R and Matlab codes for the paper “A randomlocational M-estimation problem based on the

L2-Wasserstein distance” by A. Daouia and I. VanKeilegom (2015)

Thibault LAURENT21st of April 2015

This document was generated directly from RStudio using Markdown tool. It presents the R and Matlabcodes used to obtain the figures included in the paper “A random locational M-estimation problem based onthe L2-Wasserstein distance” written by A. Daouia and I. Van Keilegom (2015). The .pdf of this page isavailable here.

Before starting

Install the necessary packages:

install.packages(c("rgdal", "gpclib", "maptools", "rgeos","OpenStreetMap", "GISTools"))

Load the packages :

require("rgdal")require("maptools")require("rgeos")require("OpenStreetMap")require("GISTools")

1. Representation of the map of Sahel region - North Africa

Import the world country boundaries into R :

world <- readOGR(dsn="world", layer="Pays_WGS84")

Perhaps you will need to run the following code the first time :

require("gpclib")gpclibPermit()

We consider some border cities located in the Algerian and Moroccan territories and characterized by thefollowing Longitute/Latitude coordinates (given in decimal degrees):

1

provinces<-SpatialPoints(rbind(c(-2.938114,35.292437), c(-6.855839,33.985752),c(-5.353647,36.142444), c(-5.339725,35.885957), c(-1.914259,34.685276),c(-2.211851,35.085182),c(-1.232229,32.085182), c(-2.103646,34.951092), c(-1.735685,34.852832), c(-5.834646,35.757976),c(-2.163441,34.311307), c(-2.302420,34.972376), c(-1.329303,34.915277), c(-1.967375,32.534366),c(-1.998600,33.048943), c(-0.583262,32.750140), c(-2.032403,34.000785), c(-1.329063,34.624857),c(-0.331307,33.249668), c(-0.290108,33.549825), c(-1.259651,34.223006)),proj4string=CRS("+proj=longlat +datum=WGS84 +no_defs +ellps=WGS84 +towgs84=0,0,0"))

The names of these cities are:

nom.prov <- c("Melilla", "Rabat", "Gibraltar", "Ceuta", "Oujda", "Saidia","Figuig","Ahfir","Maghnia","Tangier", "Jerada", "Berkane", "Tlemcen", "Bouarfa", "Tendrara","Ain Sefra", "Ain-Beni-Mathar", "Sebdou", "Naama", "Mecheria", "El Aricha")

We only keep the countries that we want to represent in the Sahel region:

carte<-world[world@data$NOM%in%c("Algeria","Morocco","Western Sahara","Mauritania","Mali","Niger","Tunisia","Libya", "Chad","Italy","Spain","France","Portugal","Senegal","Gambia, The","Guinea","Guinea-Bissau","Burkina Faso","Nigeria","Malta", "Sudan", "Somalia","Uganda","Cameroon", "Egypt","Greece", "Ivory Coast","Ghana", "Turkey","Albania", "Montenegro","Macedonia", "Bulgaria","Croatia","Bosnia and Herzegovina", "Andorra", "Benin"), ]

we precise the names of the selected countries:

nom<-carte@data$NOMrow.names(carte)<-as.character(nom)nom[nom=="Western Sahara"]="Morocco"carte<-unionSpatialPolygons(carte, nom)

We change the Coordinate Reference System of the countries and keep the one given by OpenStreetMap:

proj.OSM<-"+proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0+k=1.0 +units=m +nadgrids=@null +no_defs"carte<-spTransform(carte, CRS(proj.OSM))provinces.OSM<-spTransform(provinces, CRS(proj.OSM))

We import the relief map from OpenStreetMap:

map = openmap(c(lat= 35.60583, lon= -3.881389),c(lat= 31.96583, lon= 0.4601385),

minNumTiles=9,type="nps")

We are now in position to represent the map of interest:

op<-par(oma=c(0,0,0,0), omi=c(0,0,0,0), mar=c(0,0,0,0), mai=c(0,0,0,0))plot(carte, xlim=c(-1900000,2200000),ylim=c(3200000,3600000))masker = poly.outer(matrix(par()$usr,2,2,byrow=TRUE), carte,extend=10000000);add.masking(masker,"lightblue")plot(carte, col="lightyellow",lty=1,add=TRUE)text(coordinates(carte)[,1], coordinates(carte)[,2], row.names(carte), font=3, cex=0.9)

