library(ca)
data(bitterling)
data<-bitterling
total <- sum(data)
nrows <- nrow(data)
ncols <- ncol(data)
#dimensionality
a <- min(ncol(data)-1, nrow(data)-1) 
labs<-c(1:a) #x- axis lables
# create the contingency table
data_matrix<-as.matrix(data)
# Add row column profile to contingency table
data_rowsum<-addmargins(data_matrix,1)
data_colsum<-addmargins(data_matrix,2)
# Apply average rule ,to get number of dimensions
col_dim<-round(100/(ncols-1), digits=1)
row_dim<-round(100/(nrows-1), digits=1)
thresh_dim<-(max(col_dim, row_dim))
data_ca<- summary(ca(data))
n_dim<- length(which(data_ca$scree[,3]>=thresh_dim))
# Malinvaud's Test
mal_ca<-CA(data, ncp=a, graph=FALSE)
mal_trow <- a
mal_tcol <- 6
mal_out <-matrix(ncol= mal_tcol, nrow=mal_trow)
names(mal_out) <- c("K", "Dimension", "Eigen value", "Chi-square", "df", "p value")
mal_out[,1] <- c(0:(a-1))
mal_out[,2] <- c(1:a)
library(foreach)
library(doParallel)
cl <- makeCluster(4) # number of cores
registerDoParallel(cl)
foreach(i = 1:mal_trow) %dopar% {
  k <- -1+i
  mal_out[i,3] <- mal_ca$eig[i,1]
  mal_out[i,5] <- (nrows-k-1)*(ncols-k-1)
}
mal_out[,4] <- rev(cumsum(rev(mal_out[,3])))*total
mal_out[,6] <- round(pchisq(mal_out[,4], mal_out[,5], lower.tail=FALSE), digits=6)
optimal.dimensionality <- length(which(mal_out[,6]<=0.05))
# plot bar chart of correlation between rows and columns, and add reference line
dev.new()
perf_corr<-(1.0)
sqr.trace<-round(sqrt(sum(data_ca$scree[,2])), digits=3)
barplot(c(perf_corr, sqr.trace), main="Correlation coefficient between rows & columns (=square root of the inertia)", sub="reference line: threshold of important correlation ", ylab="correlation coeff.", names.arg=c("correlation coeff. range", "correlation coeff. bt rows & cols"), cex.main=0.80, cex.sub=0.80, cex.lab=0.80)
abline(h=0.20)

barplot(data_ca$scree[,3], xlab="Dimensions", ylab="% of Inertia", names.arg=data_ca$scree[,1])
abline(h=thresh.sig.dim)
title (main="Percentage of inertia attributed to the dimensions", sub="ref line: threshold of an optimal dimensionality of the solution, according to the average rule", cex.main=0.80, cex.sub=0.80)
plot(mal_out[,6], type="o", xaxt="n", xlim=c(1, a), xlab="Dimensions", ylab="p value")
axis(1, at=labs, labels=sprintf("%.0f",labs))
title(main="Malinvaud's test Plot", sub="dashed line: alpha 0.05 threshold", col.sub="RED", cex.sub=0.80)
abline(h=0.05, lty=2, col="RED")

