Multiple assemblage dissimilarity for orders q = 0N.
dissCqN( mat, q = 0:2, pairwise = FALSE, compare.sub = NULL, shared.spp = FALSE, parallel = "no", ncpus = NULL, cl = NULL )
mat  A matrix with assemblages in rows and species or species interactions in columns. Alternatively, a list of matrices, which will be interpreted as interaction networks and used to construct an assemblage x interaction matrix. 

q  Integer, the order(s) of q for which to calculate dissimilarity.
Can be any set of integers between 0 and N (the number of assemblages in

pairwise  Logical, whether to calculate pairwise, rather than multiple assemblage, dissimilarity. 
compare.sub  Subsets of assemblages to compare pairwise. These should
be supplied as a list of two sets of assemblage names or indices. If only
one set is supplied, this is compared to all other assemblages in 
shared.spp  Logical, whether to compare networks of shared species only
(if 
parallel  The type of parallel processing to use, if any. Can be one of

ncpus  Number of system cores to use for parallel processing. If 
cl  Optional cluster to use if 
A numeric vector of dissimilarities, or a pairwise dissimilarity
matrix (or list of matrices), for the orders of q
.
Dissimilarity is calculated here for multiple species assemblages
(or interaction networks) via the CqN generalisation of similarity
indices (Chao et al., 2008; Jost et al., 2011). Increasing the value of q
increases the 'depth' of the measure, that is, how much emphasis is placed
on changes in relative abundance of the most common species. Setting q = 0
represents the qualitative Sørensen index (Sørensen, 1948), where rare
and common species are treated equally. q
> 0 is more sensitive to common
species, with q = 1
representing the Shannonbased Horn index (Horn,
1966) and q = 2
the Simpsonbased MorisitaHorn index (Horn, 1966;
Morisita, 1959). For N > 2, indices are generalised to consider species
shared across multiple assemblages (Diserud & Ødegaard, 2007; eqns. 6.36.5
in Jost et al., 2011). For q
>= 2 <= N, common species increasingly
dominate the measure, and it can then be interpreted as the ratio of two
probabilities of randomly sampling q
individuals of the same species from
the N assemblages, where 1) the individuals came from at least one
different assemblage (\(^{q}G_{D}\)) and 2) they all came from the
same assemblage (\(^{q}G_{S}\)) (Jost et al., 2011). Dissimilarity
is thus:
$$1  ^{q}G_{D} / ^{q}G_{S}$$
Pairwise dissimilarity can be calculated for all or a subset of the
assemblages (or networks) in mat
, in which case a dissimilarity matrix is
returned (one for each value of q
). If comparing subsets, the names or
indices of assemblages to compare should be supplied to compare.sub
. Note
that pairwise calculation may take a long time if N is large, in which
case parallel processing may speed up results (e.g. parallel = "snow"
).
If shared.spp = TRUE
and mat
is a list of interaction networks (as
matrices), multiple or pairwise interaction dissimilarity will be
calculated for networks of shared species only (see netShared()
). This
can be useful to help partition the different components of network
dissimilarity, e.g. dissimilarity due to interaction 'rewiring' among
shared species vs. that due to species turnover (Poisot et al., 2012).
Chao, A., Jost, L., Chiang, S. C., Jiang, Y.H., & Chazdon, R. L. (2008). A TwoStage Probabilistic Approach to MultipleCommunity Similarity Indices. Biometrics, 64(4), 1178–1186. doi: 10/fcvn63
Diserud, O. H., & Ødegaard, F. (2007). A multiplesite similarity measure. Biology Letters, 3(1), 20–22. doi: 10/bwhfx6
Horn, H. S. (1966). Measurement of “Overlap” in Comparative Ecological Studies. The American Naturalist, 100(914), 419–424. doi: 10/b62ct5
Jost, L., Chao, A., & Chazdon, R. L. (2011). Compositional similarity and beta diversity. In A. E. Magurran & B. J. McGill (Eds.), Biological Diversity: Frontiers in Measurement and Assessment (pp. 66–84). Oxford University Press.
Morisita, M. (1959). Measuring of interspecific association and similarity between communities. Memoirs of the Faculty of Science, Kyushu Univ., Series E (Biology), 3, 65–80.
Poisot, T., Canard, E., Mouillot, D., Mouquet, N., & Gravel, D. (2012). The dissimilarity of species interaction networks. Ecology Letters, 15(12), 1353–1361. doi: 10/f4dv37
Sørensen, T. (1948). A method of establishing groups of equal amplitude in plant sociology based on similarity of species and its application to analyses of the vegetation on Danish commons. Kongelige Danske Videnskabernes Selskabs Biologiske Skrifter, 5, 1–34.
# Sample community data from SpadeR package (three assemblages, 120 species) data(SimilarityMultData, package = "SpadeR") d < SimilarityMultData$Abu # Multipleassemblage dissimilarity for q = 0:2 (CqN < dissCqN::dissCqN(t(d)))#> C0N C1N C2N #> 0.3411215 0.3479117 0.4198004# Compare to empirical CqN values from SpadeR::SimilarityMult() sim < SpadeR::SimilarityMult(d, datatype = "abundance", nboot = 1) CqN_2 < 1  c( "C0N" = sim$Empirical_richness["C0N(q=0,Sorensen)", "Estimate"], "C1N" = sim$Empirical_relative["C1N=U1N(q=1,Horn)", "Estimate"], "C2N" = sim$Empirical_relative["C2N(q=2,Morisita)", "Estimate"] ) stopifnot(all.equal(CqN, CqN_2)) # Pairwise dissimilarity matrices dissCqN::dissCqN(t(d), pairwise = TRUE)#> $C0N #> V1 V2 V3 #> V1 0.0000000 0.3099415 0.5357143 #> V2 0.3099415 0.0000000 0.5586207 #> V3 0.5357143 0.5586207 0.0000000 #> #> $C1N #> V1 V2 V3 #> V1 0.0000000 0.3193235 0.4497502 #> V2 0.3193235 0.0000000 0.4222305 #> V3 0.4497502 0.4222305 0.0000000 #> #> $C2N #> V1 V2 V3 #> V1 0.0000000 0.5413686 0.2558286 #> V2 0.5413686 0.0000000 0.5546613 #> V3 0.2558286 0.5546613 0.0000000 #>