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| test.interieur.polygone<-function(gr.x,gr.y,poly.x,poly.y) {
S<-length(poly.x)
I<-length(gr.x)
J<-length(gr.y)
TEST<-array(NA,c(I,J,S))
for( s in 1:S) {
xa<-poly.x[s]
ya<-poly.y[s]
xb<-poly.x[s+1-S*(s==S)]
yb<-poly.y[s+1-S*(s==S)]
for(i in 1:I) {
if(xa != xb) test1 <- gr.y < ((xb-gr.x[i])*ya + (gr.x[i]-xa)*yb)/(xb-xa)
if(xa == xb) test1 <- rep(TRUE,J)
if(xa != xb) test2 <- min(xa,xb) < gr.x[i] & gr.x[i] <= max(xa,xb)
if(xa == xb) test2 <- gr.x[i] == xa
TEST[i,1:J,s]<- test1 & test2
}}
NB.intersect<-apply(TEST,c(1,2),sum)
test.parite<- floor(NB.intersect/2)*2 != NB.intersect
return(test.parite)
}
FRANCE<- read.table(file="data-contour-France.dat",sep=" ",dec=".")
FR.x<-FRANCE[,1]
FR.x
FR.y<-FRANCE[,2]
FR.y
TAB<- read.table(file="data-pluvio.dat",sep="\t",dec=",",header=TRUE)
x<-TAB[,1]
x
y<-TAB[,2]
y
z<-TAB[,3]
z
n<-length(x)
n
V0<- rep(1,n)
V1<-x
V2<-y
V3<-x^2
V4<-x*y
V5<-y^2
V6<-x^3/1000
V7<-x^2*y/1000
V8<-x*y^2/1000
V9<-y^3/1000
# question 1
modele.simplifie<-lm(z~V1+V3+V4+V5+V6+V7+V8+V9)
coef<-modele.simplifie$coef
# cartographie
#----------------------
grid.x<-seq(-5.5,8.2,by=0.10)
grid.y<-seq(42,51.2,by=0.10)
p<-length(grid.x)
q<-length(grid.y)
MAP<-matrix(NA,p,q)
for(i in 1:p) for(j in 1:q)
{
x0<- grid.x[i]
y0<- grid.y[j]
eta<-c(1,x0,x0^2,x0*y0,y0^2,x0^3/1000,x0^2*y0/1000,x0*y0^2/1000,y0^3/1000)
MAP[i,j]<-sum(coef*eta)
}
# application du masque
MASK<-test.interieur.polygone(grid.x,grid.y,FR.x,FR.y)
MASK[MASK==FALSE]<-NA
MAP<- MAP*MASK
niveaux<- seq(40,150,by=5)
palette<-rainbow(length(niveaux)-1,start=0.4,end=0.7)
image(grid.x,grid.y,MAP,breaks=niveaux,col=palette,main="pluviometrie en France")
points(x,y,pch=16,cex=0.5)
contour(grid.x,grid.y,MAP,levels=niveaux,add=TRUE)
polygon(FRANCE)
v<-length(niveaux)
text<-paste(niveaux[-v],"-",niveaux[-1])
legend(x=-5.0,y=46.5,fill=palette,legend=text,cex=0.5)
#residus
z.predict<-predict(modele.simplifie)
modele.simplifie<-lm(z~V1+V3+V4+V5+V6+V7+V8+V9)
residus<-residuals(modele.simplifie)
residus
#question 2 analyse variographique
res<- residus
DIST<-matrix(NA,n,n)
for(i in 1:n) for(j in 1:n) {
DIST[i,j]<- sqrt((x[i]-x[j])^2+(y[i]-y[j])^2)
}
DIST
Q<-matrix(NA,n,n)
for(i in 1:n) for(j in 1:n) {
Q[i,j]<- 0.5*(res[i]-res[j])^2
}
Q
H<-as.vector(DIST)
V<-as.vector(Q)
plot(x=H,y=V)
breaks<-seq(0,14,by=1.0)
CL<-cut(H,breaks)
mH<-tapply(H,CL,mean)
mV<-tapply(V,CL,mean)
points(mH,mV,col="blue",type="b",pch=16)
abline(v=breaks,lty=2,col="green")
variog.curve<-function(tau2,sig2,a,type,...){
if(type=="lin")
curve( sig2*(x/a)*(x<a) + sig2*(x>=a) + tau2*(x>0) ,add=TRUE,...)
if(type=="spher")
curve( sig2*(1.5*x/a - 0.5*(x/a)^3)*(x<a) + sig2*(x>=a) + tau2*(x>0) ,add=TRUE,...)
if(type=="exp")
curve( sig2*(1 -exp(-(x/a) )) + tau2*(x>0),add=TRUE,...)
if(type=="gauss")
curve( sig2*(1 -exp(-(x/a)^2)) + tau2*(x>0),add=TRUE,...)