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plot(map, add=TRUE)xmin=map[[1]][[1]]$bbox$p1[1];xmax=map[[1]][[1]]$bbox$p2[1]ymin=map[[1]][[1]]$bbox$p2[2];ymax=map[[1]][[1]]$bbox$p1[2]polygon(c(xmin,xmax,xmax,xmin,xmin),c(ymin,ymin,ymax,ymax,ymin), lty=2)plot(carte, add=TRUE, lty=2)plot(provinces.OSM<-spTransform(provinces, CRS(proj.OSM))[c(1,2,3,4,7,6)], col="red", pch=15, add=TRUE, cex=0.7)text(coordinates(provinces.OSM<-spTransform(provinces, CRS(proj.OSM))[c(1,2,3,4,7,6),]), nom.prov[c(1,2,3,4,7,6)], cex=0.7, pos=c(3, 2, 3, 1, 4, 4))

Algeria

Greece

Libya

Mali

Malta

Mauritania

Morocco

Niger

PortugalSpain

Tunisia

Melilla

Rabat

Gibraltar

Ceuta

Figuig

Saidia

par(op)

2. Representation of the facilities (monitoring stations) and the clients (smug-glers, traffickers and illegal immigrants)

a. Preparation of the borderline

First, we prepare the borderland between Morocco and Algeria:

maghreb<-world[world@data$NOM%in%c("Algeria","Morocco","Western Sahara"), ]maghreb<-unionSpatialPolygons(maghreb, c("Algerie", "Maroc","Maroc"))

We provide the locations of 6 existing monitoring stations:

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villes<-SpatialPoints(rbind(c(-1.022099,32.506542),c(-1.483524,33.051516), c(-1.604374,33.611439), c(-1.659306,34.104095),c( -1.771780,34.742707), c(-1.703251,34.494356), c(-2.095231,34.957809)),proj4string=CRS("+proj=longlat +datum=WGS84 +no_defs +ellps=WGS84 +towgs84=0,0,0"))

row.names(villes)<-c("Figuig3", "Figuig2", "Figuig1", "Jerada2", "Oujda","Jerada", "Ahfir")

We are going to compute some distances between the points which delimit the frontier. For this reason, wemust convert the Spatial data into a CRS given in meters:

proj.maroc<-"+proj=lcc +lat_1=33.3 +lat_0=33.3 +lon_0=-5.4 +k_0=0.999625769+x_0=500000 +y_0=300000 +a=6378249.2 +b=6356515 +towgs84=31,146,47,0,0,0,0+units=m +no_defs"

maghreb<-spTransform(maghreb,CRS(proj.maroc))villes<-spTransform(villes,CRS(proj.maroc))

We define the borderline between Algeria and Morocco:

algerie<-maghreb[1,]maroc<-maghreb[2,]frontiere<-gIntersection(maroc, algerie)

This frontier is represented by the following number of segments:

length(frontiere@lines[[1]]@Lines)

## [1] 182

Now, we have to integrate the monitoring center sites into the borderline. For this, we first identify theclosest border segments to the stations:

ind.villes<-apply(sapply(frontiere@lines[[1]]@Lines,function(x) gDistance(SpatialLines(list(Lines(x, ID="1"))), villes, byid=TRUE),simplify = "matrix"), 1, which.min)

Then, we have to create a new Spatial object which defines the new borderline including the monitoringstations:

test<-vector("list", length=length(frontiere@lines[[1]]@Lines)+length(ind.villes))ind.old=1ind.new=1ind.villes.bis=NULLwhile(ind.old<=(length(frontiere@lines[[1]]@Lines))){if(!ind.old%in%ind.villes)

{test[[ind.new]]<-frontiere@lines[[1]]@Lines[[ind.old]]ind.new = ind.new+1}else{test[[ind.new]]<-Line(cbind(c(frontiere@lines[[1]]@Lines[[ind.old]]@coords[1,1],

coordinates(villes)[which(ind.villes==ind.old),1]),c(frontiere@lines[[1]]@Lines[[ind.old]]@coords[1,2],

