############################
# Passito
############################

# import data
Dataset=read.csv("Passito.csv",header=TRUE,sep=";")
head(Dataset)
dim(Dataset)

# keep only the variables (columns) of interest
X=Dataset[,17:21]
head(X)
summary(X)

# number of units = number of dataset rows
n=dim(X)[1]
n

# number of columns = dataset columns
k=dim(X)[2]
k

# correlations between observed variables
cor(X)

# normalization of the k=5 variables (rescaling type transformation)
Z=(X-min(X)+1/n)/(max(X)-min(X)+2/n)
head(Z)
dim(Z)
class(Z)

# change Z structure from data.frame into matrix
Z=as.matrix(Z)
class(Z)

# define the vector of weights (equal importance of the variables)
w=rep(1/k,times=k)
w

#Fisher combination as product between the matrix of transformed values and 
#vector of weights 
yF=-log(1-Z,base=2)%*%w
yF
summary(yF)

# definition of the minimum and maximum value of the composite indicator in order to normalize it (rescaling) 
ymin=0
range_x=7-1
yFmax=-log(1-(range_x+1/n)/(range_x+2/n),base=2)
yFmax

#final index normalization
yF_norm=(yF-ymin)/(yFmax-ymin)
summary(yF_norm)

# distribution of the composite indicator
hist(yF_norm,freq=FALSE)
lines(density(yF_norm), col ="blue", lty = 2, lwd = 2, add = TRUE)

hist(yF_norm,freq=FALSE,ylim=c(0,6))
lines(density(yF_norm), col ="blue", lty = 2, lwd = 2, add = TRUE)


#Additive combination 
yA=Z%*%w

# graphical comparison between the distributions of the two composite indicators (Fisher combination and additive combination) 
hist(yA,freq=FALSE,ylim=c(0,6))
lines(density(yA), col ="red", lty = 1, lwd = 2, add = TRUE)
lines(density(yF_norm), col ="blue", lty = 2, lwd = 2, add = TRUE)
plot(yA,yF_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)

# relationship between each partial indicator (normalized variable) and the normalized composite indicator (Fisher combination) 
plot(Z[,1],yF_norm)
plot(Z[,2],yF_norm)
plot(Z[,3],yF_norm)
plot(Z[,4],yF_norm)
plot(Z[,5],yF_norm)
cor(Z,yF_norm)


############################
# Mall
############################


X=read.csv("Mall.csv",header=TRUE,sep=";")
head(X)
dim(X)
head(X)
summary(X)
n=dim(X)[1]
n
k=dim(X)[2]
k
cor(X)

#normalization of the k=5 variables
Z=(X-(-100)+1/n)/(100-(-100)+2/n)
head(Z)
dim(Z)

Z=as.matrix(Z)
w=rep(1/k,times=k)
w
#Liptak combination as product between the matrix of transformed values and 
#vector of weights 
yL=qnorm(Z)%*%w
range_x=100-(-100)
summary(yL)
ymin=qnorm((1/n)/(range_x+2/n))
ymin
ymax=qnorm((range_x+1/n)/(range_x+2/n))
ymax

#final index normalization
yL_norm=(yL-ymin)/(ymax-ymin)
summary(yL_norm)
hist(yL_norm,freq=FALSE)
lines(density(yL_norm), col ="blue", lty = 2, lwd = 2, add = TRUE)

#Additive combination 
yA=Z%*%w

hist(yA,freq=FALSE)
lines(density(yA), col ="red", lty = 1, lwd = 2, add = TRUE)
lines(density(yL_norm), col ="blue", lty = 2, lwd = 2, add = TRUE)
plot(yA,yL_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)

summary(X)
hist(X[,1])
hist(X[,2])
hist(X[,3])
hist(X[,4])
hist(X[,5])

plot(Z[,1],yL_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)

plot(Z[,2],yL_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)

plot(Z[,3],yL_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)

plot(Z[,4],yL_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)

plot(Z[,5],yL_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)





############################
# Students
############################

Dataset=read.csv("Students.csv",header=TRUE,sep=";")
head(Dataset)
dim(Dataset)
X=Dataset[,-1]
head(X)
summary(X)
n=dim(X)[1]
n
k=dim(X)[2]
k
cor(X)

#normalization of the k=3 variables
Z=(X-1+1/n)/(10-1+2/n)
head(Z)
dim(Z)

Z=as.matrix(Z)
w=rep(1/k,times=k)
w

#Tippett combination as product between the matrix of transformed values and 
#vector of weights 
yT=matrix(0,nrow=n,ncol=1)
wZ=Z%*%diag(w)
for(u in 1:n){yT[u]=max(wZ[u,])}
cbind(wZ,yT)
range_x=10-1
summary(yL)
ymin=(1/n)/(range_x+2/n)*min(w)
ymin
ymax=(range_x+1/n)/(range_x+2/n)*max(w)
ymax

#final index normalization
yT_norm=(yT-ymin)/(ymax-ymin)

hist(yT_norm,freq=FALSE)
lines(density(yT_norm), col ="blue", lty = 2, lwd = 2, add = TRUE)

#Additive combination 
yA=Z%*%w

hist(yA,freq=FALSE)
lines(density(yA), col ="red", lty = 1, lwd = 2, add = TRUE)
lines(density(yT_norm), col ="blue", lty = 2, lwd = 2, add = TRUE)
plot(yA,yT_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)

summary(X)
hist(X[,1])
hist(X[,2])
hist(X[,3])

plot(Z[,1],yT_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)

plot(Z[,2],yT_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)

plot(Z[,3],yT_norm)
abline(a=0,b=1,col="green",lty=1,lwd=2)