Min and Max values of beta diversity (BD) estimated by the variance of Hellinger-transformed data.

The total variance is computed on the table of raw data and from a D matrix.

library(vegan)

========  Hellinger distance

# Minimum BD when the sites have exactly the same species composition

mat.sp1 = matrix(c(1,1,1,1,1,2,2,2,2,2,3,3,3,3,3),5,3)
n1 = nrow(mat.sp1)
mat.sp1.hel = decostand(mat.sp1, "hel")

BD1 = sum( apply(mat.sp1.hel,2,var) )
BD1
[1] 0

# Another way to compute the total variance in the matrix
# See Numerical ecology (1998), p. 329, eq. 8.5 ad 8.6

mat.sp1.D = dist(mat.sp1.hel)
BD1 = sum(mat.sp1.D^2)/(n1*(n1-1))
BD1
[1] 0

--------

# Maximum BD when the sites have totally different species compositions

mat.sp2 = matrix(c(1,0,0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,1),3,6)
n2 = nrow(mat.sp2)
mat.sp2.hel = decostand(mat.sp2, "hel")

BD2 = sum( apply(mat.sp2.hel,2,var) )
BD2
[1] 1

mat.sp2.D = dist(mat.sp2.hel)
BD2 = sum(mat.sp2.D^2) / (n2*(n2-1))
BD2
[1] 1

---

# Another example

mat.sp3 = matrix(c(1,0,0,2,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,10,0,0,100),3,8)
n3 = nrow(mat.sp3)
mat.sp3.hel = decostand(mat.sp3, "hel")

BD3 = sum( apply(mat.sp3.hel,2,var) )
BD3
[1] 1

mat.sp3.D = dist(mat.sp3.hel)
BD3 = sum(mat.sp3.D^2) / (n3*(n3-1))
BD3
[1] 1

========  Chord distance

# Chord transformation

mat.sp3 = matrix(c(1,0,0,2,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,10,0,0,100),3,8)
n3 = nrow(mat.sp3)
mat.sp3.norm = decostand(mat.sp3, "norm")

BD3 = sum( apply(mat.sp3.norm,2,var) )
BD3
[1] 1

mat.sp3.chordD = dist(mat.sp3.norm)
BD3 = sum(mat.sp3.chordD^2) / (n3*(n3-1))
BD3
[1] 1

========  Chi-square distance

# Chi-square transformation

mat.sp3 = matrix(c(1,0,0,2,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,10,0,0,100),3,8)
n3 = nrow(mat.sp3)
mat.sp3.chi = decostand(mat.sp3, "chi.sq")

BD3 = sum( apply(mat.sp3.chi,2,var) )
BD3
[1] 26.35135

mat.sp3.chisqD = dist(mat.sp3.chi)
BD3 = sum(mat.sp3.chisqD^2) / (n3*(n3-1))
BD3
[1] 26.35135

========  Bray-Curtis distance

# Bray-Curtis distance
# Beta diversity can only be computed from the distance matrix

library(PCNM)

mat.sp3 = matrix(c(1,0,0,2,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,10,0,0,100),3,8)
n3 = nrow(mat.sp3)
mat.sp3.BC = vegdist(mat.sp3, "bray")

BD3 = sum(mat.sp3.BC^2) / (n3*(n3-1))
BD3
[1] 0.5

-----

# Square root of Bray-Curtis distance
=> Simply obtained by NOT squaring the BC distances before taking the sum

BD3 = sum(mat.sp3.BC) / (n3*(n3-1))
BD3
[1] 0.5

========