userow_dimensionality <- 3
dims.to.be.plotted <- userow_dimensionality 
# CA analysis by Greenacre's package to be used later on for the Standard Biplots
res.ca <- ca(data, nd=dims.to.be.plotted)
str(res.ca)
List of 15
 $ sv        : num [1:11] 0.831 0.799 0.585 0.529 0.443 ...
 $ nd        : num 3
 $ rownames  : chr [1:12] "jk" "tu" "hb" "chs" ...
 $ rowmass   : num [1:12] 0.1748 0.0382 0.1021 0.0709 0.0853 ...
 $ rowdist   : num [1:12] 1.19 1.01 1.02 1.31 2.69 ...
 $ rowinertia: num [1:12] 0.2466 0.0386 0.1064 0.1214 0.6195 ...
 $ rowcoord  : num [1:12, 1:3] -0.04047 -0.00192 0.16761 0.20569 -3.12697 ...
 $ rowsup    : logi(0) 
 $ colnames  : chr [1:12] "jk" "tu" "hb" "chs" ...
 $ colmass   : num [1:12] 0.1983 0.0101 0.1167 0.0792 0.0864 ...
 $ coldist   : num [1:12] 1.159 0.798 1.075 1.191 2.677 ...
 $ colinertia: num [1:12] 0.26613 0.00644 0.13485 0.11225 0.6189 ...
 $ colcoord  : num [1:12, 1:3] -0.0126 0.0104 0.1632 0.2324 -3.113 ...
 $ colsup    : logi(0) 
 $ call      : language ca.matrix(obj = as.matrix(obj), nd = ..1)
 - attr(*, "class")= chr "ca"
# CA output as dataframe to be used for the some graphs to come
cadataframe<-summary(ca(data, nd=dims.to.be.plotted))
# plot the quality of the display of categories on successive pairs of dimensions
#row categories
dev.new()
counter <- 1
for(i in seq(9, ncol(cadataframe$rows), 3)){  
  counter <- counter +1
  quality.rows <- (cadataframe$rows[,6]+cadataframe$rows[,i])/10
  barplot(quality.rows, ylim=c(0,100), xlab="Row categories", ylab=paste("Quality of the display (% of inertia) on Dim. 1+", counter), names.arg=cadataframe$rows[,1], cex.lab=0.80)
}

#column categories
dev.new()
counter <- 1
for(i in seq(9, ncol(cadataframe$columns), 3)){  
  counter <- counter +1
  quality.cols <- (cadataframe$columns[,6]+cadataframe$columns[,i])/10
  barplot(quality.cols, ylim=c(0,100), xlab="Column categories", ylab=paste("Quality of the display (% of inertia) on Dim. 1+", counter), names.arg=cadataframe$columns[,1], cex.lab=0.80)
}
# charts of categories contribution
# plot bar charts of contribution of row categories to the axes, and add a reference line
dev.new()
counter <- 0
for(i in seq(7, ncol(cadataframe$rows), 3)){  
  counter <- counter +1
  barplot(cadataframe$rows[,i], ylim=c(0,1000), xlab="Row categories", ylab=paste("Contribution to Dim. ",counter," (in permills)"), names.arg=cadataframe$rows[,1], cex.lab=0.80)
  abline(h=round(((100/nrows)*10), digits=0))
}
# plot bar charts of contribution of column categories to the axes, and add a reference line
dev.new()
counter <- 0
for(i in seq(7, ncol(cadataframe$columns), 3)){  
  counter <- counter +1
  barplot(cadataframe$columns[,i], ylim=c(0,1000), xlab="Column categories", ylab=paste("Contribution to Dim. ",counter," (in permills)"), names.arg=cadataframe$columns[,1], cex.lab=0.80)
  abline(h=round(((100/ncols)*10), digits=0))
}
# let us estimate the correlation of categories to dimensions
# row categories
dev.new()
counter <- 0
for(i in seq(6, ncol(cadataframe$rows), 3)){  
  counter <- counter +1
  correl.rows <- round(sqrt((cadataframe$rows[,i]/1000)), digits=3)
  barplot(correl.rows, ylim=c(0,1), xlab="Row categories", ylab=paste("Correlation with Dim. ", counter), names.arg=cadataframe$rows[,1], cex.lab=0.80)
}
#column categories
dev.new()
counter <- 0
for(i in seq(6, ncol(cadataframe$columns), 3)){  
  counter <- counter +1
  correl.cols <- round(sqrt((cadataframe$columns[,i]/1000)), digits=3)
  barplot(correl.cols, ylim=c(0,1), xlab="Column categories", ylab=paste("Correlation with Dim. ", counter), names.arg=cadataframe$columns[,1], cex.lab=0.80)
}
  #let us check the Contingency Table
print(addmargins(data_matrix))
# Association coefficients can be estimated by 
library(vcd)
print(assocstats(data_matrix))
                      X^2  df P(> X^2)
Likelihood Ratio 9251 121        0
Pearson                14589 121        0
Phi-Coefficient: 1.581 
Contingency Coeff.: 0.845 
Cramer's V        : 0.477