Compute dissimilarity indices for ecological data matrices. The dissimilarity
indices computed by this function are those described in Legendre and De
Cáceres (2013). In the name of the function, 'ldc' stands for the author's
names. Twelve of these 21 indices are not readily available in other R
package functions; four of them can, however, be computed in two computation
steps in vegan.
dist.ldc(Y, method = "hellinger", binary = FALSE, samp = TRUE, silent = FALSE)Community composition data. The object class can be either
data.frame or matrix.
One of the 21 dissimilarity coefficients available in the
function: "hellinger", "chord", "log.chord",
"chisquare", "profiles", "percentdiff",
"ruzicka", "divergence", "canberra",
"whittaker", "wishart", "kulczynski",
"jaccard", "sorensen", "ochiai", "ab.jaccard",
"ab.sorensen", "ab.ochiai", "ab.simpson",
"euclidean", "manhattan", "modmeanchardiff". See
Details. Names can be abbreviated to a non-ambiguous set of first letters.
Default: method="hellinger".
If binary=TRUE, the data are transformed to
presence-absence form before computation of the dissimilarities. Default
value: binary=FALSE, except for the Jaccard, Sørensen and Ochiai
indices where binary=TRUE.
If samp=TRUE, the abundance-based distances (ab.jaccard,
ab.sorensen, ab.ochiai, ab.simpson) are computed for sample data. If
samp=FALSE, binary indices are computed for true population data.
If silent=FALSE, informative messages sent to users will
be printed to the R console. Use silent=TRUE is called on a
numerical simulation loop, for example.
A dissimilarity matrix, with class dist.
The dissimilarities computed by this function are the following. Indices i and k designate two rows (sites) of matrix Y, j designates a column (species). D[ik] is the dissimilarity between rows i and k. p is the number of columns (species) in Y; pp is the number of species present in one or the other site, or in both. y[i+] is the sum of values in row i; same for y[k+]. y[+j] is the sum of values in column j. y[++] is the total sum of values in Y. The indices are computed by functions written in C for greater computation speed with large data matrices.
Group 1 - D computed by transformation of Y followed by Euclidean distance
Hellinger D, D[ik] = sqrt(sum((sqrt(y[ij]/y[i+])-sqrt(y[kj]/y[k+]))^2)) (Rao 1995)
chord D, D[ik] = sqrt(sum((y[ij]/sqrt(sum(y[ij]^2))-y[kj]/sqrt(sum(y[kj]^2)))^2)) (Orlóci 1967)
log-chord D, D[ik] = chord D[ik] computed on log(y[ij]+1)-transformed data (Legendre & Borcard 2018)
chi-square D, D[ik] = sqrt(y[++] sum((1/j[+j])(y[ij]/y[i+]-y[kj]/y[k+])^2)) (Lebart & Fénelon 1971)
species profiles D, D[ik] = sqrt(sum((y[ij]/y[i+]-y[kj]/y[k+])^2)) (Legendre & Legendre (2012)
Group 2 - Other D functions appropriate for beta diversity studies where A = sum(min(y[ij],y[kj])), B = y[i+]-A, C = y[k+]-A
percentage difference D (aka Bray-Curtis), D[ik] = (sum(abs(y[ij]-y[k,j])))/(y[i+]+y[k+]) or else, D[ik] = (B+C)/(2A+B+C) (Odum 1950)
Ružička D, D[ik] = 1-(sum(min(y[ij],y[kj])/sum(max(y[ij],y[kj])) or else, D[ik] = (B+C)/(A+B+C) (Ružička 1958)
coefficient of divergence D, D[ik] = sqrt((1/pp)sum(((y[ij]-y(kj])/(y[ij]+y(kj]))^2)) (Clark 1952)
Canberra metric D, D[ik] = (1/pp)sum(abs(y[ij]-y(kj])/(y[ij]+y(kj])) (Lance & Williams 1967)
Whittaker D, D[ik] = 0.5*sum(abs(y[ij]/y[i+]-y(kj]/y[k+])) (Whittaker 1952)
Wishart D, D[ik] = 1-sum(y[ij]y[kj]) /(sum(y[ij]^2)+sum(y[kj]^2)-sum(y[ij]y[kj])) (Wishart 1969)
Kulczynski D, D[ik] = 1-0.5((sum(min(y[ij],y[kj])/y[i+]+sum(min(y[ij],y[kj])/y[k+]))
Group 3 - Classical indices for binary data; they are appropriate for beta diversity studies. Value a is the number of species found in both i and k, b is the number of species in site i not found in k, and c is the number of species found in site k but not in i. The D matrices are square-root transformed, as in dist.binary of ade4; the user-oriented reason for this transformation is explained below.
