contrasting the empirically calculated statistic against the frequency of values obtained from the neutral landscapes. The results of the likelihood ratio serve to identify the most likely cluster, which is followed by secondary clusters by the expedient of sorting them according to the magnitude of the test. As usual, significance is assigned by the analyst, and the cutoff value for significance reflects the confidence of the analyst, or tolerance for error.
When implementing the statistic, the analyst must decide the shape of the window and the maximum number of cases that any given window can cover. Currently, analysis can be done using circular or elliptical windows.
Elliptical windows are more time consuming to evaluate but provide greater flexibility to contrast the distribution of events inside and outside the window, and are our selected shape in the analyses to follow. Furthermore, it is recommended that the maximum number of cases entering any given window does not exceed 50% of all available cases.

Value

A object of the htest and scantest class

method	The type of test applied ().
fx	Factor included as input to get the scan test.
MLC	Observations included into the Most Likelihood Cluster (MLC).
statistic	Value of the scan test (maximum Log-likelihood ratio).
N	Total number of observations.
nn	Windows used to get the cluster.
nv	Maximum number of observations into the cluster.
data.name	A character string giving the name of the factor.
coor	coordinates.
alternative	Only for bernoulli spatial process. A character string describing the alternative hypothesis select by the user.
p.value	p-value of the scan test.
cases.expect	Expected cases into the MLC.
cases.observ	Observed cases into the MLC.
nsim	Number of permutations.
scan.mc	a (nsim x 1) vector with the loglik values under bootstrap permutation.
secondary.clusters	a list with the observations included into the secondary clusters.
loglik.second	a vector with the value of the secondary scan tests (maximum Log-likelihood ratio).
p.value.secondary	a vector with the p-value of the secondary scan tests.
Alternative.MLC	A vector with the observations included in another cluster with the same loglik than MLC.

Control arguments

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

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Kulldorff M, Nagarwalla N. (1995). Spatial disease clusters: Detection and Inference. Statistics in Medicine. 14:799-810

• Jung I, Kulldorff M, Richard OJ (2010). A spatial scan statistic for multinomial data. Statistics in Medicine. 29(18), 1910-1918

• Páez, A., López-Hernández, F.A., Ortega-García, J.A., Ruiz, M. (2016). Clustering and co-occurrence of cancer types: A comparison of techniques with an application to pediatric cancer in Murcia, Spain. Spatial Analysis in Health Geography, 69-90.

• Tango T., Takahashi K. (2005). A flexibly shaped spatial scan statistic for detecting clusters, International Journal of Health Geographics 4:11.

See Also

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

Examples

# Case 1: Scan test bernoulli
data(provinces_spain)
sf::sf_use_s2(FALSE)
provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman")
formula <- ~ Male2Female
scan <- scan.test(formula = formula, data = provinces_spain, case="men",
nsim = 99, distr = "bernoulli")
print(scan)
summary(scan)
plot(scan, sf = provinces_spain)

## With maximum number of neighborhood
scan <- scan.test(formula = formula, data = provinces_spain, case = "woman",
nsim = 99, distr = "bernoulli")
print(scan)
plot(scan, sf = provinces_spain)


## With elliptic windows
scan <- scan.test(formula = formula, data = provinces_spain, case = "men", nv = 25,
nsim = 99, distr = "bernoulli", windows ="elliptic")
print(scan)
scan <- scan.test(formula = formula, data = provinces_spain, case = "men", nv = 15,
nsim = 99, distr = "bernoulli", windows ="elliptic", alternative = "Low")
print(scan)
plot(scan, sf = provinces_spain)

# Case 2: scan test multinomial
data(provinces_spain)
provinces_spain$Older <- cut(provinces_spain$Older, breaks = c(-Inf,19,22.5,Inf))
levels(provinces_spain$Older) = c("low","middle","high")
formula <- ~ Older
scan <- scan.test(formula = formula, data = provinces_spain, nsim = 99, distr = "multinomial")
print(scan)
plot(scan, sf = provinces_spain)

# Case 3: scan test multinomial
data(FastFood.sf)
sf::sf_use_s2(FALSE)
formula <- ~ Type
scan <- scan.test(formula = formula, data = FastFood.sf, nsim = 99,
distr = "multinomial", windows="elliptic", nv = 50)
print(scan)
summary(scan)
plot(scan, sf = FastFood.sf)

# Case 4: DGP two categories
N <- 150
cx <- runif(N)
cy <- runif(N)
listw <- spdep::knearneigh(cbind(cx,cy), k = 10)
p <- c(1/2,1/2)
rho <- 0.5
fx <- dgp.spq(p = p, listw = listw, rho = rho)
scan <- scan.test(fx = fx, nsim = 99, case = "A", nv = 50, coor = cbind(cx,cy),
distr = "bernoulli",windows="circular")
print(scan)
plot(scan)

