# Hellinger transformation
# First calculate the total species abundances by site
(site.totals=apply(spe, 1, sum))

# Scale species abundances by dividing them by site totals
(scale.spe<-spe/site.totals)

# Calculate the square root of scaled species abundances
(sqrt.scale.spe<-sqrt(scale.spe))

# Compare the results
sqrt.scale.spe
spe.hel
sqrt.scale.spe-spe.hel # or: sqrt.scale.spe/spe.hel

# Chi-square transformation
# First calculate the total species abundances by site
(site.totals<-apply(spe, 1, sum))

# Then calculate the square root of total species abundances
(sqrt.spe.totals<-sqrt(apply(spe, 2, sum)))

# Scale species abundances by dividing them by the site totals and the species totals
scale.spe2<-spe
for (i in 1:nrow(spe)) {
  for (j in 1:ncol(spe)) {
   (scale.spe2[i,j]=scale.spe2[i,j]/(site.totals[i]*sqrt.spe.totals[j]))   }}

#Adjust the scale abundance species by multiplying by the square root of the species matrix total                                         
(adjust.scale.spe2<-scale.spe2*sqrt(sum(rowSums(spe))))

# Compare the results
adjust.scale.spe2
spe.chi
adjust.scale.spe2-spe.chi # or: adjust.scale.spe2/spe.chi