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coordinates(villes)[which(ind.villes==ind.old),2])))

ind.villes.bis=c(ind.villes.bis,ind.new)

test[[ind.new+1]]<-Line(cbind(c(coordinates(villes)[which(ind.villes==ind.old),1],frontiere@lines[[1]]@Lines[[ind.old]]@coords[2,1]),c(coordinates(villes)[which(ind.villes==ind.old),2],

frontiere@lines[[1]]@Lines[[ind.old]]@coords[2,2])))ind.new=ind.new+2}ind.old = ind.old+1

}

Finally, we define the resulting frontier:

frontiere<-SpatialLines(list(Lines(test, ID="1")))proj4string(frontiere) <- CRS(proj.maroc)

b. Computing the cumulative distance of the data points lying on the frontier

We shall need compute the cumulative distance of the data points Xi:

coord.front<-t(sapply(frontiere@lines[[1]]@Lines,function(x) x@coords))dist.front<-sapply(frontiere@lines[[1]]@Lines,function(x) dist(x@coords))don<-data.frame(coord.front, dist.front)names(don)<-c("xA","xB","yA","yB","dist")

We sort the data points Xi on the borderline in order to facilitate the computation the cumulative distances:

don.int<-don[1:3,]don<-rbind(don[4:189,],don.int)row.names(don)<-1:189

for(k in 1:length(villes)){ind.k <- which(don$xB%in%coordinates(villes)[k,1])row.names(don)[ind.k]<-row.names(villes)[k]

}

Now, we want to restrict ourselves to the part of the borderline which stretches some 450 km starting fromthe Mediterranean sea:

don<-don[cumsum(don$dist)<=450000,]

We compute now the cumulative distances of the Xi’s starting from the coastal city of Saidia until the city ofFiguig.

don$dist.cum.A<-cumsum(c(0,don$dist[-length(don$dist)]))/sum(don$dist)don$dist.cum.B<-cumsum(don$dist)/sum(don$dist)

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c. Preparation of the data

Historical data before the placement of the new station When converting the original borderlineinto a straight line with lower and upper endpoints a and b, representing respectively the coastal city of Saidiaand the eastern city of Figuig, the resulting client’s locations Xi and facility’s locations yj , along with theircorresponding capacities zj and allocation zones, are available in the file here that can be imported as follows:

vec.exist <- as.numeric(read.table("data/table_exist_loc.txt", dec=".",sep=","))

The endpoints of the transformed straight line:

a <- vec.exist[1]b <- vec.exist[2]

The locations yj and capacities zj of the monitoring stations:

postes <- vec.exist[3:8]taille.postes <- vec.exist[9:14]

The location of incidents affected by guardhouse:

incident.p1 <- vec.exist[15:214]; (n.p1 <- length(incident.p1))

## [1] 200

incident.p2 <- vec.exist[215:414]; (n.p2 <- length(incident.p2))

## [1] 200

incident.p3 <- vec.exist[415:484]; (n.p3 <- length(incident.p3))

## [1] 70

incident.p4 <- vec.exist[485:514]; (n.p4 <- length(incident.p4))

## [1] 30

incident.p5 <- vec.exist[515:549]; (n.p5 <- length(incident.p5))

## [1] 35

incident.p6 <- vec.exist[550:614]; (n.p6 <- length(incident.p6))

## [1] 65

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incidents <- c(incident.p1, incident.p2, incident.p3, incident.p4,incident.p5, incident.p6)

The new situation after positioning the 7th station The estimated optimal location and capacity ofthe new monitoring station were obtained by using Matlab codes. The resulting optimal allocation zonesafter positioning the seventh monitoring station were determined as well. The file containing all these data isavailable here and can be imported as follows:

vec.new <- as.numeric(read.table("data/table_new_loc.txt", dec=".",sep=","))

The new allocation zones:

incident.p1.new <- vec.new[15:148] ; n.p1.new <- length(incident.p1.new)incident.p2.new <- vec.new[149:292]; n.p2.new <- length(incident.p2.new)incident.p3.new <- vec.new[293:364]; n.p3.new <- length(incident.p3.new)incident.p4.new <- vec.new[365:393]; n.p4.new <- length(incident.p4.new)incident.p5.new <- vec.new[394:422]; n.p5.new <- length(incident.p5.new)incident.p6.new <- vec.new[423:494]; n.p6.new <- length(incident.p6.new)

The location and capacity of the new monitoring station:

optim.poste <- vec.new[495]postes.new <- c(postes, optim.poste)taille.postes.new <- c(vec.new[9:14], vec.new[496])