Jaccard coefficient of community (CC), D[ik] = sqrt((b+c)/(a+b+c)) (Jaccard 1901, CC described in a sentence; 1908, CC shown as an equation)
Sørensen D, D[ik] = sqrt((b+c)/(2a+b+c)) (Sørensen 1948)
Ochiai D, D[ik] = sqrt(1 - a/sqrt((a+b)(a+c))) (Ochiai 1957)
Group 4 - Abundance-based indices of Chao et al. (2006) for
quantitative abundance data. These functions correct the index for species
that have not been observed due to sampling errors. For the meaning of the
U and V notations, see Chao et al. (2006, section 3). When
samp=TRUE, the abundance-based distances (ab.jaccard, ab.sorensen,
ab.ochiai, ab.simpson) are computed for sample data. If samp=FALSE,
indices are computed for true population data. - Do not use indices of
group 4 with samp=TRUE on presence-absence data; the indices are not
meant to accommodate this type of data. If samp=FALSE is used with
presence-absence data, the indices are the regular Jaccard, Sørensen,
Ochiai and Simpson indices. On output, however, the D matrices are not
square-rooted, contrary to the Jaccard, Sørensen and Ochiai indices in
section 3 which are square-rooted.
abundance-based Jaccard D, D[ik] = 1-(UV/(U+V-UV))
abundance-based Sørensen D, D[ik] = 1-(2UV/(U+V))
abundance-based Ochiai D, D[ik] = 1-sqrt(UV)
abundance-based Simpson D, D[ik] = 1-(UV/(UV+min((U-UV),(V-UV))))
Group 5 - General-purpose dissimilarities that do not have an upper bound (maximum D value). They are inappropriate for beta diversity studies.
Euclidean D, D[ik] = sqrt(sum(y[ij]-y[kj])^2)
Manhattan D, D[ik] = sum(abs(y[ij] - y[ik]))
modified mean character difference, D[ik] = (1/pp) sum(abs(y[ij] - y[ik])) (Legendre & Legendre (2012)
The properties of
all dissimilarities available in this function (except Ružička D) were
described and compared in Legendre & De Cáceres (2013), who showed that
most of these dissimilarities are appropriate for beta diversity studies.
Inappropriate are the Euclidean, Manhattan, modified mean character
difference, species profile and chi-square distances. Most of these
dissimilarities have a maximum value of either 1 or sqrt(2). Three
dissimilarities (Euclidean, Manhattan, Modified mean character difference)
do not have an upper bound and are thus inappropriate for beta diversity
studies. The chi-square distance has an upper bound of sqrt(2*(sum(Y))).
The Euclidean, Hellinger, chord, chi-square and
species profiles dissimilarities have the property of being Euclidean,
meaning that they never produce negative eigenvalues in principal
coordinate analysis. The Canberra, Whittaker, percentage difference,
Wishart and Manhattan coefficients are Euclidean when they are square-root
transformed (Legendre & De Cáceres 2013, Table 2). The distance forms (1-S)
of the Jaccard, Sørensen and Ochiai similarity (S) coefficients are
Euclidean after taking the square root of (1-S) (Legendre & Legendre 2012,
Table 7.2). The D matrices resulting from these three coefficients are
outputted in the form sqrt(1-S), as in function dist.binary of ade4,
because that form is Euclidean and will thus produce no negative
eigenvalues in principal coordinate analysis.
The Hellinger, chord,
chi-square and species profile dissimilarities are computed using the
two-step procedure developed by Legendre & Gallagher (2001). The data are
first transformed using either the row marginals, or the row and column
marginals in the case of the chi-square distance. The dissimilarities are
then computed from the transformed data using the Euclidean distance
formula. As a consequence, these four dissimilarities are necessarily
Euclidean. D matrices for other binary coefficients can be computed in two
ways: either by using function dist.binary of ade4, or by choosing
option binary=TRUE, which transforms the abundance data to binary
form, and using one of the quantitative indices of the present function.
Table 1 of Legendre & De Cáceres (2013) shows the incidence-based
(presence-absence-based) indices computed by the various indices using
binary data.
