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

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.

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.

Details

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

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.

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.

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:
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)
#> [[1]]
#> 
#> 	Q-Map test of Equivalence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 9.1
#> 
#> data:  QY1 and QY2
#> QE = 270.46, df = 9, p-value < 2.2e-16
#> alternative hypothesis: two.sided
#> 
#> 
#> [[2]]
#> 
#> 	Q-Map test of Independence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 9.1
#> 
#> data:  QY1 and QY2
#> QI = 48.865, df = 81, p-value = 0.9982
#> alternative hypothesis: two.sided
#> 
#> 
#> attr(,"class")
#> [1] "qmap" "list"
plot(qmap)

plot(qmap, ci=.6)

plot(qmap[[1]]$mh)

summary(qmap[[1]]$mh)
#> 
#> Characteristics of m-surrounding:
#> 
#> Number of m-surrounding (R): 91
#> Length of m-surrounding (m): 3
#> Number no-symbolized observations: 11
#> 
#> List of no-symbolized observations:
#> 7 13 18 23 65 95 115 123 131 139 142
#> 
#> List of the degree overlaping:
#>     There are 2 m-surrounding that have intersection with 0 m-surrounding
#>     There are 10 m-surrounding that have intersection with 1 m-surrounding
#>     There are 79 m-surrounding that have intersection with 2 m-surrounding
#> Mean degree of overlaping: 1.8462

control <- list(dtmaxknn = 20)
qmap <- Q.map.test(formula = formula, data = dt, coor = x, m = m, r = r,
                   type ="permutations", control = control)
#> Warning: The ratio between the number of symbolized observations and the number of symbols is lower than 5.
print(qmap)
#> [[1]]
#> 
#> 	Q-Map test of Equivalence for qualitative data.
#> 	
#> 	Symbols type: permutations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 3.44
#> 
#> data:  QY1 and QY2
#> QE = 291.99, df = 26, p-value < 2.2e-16
#> alternative hypothesis: two.sided
#> 
#> 
#> [[2]]
#> 
#> 	Q-Map test of Independence for qualitative data.
#> 	
#> 	Symbols type: permutations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 3.44
#> 
#> data:  QY1 and QY2
#> QI = 212.1, df = 676, p-value = 1
#> alternative hypothesis: two.sided
#> 
#> 
#> attr(,"class")
#> [1] "qmap" "list"
plot(qmap)

plot(qmap[[1]]$mh)

qmap <- Q.map.test(formula = formula, data = dt, coor = x, m = m, r = r,
                   type ="combinations")
print(qmap)
#> [[1]]
#> 
#> 	Q-Map test of Equivalence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 9.9
#> 
#> data:  QY1 and QY2
#> QE = 295.79, df = 9, p-value < 2.2e-16
#> alternative hypothesis: two.sided
#> 
#> 
#> [[2]]
#> 
#> 	Q-Map test of Independence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 9.9
#> 
#> data:  QY1 and QY2
#> QI = 49.016, df = 81, p-value = 0.9981
#> alternative hypothesis: two.sided
#> 
#> 
#> attr(,"class")
#> [1] "qmap" "list"
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)
#> [[1]]
#> 
#> 	Q-Map test of Equivalence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 9.1
#> 
#> data:  QY1 and QY2
#> QE = 270.46, df = 9, p-value < 2.2e-16
#> alternative hypothesis: two.sided
#> 
#> 
#> [[2]]
#> 
#> 	Q-Map test of Independence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 9.1
#> 
#> data:  QY1 and QY2
#> QI = 48.865, df = 81, p-value = 0.9982
#> alternative hypothesis: two.sided
#> 
#> 
#> attr(,"class")
#> [1] "qmap" "list"
plot(qmap)


# Case 2:
data(provinces_spain)
sf::sf_use_s2(FALSE)
#> Spherical geometry (s2) switched off
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)
#> [[1]]
#> 
#> 	Q-Map test of Equivalence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 6
#> 
#> data:  Coast and Male2Female
#> QE = 71.83, df = 3, p-value = 1.731e-15
#> alternative hypothesis: two.sided
#> 
#> 
#> [[2]]
#> 
#> 	Q-Map test of Independence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 6
#> 
#> data:  Coast and Male2Female
#> QI = 7.5057, df = 9, p-value = 0.5846
#> alternative hypothesis: two.sided
#> 
#> 
#> attr(,"class")
#> [1] "qmap" "list"
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)
#> [[1]]
#> 
#> 	Q-Map test of Equivalence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 5
#> 
#> data:  Coast and Male2Female
#> QE = 58.615, df = 3, p-value = 1.162e-12
#> alternative hypothesis: two.sided
#> 
#> 
#> [[2]]
#> 
#> 	Q-Map test of Independence for qualitative data.
#> 	
#> 	Symbols type: combinations
#> 	
#> 	Ratio Symbolized observations/Num symbols = 5
#> 
#> data:  Coast and Male2Female
#> QI = 7.1454, df = 9, p-value = 0.622
#> alternative hypothesis: two.sided
#> 
#> 
#> attr(,"class")
#> [1] "qmap" "list"
plot(qmap[[1]]$mh)

summary(qmap[[1]]$mh)
#> 
#> Characteristics of m-surrounding:
#> 
#> Number of m-surrounding (R): 20
#> Length of m-surrounding (m): 3
#> Number no-symbolized observations: 8
#> 
#> List of no-symbolized observations:
#> 7 13 19 20 34 35 37 48
#> 
#> List of the degree overlaping:
#>     There are 4 m-surrounding that have intersection with 1 m-surrounding
#>     There are 16 m-surrounding that have intersection with 2 m-surrounding
#> Mean degree of overlaping: 1.8