The clients allocated to the new station:

incident.optim.poste <- vec.new[497:616]; n.optim.new <- length(incident.optim.poste)incidents.new <- c(incident.p1.new, incident.p2.new, incident.p3.new,incident.p4.new, incident.p5.new, incident.p6.new, incident.optim.poste)

The 95% confidence interval of the true optimal location

LB <- vec.new[617]UB <- vec.new[618]

d. Representation of the data

col.postes <- c("mediumorchid1", "aquamarine2","chocolate2","indianred1","darkgoldenrod2","deeppink1", "red")

cex.postes = as.vector(7 * sqrt(taille.postes)/max(sqrt(taille.postes)))cex.postes.new = as.vector(7 * sqrt(taille.postes.new)/max(sqrt(taille.postes.new)))

Graphical parameters

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#pdf(file="droite_avant.pdf", width=12, height=4)plot(incidents,rep(1,length(incidents)), xlim=c(a,b), xlab="", ylab="",axes=FALSE, pch=3, cex=2,col=col.postes[c(rep(1,n.p1),rep(2,n.p2),rep(3,n.p3),rep(4,n.p4),rep(5,n.p5),rep(6,n.p6))])points(postes,rep(1,length(postes)), pch=15)points(postes,rep(1,length(postes)), col=col.postes, cex=cex.postes, lwd=2)arrows(a,1,b,1)text(postes, rep(1.2,6), as.expression(c(bquote(y[1]), bquote(y[2]), bquote(y[3]),bquote(y[4]), bquote(y[5]), bquote(y[6]))),cex=1.6)

1st graphic (historical data before the placement of the new station)

y1 y2 y3 y4 y5 y6

#dev.off()

8

● ●

y1 y2 y3 y4 y5 y6

#pdf(file="droite_apres.pdf", width=12, height=4)plot(incidents.new, rep(1,length(incidents.new)), xlim=c(a,b), xlab="", ylab="",

axes=FALSE, pch=3, cex=2, col=col.postes[c(rep(1,n.p1.new),rep(2,n.p2.new),rep(3,n.p3.new),rep(4,n.p4.new),rep(5,n.p5.new),rep(6,n.p6.new), rep(7,n.optim.new))])

points(postes.new, rep(1,length(postes.new)), pch=15)points(postes.new,rep(1,length(postes.new)), col=col.postes, cex=cex.postes.new, lwd=2)arrows(a,1,b,1)text(optim.poste , 1.23, as.expression(bquote(hat(y)[list(7,n)])), col="red",cex=1.6)text(postes, rep(1.2,6), as.expression(c(bquote(y[1]), bquote(y[2]), bquote(y[3]),

bquote(y[4]), bquote(y[5]), bquote(y[6]))),cex=1.6)text(optim.poste, 1.28, "*", col="red")arrows(LB,0.87,UB,0.87, lwd=2, angle=90, col="red", code=3, length=0.04)text((LB+UB)/2, 0.8, "CI", col="red", cex=1.6)

9

2nd graphic (the new situation after positioning the 7th station)

y7, ny1 y2 y3 y4 y5 y6*

CI

#dev.off()

● ●

y7, ny1 y2 y3 y4 y5 y6*

CI

3. Representation of the monitoring stations and associated clients on a map

The idea is to convert the segment [ab] into the original borderline on the map by considering the fact that acorresponds to the coastal city of Saidia and b to the city of Figuig. First, we define the locations of themonitoring centers and their allocations zones in a vector and then we sort this vector:

10

x <- c(a, b, incidents,postes)x.new <- c(a, b, incidents.new, postes.new, LB, UB)x.ord<-sort(x)x.ord.new<-sort(x.new)

We compute the distances and cumulative distances between consecutive endpoints of the segments [ab]:

x.ord <- diff(x.ord)x.ord.new <- diff(x.ord.new)x.cum<-cumsum(x.ord)/sum(x.ord)x.cum.new<-cumsum(x.ord.new)/sum(x.ord.new)x.final.cum<-x.cum[-c(length(x.cum))]x.final.cum.new<-x.cum.new[-c(length(x.cum.new))]