}
tau2<- 100
sig2<- 50
a <- 0.5
variog.curve(tau2,sig2,a,type="gauss",col="red",lwd=2)
# 4. modele variographique le plus vraisemblable
type<-"gauss"
gr.tau2<-seq(0.01,100,length=25) ; p<-length(gr.tau2)
gr.sig2<-seq(0.10,50,length=25) ; q<-length(gr.sig2)
gr.a<-seq(0.10,0.50,length=25) ; r<-length(gr.a)
calcul.cov<-function(D,tau2,sig2,a,type){
if(type=="lin") COV <- sig2 * (1- D/a )*(D<a) + diag(rep(tau2,n))
if(type=="spher") COV <- sig2 * (1- 1.5*D/a + 0.5*(D/a)^3)*(D<a) + diag(rep(tau2,n))
if(type=="exp") COV <- sig2 * exp (-(D/a) ) + diag(rep(tau2,n))
if(type=="gauss") COV <- sig2 * exp (-(D/a)^2) + diag(rep(tau2,n))
return(COV)
}
loglik <- function(tau2,sig2,a,type,res,DIST){
COV<-calcul.cov(DIST,tau2,sig2,a,type)
ll<- -0.5*log(det(COV)) - 0.5*t(res)%*%solve(COV)%*%res
return(ll)
}
CUBE<-array(NA,dim=c(p,q,r))
for(i in 1:p) for(j in 1:q) for(k in 1:r)
CUBE[i,j,k]<-loglik(gr.tau2[i],gr.sig2[j],gr.a[k],type,res,DIST)
i.opt<-which(CUBE==max(CUBE),arr.ind=TRUE) ; i.opt
tau2.optimal<-gr.tau2[i.opt[1]] ; tau2.optimal
sig2.optimal<-gr.sig2[i.opt[2]] ; sig2.optimal
a.optimal <-gr.a[i.opt[3]] ; a.optimal
loglik.max <- loglik(tau2.optimal,sig2.optimal,a.optimal,type,res,DIST) ; loglik.max
# question 3
COV<-calcul.cov(DIST,tau2,sig2,a,type)
COV
INVCOV<-solve(COV)
INVCOV
gr.x<-seq(0,1,length=200)
gr.y<-seq(0,1,length=200)
MAP<-matrix(NA,length(gr.x),length(gr.y))
for(i in 1:length(gr.x))
for(j in 1:length(gr.y)){
x0<-gr.x[i]
y0<-gr.y[j]
dist<- sqrt((x-x0)^2+(y-y0)^2)
co<- diag(calcul.cov(diag(dist),tau2,sig2,a,type))
lambda<- INVCOV %*% co
MAP[i,j]<-sum(lambda*res)
}
#question 4
#cartographie du krigeage
#~~~~~~~~~~~~~~~~~~~~~~~~
niveaux<-seq(-2,2,length=20)
palette<-topo.colors(length(niveaux)-1)
image(MAP,breaks=niveaux,col=palette,main="krigeage")
contour(MAP,levels=niveaux,add=TRUE)
points(x[res>0],y[res>0],col="red",pch=16)
points(x[res<0],y[res<0],col="black",pch=16)
#cartographie idw des résidus
interp.inv.dist <-function(x,y,z,gr.x, gr.y,p) {
I<-length(gr.x)
J<-length(gr.y)
MAP<-matrix(NA,I,J)
for(i in 1:I) for(j in 1:J) {
x0<- gr.x[i]
y0<- gr.y[j]
d<-sqrt((x - x0)^2 + (y - y0)^2)
w<- 1 / (d+0.000001)^p
MAP[i,j]<- sum(w*z)/sum(w)
}
return(MAP)
}
par(mfrow=c(1,2))
MAP2<-interp.inv.dist(x,y,res,gr.x,gr.y,2)
niveaux<-seq(-2,2,length=20)
palette<-topo.colors(length(niveaux)-1)
image(MAP2,breaks=niveaux,col=palette, main="IDW")
contour(MAP2,levels=niveaux,add=TRUE)
points(x[res>0],y[res>0],col="red",pch=16)
points(x[res<0],y[res<0],col="black",pch=16)
# 5d/ cartographie du krigeage
#~~~~~~~~~~~~~~~~~~~~~~~~
niveaux<-seq(-2,2,length=20)
palette<-topo.colors(length(niveaux)-1)
image(MAP,breaks=niveaux,col=palette,main="krigeage")
contour(MAP,levels=niveaux,add=TRUE)
points(x[res>0],y[res>0],col="red",pch=16)
points(x[res<0],y[res<0],col="black",pch=16) |
