Design your own Dissimilarities

Function designdist lets you define your own dissimilarities using terms for shared and total quantities, number of rows and number of columns. The shared and total quantities can be binary, quadratic or minimum terms. In binary terms, the shared component is number of shared species, and totals are numbers of species on sites. The quadratic terms are cross-products and sums of squares, and minimum terms are sums of parallel minima and row totals. Function chaodist lets you define your own dissimilarities using terms that are supposed to take into account the “unseen species” (see Chao et al., 2005 and Details in vegdist).

designdist(x, method = "(A+B-2*J)/(A+B)",
           terms = c("binary", "quadratic", "minimum"),
           abcd = FALSE, alphagamma = FALSE, name)
chaodist(x, method = "1 - 2*U*V/(U+V)", name)

Arguments

x

Input data.
method

Equation for your dissimilarities. This can use terms J for shared quantity, A and B for totals, N for the number of rows (sites) and P for the number of columns (species) or in chaodist it can use terms U and V. The equation can also contain any R functions that accepts vector arguments and returns vectors of the same length.
terms

How shared and total components are found. For vectors x and y the "quadratic" terms are J = sum(x*y), A = sum(x^2), B = sum(y^2), and "minimum" terms are J = sum(pmin(x,y)), A = sum(x) and B = sum(y), and "binary" terms are either of these after transforming data into binary form (shared number of species, and number of species for each row).
abcd

Use 2x2 contingency table notation for binary data: \(a\) is the number of shared species, \(b\) and \(c\) are the numbers of species occurring only one of the sites but not in both, and \(d\) is the number of species that occur on neither of the sites.
alphagamma

Use beta diversity notation with terms alpha for average alpha diversity for compared sites, gamma for diversity in pooled sites, and delta for the absolute value of difference of average alpha and alpha diversities of compared sites. Terms A and B refer to alpha diversities of compared sites.
name

The name you want to use for your index. The default is to combine the method equation and terms argument.

Details

Most popular dissimilarity measures in ecology can be expressed with the help of terms J, A and B, and some also involve matrix dimensions N and P. Some examples you can define in designdist are:

A+B-2*J"quadratic"squared Euclidean
A+B-2*J"minimum"Manhattan
(A+B-2*J)/(A+B)"minimum"Bray-Curtis
(A+B-2*J)/(A+B)"binary"Sørensen
(A+B-2*J)/(A+B-J)"binary"Jaccard
(A+B-2*J)/(A+B-J)"minimum"Ružička
(A+B-2*J)/(A+B-J)"quadratic"(dis)similarity ratio
1-J/sqrt(A*B)"binary"Ochiai
1-J/sqrt(A*B)"quadratic"cosine complement
1-phyper(J-1, A, P-A, B)"binary"Raup-Crick (but see raupcrick)

The function designdist can implement most dissimilarity indices in vegdist or elsewhere, and it can also be used to implement many other indices, amongst them, most of those described in Legendre & Legendre (2012). It can also be used to implement all indices of beta diversity described in Koleff et al. (2003), but there also is a specific function betadiver for the purpose.

If you want to implement binary dissimilarities based on the 2x2 contingency table notation, you can set abcd = TRUE. In this notation a = J, b = A-J, c = B-J, d = P-A-B+J. This notation is often used instead of the more more tangible default notation for reasons that are opaque to me.

With alphagamma = TRUE it is possible to use beta diversity notation with terms alpha for average alpha diversity and gamma for gamma diversity in two compared sites. The terms are calculated as alpha = (A+B)/2, gamma = A+B-J and delta = abs(A-B)/2. Terms A and B are also available and give the alpha diversities of the individual compared sites. The beta diversity terms may make sense only for binary terms (so that diversities are expressed in numbers of species), but they are calculated for quadratic and minimum terms as well (with a warning).

Function chaodist is similar to designgist, but uses terms U and V of Chao et al. (2005). These terms are supposed to take into account the effects of unseen species. Both U and V are scaled to range \(0 \dots 1\). They take the place of A and B and the product U*V is used in the place of J of designdist. Function chaodist can implement any commonly used Chao et al. (2005) style dissimilarity:

1 - 2*U*V/(U+V)Sørensen type
1 - U*V/(U+V-U*V)Jaccard type
1 - sqrt(U*V)Ochiai type
(pmin(U,V) - U*V)/pmin(U,V)Simpson type

Function vegdist implements Jaccard-type Chao distance, and its documentation contains more complete discussion on the calculation of the terms.

Value

designdist returns an object of class dist.

References

Chao, A., Chazdon, R. L., Colwell, R. K. and Shen, T. (2005) A new statistical approach for assessing similarity of species composition with incidence and abundance data. Ecology Letters 8, 148--159.

Koleff, P., Gaston, K.J. and Lennon, J.J. (2003) Measuring beta diversity for presence--absence data. J. Animal Ecol. 72, 367--382.

Legendre, P. and Legendre, L. (2012) Numerical Ecology. 3rd English ed. Elsevier

Note

designdist does not use compiled code, but it is based on vectorized R code. The designdist function can be much faster than vegdist, although the latter uses compiled code. However, designdist cannot skip missing values and uses much more memory during calculations.

The use of sum terms can be numerically unstable. In particularly, when these terms are large, the precision may be lost. The risk is large when the number of columns is high, and particularly large with quadratic terms. For precise calculations it is better to use functions like dist and vegdist which are more robust against numerical problems.

See also

vegdist, betadiver, dist, raupcrick.

Examples

data(BCI)
## Four ways of calculating the same Sørensen dissimilarity
d0 <- vegdist(BCI, "bray", binary = TRUE)
d1 <- designdist(BCI, "(A+B-2*J)/(A+B)")
d2 <- designdist(BCI, "(b+c)/(2*a+b+c)", abcd = TRUE)
d3 <- designdist(BCI, "gamma/alpha - 1", alphagamma = TRUE)
## Arrhenius dissimilarity: the value of z in the species-area model
## S = c*A^z when combining two sites of equal areas, where S is the
## number of species, A is the area, and c and z are model parameters.
## The A below is not the area (which cancels out), but number of
## species in one of the sites, as defined in designdist().
dis <- designdist(BCI, "(log(A+B-J)-log(A+B)+log(2))/log(2)")
## This can be used in clustering or ordination...
ordiplot(cmdscale(dis))
#> species scores not available
## ... or in analysing beta diversity (without gradients)
summary(dis)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  0.2733  0.3895  0.4192  0.4213  0.4537  0.5906