mc_spGini: Monte Carlo simulation for the significance of the Spatial...
In lctools: Local Correlation, Spatial Inequalities, Geographically Weighted Regression and Other Tools

Description Usage Arguments Details Value Note Author(s) References Examples

This function provides one approach for inference on the spatial Gini inequality measure. This is a small Monte Carlo simulation according to which: a) the data are spatially reallocated in a random way; b) the share of overall inequality that is associated with non-neighbour pairs of locations - SG (Eq. 5 in Rey & Smith, 2013) - is calculated for the original and simulated spatial data sets; c) a pseudo p-value is calculated as p=(1+C)/(1+M) where C is the number of the permutation data sets that generated SG values that were as extreme as the observed SG value for the original data (Eq. 6 in Rey & Smith, 2013). If p<=0.05 it can be argued that the component of the Gini for non-neighbour inequality is statistically significant. For this approach, a minimum of 19 simulations is required.

1	mc.spGini(Nsim=99,Bandwidth,x,Coord.X,Coord.Y,WType='Binary')

`Nsim`	a positive integer that defines the number of the simulation's iterations
`Bandwidth`	a positive integer that defines the number of nearest neighbours for the calculation of the weights
`x`	a numeric vector of a variable
`Coord.X`	a numeric vector giving the X coordinates of the observations (data points or geometric centroids)
`Coord.Y`	a numeric vector giving the Y coordinates of the observations (data points or geometric centroids)
`WType`	string giving the weighting scheme used to compute the weights matrix. Options are: "Binary", "Bi-square", "RSBi-square". Default is "Binary". Binary: weight = 1 for distances less than or equal to the distance of the furthest neighbour (H), 0 otherwise; Bi-square: weight = (1-(ndist/H)^2)^2 for distances less than or equal to H, 0 otherwise; RSBi-square: weight = Bi-square weights / sum (Bi-square weights) for each row in the weights matrix

For 0.05 level of significance in social sciences, a minimum number of 19 simulations (Nsim>=19) is required. We recommend at least 99 and at best 999 iterations

Returns a list of the simulated values, the observed Gini and its spatial decomposition, the pseudo p-value of significance

`SIM`	a dataframe with simulated values: SIM.ID is the simulation ID, SIM.gwGini is the simulated Gini of neighbours, SIM.nsGini is the simulated Gini of non-neighbours, SIM.SG is the simulated share of the overall Gini that is associated with non-neighbour pairs of locations, SIM.Extr = 1 if the simulated SG is greater than or equal to the observed SG
`spGini.Observed`	Observed Gini (Gini) and its spatial components (gwGini, nsGini)
`pseudo.p`	pseudo p-value: if this is lower than or equal to 0.05 it can be argued that the component of the Gini for non-neighbour inequality is statistically significant.

Acknowledgement: I would like to thank LI Zai-jun, PhD student at Nanjing Normal University, China for encouraging me to develop this function and for testing this package.

Stamatis Kalogirou <stamatis@lctools.science>

Rey, S.J., Smith, R. J. (2013) A spatial decomposition of the Gini coefficient, Letters in Spatial and Resource Sciences, 6 (2), pp. 55-70.

Kalogirou, S. (2015) Spatial Analysis: Methodology and Applications with R. [ebook] Athens: Hellenic Academic Libraries Link. ISBN: 978-960-603-285-1 (in Greek). https://repository.kallipos.gr/handle/11419/5029?locale=en

data(GR.Municipalities)
Nsim=19
Bd1<-4
x1<-GR.Municipalities@data$Income01[1:45]
WType1<-'Binary'

SIM20<-mc.spGini(Nsim,Bd1,x1,GR.Municipalities@data$X[1:45], GR.Municipalities@data$Y[1:45],WType1)
SIM20

hist(SIM20$SIM$SIM.nsGini,col = "lightblue", main = "Observed and simulated nsGini",
xlab = "Simulated nsGini", ylab = "Frequency",xlim = c(min(SIM20$SIM$SIM.nsGini),
SIM20$spGini.Observed[[3]]))
abline(v=SIM20$spGini.Observed[[3]], col = 'red')