We report the cumulative distances computed on the straight line on the borderline, then we determine thecartesian coordinates of the data points:

coord.pts <- data.frame(prop=x.final.cum, indice=apply(t(x.final.cum),2,function(x) which(x>don$dist.cum.A & x<=don$dist.cum.B)))

coord.pts$coef <- (coord.pts$prop-don$dist.cum.A[coord.pts$indice])/(don$dist.cum.B[coord.pts$indice]-don$dist.cum.A[coord.pts$indice])

coord.pts$x <- don$xA[coord.pts$indice] + coord.pts$coef*(don$xB[coord.pts$indice]-don$xA[coord.pts$indice])

coord.pts$y <- don$yA[coord.pts$indice] + coord.pts$coef*(don$yB[coord.pts$indice]-don$yA[coord.pts$indice])

coord.pts.new <- data.frame(prop=x.final.cum.new, indice =apply(t(x.final.cum.new),2,function(x) which(x>don$dist.cum.A & x<=don$dist.cum.B)))

coord.pts.new$coef<-(coord.pts.new$prop-don$dist.cum.A[coord.pts.new$indice])/(don$dist.cum.B[coord.pts.new$indice]-don$dist.cum.A[coord.pts.new$indice])

coord.pts.new$x <- don$xA[coord.pts.new$indice] + coord.pts.new$coef*(don$xB[coord.pts.new$indice]-don$xA[coord.pts.new$indice])

coord.pts.new$y <- don$yA[coord.pts.new$indice] + coord.pts.new$coef*(don$yB[coord.pts.new$indice]-don$yA[coord.pts.new$indice])

We identify the monitoring stations and their associated clients:

order.data<-order(c(incidents,postes))ind.optimaux<-which(order.data>length(incidents))ind.incidents<-which(!order.data>length(incidents))

order.data.new<-order(c(incidents.new,LB,UB,postes.new))ind.optimaux.new<-which(order.data.new>length(incidents.new)+2)ind.incidents.new<-which(!order.data.new>length(incidents.new))ind.IC.new<-which(order.data.new%in%c(length(incidents.new)+1,length(incidents.new)+2))

We create the following Spatial objects:

incidents.SP <- SpatialPoints(cbind(coord.pts$x[ind.incidents],coord.pts$y[ind.incidents]), proj4string=CRS(proj.maroc))

incidents.SP.new <- SpatialPoints(cbind(coord.pts.new$x[ind.incidents.new],coord.pts.new$y[ind.incidents.new]), proj4string=CRS(proj.maroc))

postes.SP <- SpatialPoints(cbind(coord.pts$x[ind.optimaux],coord.pts$y[ind.optimaux]),

11

proj4string=CRS(proj.maroc))postes.SP.new <- SpatialPoints(cbind(coord.pts.new$x[ind.optimaux.new],coord.pts.new$y[ind.optimaux.new]), proj4string=CRS(proj.maroc))

IC.SP.new <- SpatialPoints(cbind(coord.pts.new$x[ind.IC.new],coord.pts.new$y[ind.IC.new]),proj4string=CRS(proj.maroc))

We convert the coordinates in the appropriate CRS :

incidents.SP <- spTransform(incidents.SP, CRS(proj.OSM))incidents.SP.new <- spTransform(incidents.SP.new, CRS(proj.OSM))postes.SP <- spTransform(postes.SP, CRS(proj.OSM))postes.SP.new <- spTransform(postes.SP.new , CRS(proj.OSM))IC.SP.new <- spTransform(IC.SP.new , CRS(proj.OSM))frontiere.OSM <- spTransform(frontiere, CRS(proj.OSM))provinces.OSM<-spTransform(provinces, CRS(proj.OSM))

#pdf(file="carte_avant.pdf")plot(map)plot(incidents.SP, add=TRUE, pch=3, cex=1.6,

col=col.postes[c(rep(1,n.p1),rep(2,n.p2),rep(3,n.p3),rep(4,n.p4),rep(5,n.p5),rep(6,n.p6))])

plot(postes.SP,add=TRUE, pch=22, cex=0.6)points(postes.SP, col=col.postes, cex=cex.postes, lwd=3)plot(frontiere.OSM, add=TRUE)text(coordinates(postes.SP), as.expression(c(bquote(y[1]), bquote(y[2]), bquote(y[3]),

bquote(y[4]), bquote(y[5]), bquote(y[6]))), font=3, pos=4, cex=1.2, offset=0.6)plot(provinces, pch=1, add=TRUE, cex=0.8)text(coordinates(provinces), nom.prov, cex=0.95,

pos=c(4, 1, 1, 1, 2, 3, 1, 1, 4, 1, 2, 2, 4, 2, 2, 4, 2, 4, 4, 4, 4), offset=0.4)