# Case 5: DGP three categories
N <- 200
cx <- runif(N)
cy <- runif(N)
listw <- spdep::knearneigh(cbind(cx,cy), k = 10)
p <- c(1/3,1/3,1/3)
rho <- 0.5
fx <- dgp.spq(p = p, listw = listw, rho = rho)
scan <- scan.test(fx = fx, nsim = 19, coor = cbind(cx,cy), nv = 30,
distr = "multinomial", windows = "elliptic")
print(scan)
plot(scan)

# Case 6: Flexible windows
data(provinces_spain)
sf::sf_use_s2(FALSE)
provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman")
formula <- ~ Male2Female
listw <- spdep::poly2nb(provinces_spain, queen = FALSE)
scan <- scan.test(formula = formula, data = provinces_spain, case="men", listw = listw, nv = 6,
                  nsim = 99, distr = "bernoulli", windows = "flexible")
print(scan)
summary(scan)
plot(scan, sf = provinces_spain)

---

Compute the similarity test.

Description

This function compute the nonparametric test for spatial independence using symbolic analysis for categorical/qualitative spatial process.

Usage

similarity.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 listw object
alternative	a character string specifying the type of cluster, must be one of "High" (default), "Both" or "Low".
distr	A string. Distribution of the test "asymptotic" (default) or "bootstrap"
nsim	Number of permutations.
control	List of additional control arguments.

Details

This testing approach for spatial independence that extends some of the properties of the join count statistic. The premise of the tests is similar to the join count statistic in that they use the concept of similarity between neighbouring spatial entities (Dacey 1968; Cliff and Ord 1973, 1981). However, it differs by taking into consideration the number of joins belonging locally to each spatial unit, rather than the total number of joins in the entire spatial system. The approach proposed here is applicable to spatial and network contiguity structures, and the number of neighbors belonging to observations need not be constant.

We define an equivalence relation $\sim$ in the set of locations S. An equivalence relation satisfies the following properties:

Reflexive: $s_i \sim s_i$ for all $s_i \in S$,
Symmetric: If $s_i \sim s_j$, then $s_j \sim s_i$ for all $s_i,\ s_j \in S$ and
Transitive: If $s_i \sim s_j$ and $s_j \sim s_k$, then $s_i \sim s_k$ for all $s_i, \ s_j, \ s_k \in S$

We call the relation $\sim$ a similarity relation. Then, the null hypothesis that we are interested in is

$H_0: \{X_s\}_{s \in S} \ \ is\ \ iid$

Assume that, under the null hypothesis, $P(s_i \sim s_{ji}) = p_i$ for all $s_{ji} \in N_{s_i}$.
Define

$I_{ij}=1 \ \ if \ \ s_i \sim s_{ji} \ \ ; 0 \ \ otherwise$

Then, for a fixed degree d and for all location si with degree d, the variable d

$\Lambda_{(d,i)}=\sum_{j=1}^d I_{ij}$

gives the number of nearest neighbours to si that are similar to si. Therefore, under the null hypothesis, $H_0$, $\Lambda(d,i)$ follows a binomial distribution $B(d, p_i)$.

Value

A object of the htest

data.name	a character string giving the names of the data.
statistic	Value of the similarity test
N	total number of observations.
Zmlc	Elements in the Most Likelihood Cluster.
alternative	a character string describing the alternative hypothesis.
p.value	p-value of the similarity test
similiarity.mc	values of the similarity test in each permutation.

Control arguments

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

Author(s)

Fernando López	[email protected]
Román Mínguez	[email protected]
Antonio Páez	[email protected]
Manuel Ruiz	[email protected]

References

• Farber, S., Marin, M. R., & Paez, A. (2015). Testing for spatial independence using similarity relations. Geographical Analysis. 47(2), 97-120.

See Also

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

Examples

# Case 1:
N <- 100
cx <- runif(N)
cy <- runif(N)
listw <- spdep::knearneigh(cbind(cx,cy), k = 3)
p <- c(1/4,1/4,1/4,1/4)
rho <- 0.5
fx <- dgp.spq(p = p, listw = listw, rho = rho)
W <- (spdep::nb2mat(spdep::knn2nb(listw)) >0)*1
similarity <- similarity.test(fx = fx, data = FastFood.sf, listw = listw)
print(similarity)

# Case 2: test with formula, a sf object (points) and knn
data("FastFood.sf")
coor <- sf::st_coordinates(sf::st_centroid(FastFood.sf))
listw <- spdep::knearneigh(coor, k = 4)
formula <- ~ Type
similarity <- similarity.test(formula = formula, data = FastFood.sf, listw = listw)
print(similarity)

# Case 3:
data(provinces_spain)
listw <- spdep::poly2nb(as(provinces_spain,"Spatial"), queen = FALSE)
provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman")
formula <- ~ Male2Female
similarity <- similarity.test(formula = formula, data = provinces_spain, listw = listw)
print(similarity)

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