The Euclidean distance computed on untransformed
presence-absence or abundance data produces non-informative and incorrect
ordinations, as shown in Legendre & Legendre (2012, p. 300) and in Legendre
& De Cáceres (2013). However, the Euclidean distance computed on
log-transformed abundance data produces meaningful ordinations in principal
coordinate analysis (PCoA). Nonetheless, it is easier to compute a PCA of
log-transformed abundance data instead of a PCoA; the resulting ordination
with scaling 1 will be meaningful. Messages are printed to the R console
indicating the Euclidean status of the computed dissimilarity matrices.
Note that for the chi-square distance, the columns that sum to zero are
eliminated before calculation of the distances, thus preventing divisions
by zero in the calculation of the chi-square transformation.
Chao, A., R.L. Chazdon, R.K. Colwell and T.J. Shen. 2006. Abundance-based similarity indices and their estimation when there are unseen species in samples. Biometrics 62: 361-371.
Clark, P. J. 1952. An extension of the coefficient of divergence for use with multiple characters. Copeia 1952: 61-64.
Jaccard, P. 1901. Étude comparative de la distribution florale dans dune portiones Alpes et du Jura. Bulletin de la Société vaudoise des sciences naturelles 37: 547-579.
Jaccard, P. 1908. Nouvelles recherches sur la distribution florale. Bulletin de la Société vaudoise des sciences naturelles 44: 223-270.
Kulczynski, S. 1928. Die Pflanzenassoziationen der Pieninen. Bull. Int. Acad. Pol. Sci. Lett. Cl. Sci. Math. Nat. Ser. B, Suppl. II (1927): 57–203.
Lance, G. N. and W. T. Williams. 1967. Mixed-data classificatory programs. I. Agglomerative systems. Australian Computer Journal 1: 15-20.
Lebart, L. and J. P. Fénelon. 1971. Statistique et informatique appliquées. Dunod, Paris.
Legendre, P. and D. Borcard. 2018. Box-Cox-chord transformations for community composition data prior to beta diversity analysis. Ecography 41: 1820-1824.
Legendre, P. and M. De Cáceres. 2013. Beta diversity as the variance of community data: dissimilarity coefficients and partitioning. Ecology Letters 16: 951-963.
Legendre, P. and E. D. Gallagher, E.D. 2001. Ecologically meaningful transformations for ordination of species data. Oecologia 129: 271-280.
Legendre, P. and Legendre, L. 2012. Numerical Ecology, 3rd English edition. Elsevier Science BV, Amsterdam.
Ochiai, A. 1957. Zoogeographic studies on the soleoid fishes found in Japan and its neighbouring regions–II. Bulletin of the Japanese Society of Scientific Fisheries 22: 526-530.
Odum, E. P. 1950. Bird populations of the Highlands (North Carolina) plateau in relation to plant succession and avian invasion. Ecology 31: 587-605.
Orlóci, L. 1967. An agglomerative method for classification of plant communities. Journal of Ecology 55: 193-206.
Rao, C. R. 1995. A review of canonical coordinates and an alternative to correspondence analysis using Hellinger distance. Qüestiió (Quaderns d’Estadística i Investigació Operativa) 19: 23-63.
Ružička, Milan. 1958. Anwendung mathematisch-statistiker Methoden in Geobotanik (synthetische Bearbeitung von Aufnahmen). Biologia (Bratislava) 13: 647–661.
Sørensen, T. 1948. A method of establishing groups of equal amplitude in plant sociology based on similarity of species content and its application to analysis of the vegetation on Danish commons. Biologiske Skrifter 5: 1-34.
Whittaker, R. H. 1952. A study of summer foliage insect communities in the Great Smoky Mountains. Ecological Monographs 22: 1–44.
Wishart, D. (1969). CLUSTAN 1a User Manual. Computing Laboratory, University of St. Andrews, St. Andrews, Fife, Scotland.
if(require("vegan", quietly = TRUE)) {
data(mite)
mat1 = as.matrix(mite[1:10, 1:15]) # No column has a sum of 0
mat2 = as.matrix(mite[61:70, 1:15]) # 7 of the 15 columns have a sum of 0
#Example 1: compute Hellinger distance for mat1
D.out = dist.ldc(mat1,"hellinger")
#Example 2: compute chi-square distance for mat2
D.out = dist.ldc(mat2,"chisquare")
#Example 3: compute percentage difference dissimilarity for mat2
D.out = dist.ldc(mat2,"percentdiff")
}
#> Info -- This coefficient is Euclidean
#> Info -- This coefficient is Euclidean
#> Info -- For this coefficient, sqrt(D) would be Euclidean