12

1st graphic (historical data before the placement of the new station)

y1

y2

y3

y4

y5

y6

#dev.off()

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y1

y2

y3

y4

y5

y6

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Melilla

Oujda

Saidia

Figuig

Ahfir Maghnia

Jerada

BerkaneTlemcen

Bouarfa

Tendrara

Ain Sefra

Ain−Beni−Mathar

Sebdou

Naama

Mecheria

El Aricha

#pdf(file="carte_apres.pdf")plot(map)plot(incidents.SP.new, add=TRUE, pch=3, cex=1.6,

col=col.postes[c(rep(1,n.p1.new), rep(7,n.optim.new), rep(2,n.p2.new),rep(3,n.p3.new),rep(4,n.p4.new),rep(5,n.p5.new),rep(6,n.p6.new))])

plot(postes.SP.new, add=TRUE, pch=22, cex=0.6)points(postes.SP.new, col=col.postes[c(1,7,2:6)], cex=cex.postes.new[c(1,7,2:6)], lwd=3)plot(frontiere.OSM, add=TRUE)plot(IC.SP.new, add=TRUE, pch=5, cex=0.6)text(coordinates(postes.SP.new[c(1,3:7),]), as.expression(c(bquote(y[1]), bquote(y[2]),

bquote(y[3]), bquote(y[4]), bquote(y[5]), bquote(y[6]))),

14

font=3, pos=4, cex=1.2, offset=0.6)text(coordinates(postes.SP.new[2,])[1]-5000,

coordinates(postes.SP.new[2,])[2]+16000, as.expression(bquote(hat(y)[list(7,n)])),font=3, pos=4, cex=1.2, col="red", offset=0.9)

text(coordinates(postes.SP.new[2,])[1]-3800,coordinates(postes.SP.new[2,])[2]+18000,"*", font=3, pos=4, cex=0.8, col="red", offset=1.2)

plot(provinces.OSM, pch=1, add=TRUE, cex=0.8)text(coordinates(provinces.OSM), nom.prov, cex=0.95,

pos=c(4, 1, 1, 1, 2, 3, 1, 1, 4, 1, 2, 2, 4, 2, 2, 4, 2, 4, 4, 4, 4), offset=0.4)

2nd graphic (the new situation after positioning the 7th station)

y1

y2

y3

y4

y5

y6

y7, n*

Melilla

Oujda

Saidia

Figuig

Ahfir Maghnia

Jerada

Berkane Tlemcen

Bouarfa

Tendrara

Ain Sefra

Ain−Beni−Mathar

Sebdou

Naama

Mecheria

El Aricha

#dev.off()

15

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y1

y2

y3

y4

y5

y6

y7, n*

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Melilla

Oujda

Saidia

Figuig

Ahfir Maghnia

Jerada

BerkaneTlemcen

Bouarfa

Tendrara

Ain Sefra

Ain−Beni−Mathar

Sebdou

Naama

Mecheria

El Aricha

4. Animation

In this application, we actually consider a dynamic setting where the patrol agents’ intervention sites {xi}vary over time, but the dataset at hand upon which we base the long term decision of positioning a newborder monitoring station is just a picture of the situation over a given period of time i = 1, . . . , 600.This kind of static location modeling is based on the implicit assumption that the time or effort neededfor the illegal immigration, drug trafficking and contraband smuggling to react is long enough to harvestthe main benefits of the new monitoring site. One way to check this assumption is by considering theevolution of the estimated optimal location and capacity (y∗

7,n, z∗7,n) starting from the subsample of the

first 300 recorded observations {x1, . . . , x300}. For each new patrol agents’ intervention i = 301, . . . , 600,we add the corresponding observation xi to the previous sample and then display the obtained resultssimilarly to the second graphic above. This leads to an animation on the straight line that can be found athttps://gremaq.univ-tlse1.fr/perso/laurent/code/DVK.html, and its conversion into the true borderline at

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https://gremaq.univ-tlse1.fr/perso/laurent/code/DVK2.html.

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