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fundiversity lets you compute functional diversity indices. Currently it can compute five indices:

  • functional richness (FRic),
  • functional volume intersections (FRic_intersect),
  • functional divergence (FDiv),
  • functional evenness (FEve),
  • functional dispersion (FDis)
  • Rao’s quadratic entropy (Q).

This vignette will introduce you to the data needed as well as how to compute and interpret each index.

Required data

To compute functional diversity indices, you will need at least a dataset describing species traits, i.e. species characteristics. Note that we here talk about species but the reasoning could apply on whatever unit you’re interested in whether it’s individual organisms, ecological plots, or even entire ecosystems. The traits are the features that describe these units.

fundiversity comes with one example trait dataset. The dataset comes from Nowak, Kissling, Bender, Dehling, Töpfer, Böhning-Gaese, et al. (2019) and describe the traits of birds and plants along a tropical gradient (Nowak, Kissling, Bender, Dehling, Töpfer, Böhning-Gaese, et al. 2019). You can see the datasets available in fundiversity using the data() function:

data(package = "fundiversity")

To load them use their names into the data() function:

data("traits_birds", package = "fundiversity")

head(traits_birds)
#>                      Bill.width..mm. Bill.length..mm. Kipp.s.index Bodymass..g.
#> Aburria_aburri                 18.35            35.48         0.18       1407.5
#> Amazona_farinosa               26.50            38.81         0.29        626.0
#> Amazona_mercenaria             17.51            26.30         0.33        340.0
#> Amazona_ochrocephala           20.17            31.40         0.26        440.0
#> Ampelioides_tschudii           16.53            24.58         0.24         78.4
#> Ampelion_rufaxilla             16.97            21.89         0.28         73.9

data("traits_plants", package = "fundiversity")

head(traits_plants)
#>                        Fruit.length..mm. Fruit.width..mm. Plant.height..m.
#> Abuta_grandifolia                  17.48             9.38             6.00
#> Alchornea_latifolia                 6.82             8.23            11.53
#> Alchornea_triplinervia              8.18            10.26            14.72
#> Allophylus_punctatus               13.58            12.49             4.40
#> Alnus_acuminata                    19.32            11.48            17.54
#> Aniba_guianensis                   15.65            13.43             8.00
#>                        Crop.mass..g.
#> Abuta_grandifolia             341.25
#> Alchornea_latifolia          1290.48
#> Alchornea_triplinervia       1156.67
#> Allophylus_punctatus           33.34
#> Alnus_acuminata              5009.33
#> Aniba_guianensis               25.50

Note that in these datasets the species are shown in rows, with species names as row names, and traits are in columns.

Functional diversity indices are generally computed at different locations that we hereafter call sites. We thus need a description of which species is in which site in the form of a site-species matrix. Again, we’re calling it a site-species matrix but the granularity of both your “species” and “site” units can vary depending on what you want to compute functional diversity on.

fundiversity contains the corresponding site-species matrices to the above-mentioned trait dataset (Nowak, Kissling, Bender, Dehling, Töpfer, Böhning-Gaese, et al. 2019):

# Site-species matrix for birds
data("site_sp_birds", package = "fundiversity")

head(site_sp_birds)[, 1:5]
#>           Aburria_aburri Amazona_farinosa Amazona_mercenaria
#> elev_250               0                1                  0
#> elev_500               0                1                  0
#> elev_1000              1                1                  1
#> elev_1500              1                0                  1
#> elev_2000              0                0                  1
#> elev_2500              0                0                  1
#>           Amazona_ochrocephala Ampelioides_tschudii
#> elev_250                     1                    0
#> elev_500                     1                    0
#> elev_1000                    0                    1
#> elev_1500                    0                    0
#> elev_2000                    0                    0
#> elev_2500                    0                    0

# Site-species matrix for plants
data("site_sp_plants", package = "fundiversity")

head(site_sp_plants)[, 1:5]
#>           Abuta_grandifolia Alchornea_latifolia Alchornea_triplinervia
#> elev_250                  1                   1                      1
#> elev_500                  1                   1                      1
#> elev_1000                 0                   0                      1
#> elev_1500                 0                   0                      1
#> elev_2000                 0                   0                      1
#> elev_2500                 0                   0                      1
#>           Allophylus_punctatus Alnus_acuminata
#> elev_250                     1               1
#> elev_500                     1               1
#> elev_1000                    1               1
#> elev_1500                    1               1
#> elev_2000                    1               1
#> elev_2500                    1               1

The site species matrix represent the presence of a given species (in column) in a given site (in row), similar to the format used in the vegan package. Here the site-species matrix contains only 0 (absence) and 1 (presence), but fundiversity can also use matrices that contain abundances for some functional diversity indices (FDiv and Q).

To ensure the good computation of functional diversity indices, at least some of species names (row names) in the trait data need to be in to the column names of the site species matrix:

# Fewer species in trait dataset than species in the site-species matrix
fd_fric(traits_birds[2:217,], site_sp_birds)
#> Differing number of species between trait dataset and site-species matrix
#> Taking subset of species
#>        site       FRic
#> 1  elev_250 171543.730
#> 2  elev_500 185612.548
#> 3 elev_1000 109615.330
#> 4 elev_1500  63992.817
#> 5 elev_2000  20065.764
#> 6 elev_2500  18301.176
#> 7 elev_3000  17530.651
#> 8 elev_3500   3708.735

# Fewer species in the site-species matrix than in the traits
fd_fric(traits_birds, site_sp_birds[, 1:5])
#> Differing number of species between trait dataset and site-species matrix
#> Taking subset of species
#>        site FRic
#> 1  elev_250   NA
#> 2  elev_500   NA
#> 3 elev_1000   NA
#> 4 elev_1500   NA
#> 5 elev_2000   NA
#> 6 elev_2500   NA
#> 7 elev_3000   NA
#> 8 elev_3500   NA

# No species in common between both dataset
fd_fric(traits_birds[1:5,], site_sp_birds[, 6:10])
#> Error: No species in common found between trait dataset and site-species matrix

Functional Richness (FRic) - fd_fric()

Functional Richness (FRic) represents the total amount of functional space filed by a community in a dataset (Villéger, Mason, and Mouillot 2008). You can compute FRic in fundiversity using the fd_fric() function.

For a single trait range FRic is the range of trait observed in the dataset:

# Range of bill width in the birds dataset
diff(range(traits_birds[, "Bill.width..mm."]))
#> [1] 33.64

# Using fundiversity::fd_fric()
fd_fric(traits_birds)
#>   site     FRic
#> 1   s1 230967.7

The first column site describes the site on which FRic has been computed while the FRic column contains the computed FRic values. If no site-species matrix has been provided the site is named by default s1.

For multiple traits, FRic can be thought as a multi-dimensional range which is computed as the convex hull volume of the considered species (Villéger, Mason, and Mouillot 2008):

fd_fric(traits_birds)
#>   site     FRic
#> 1   s1 230967.7

If you provide only a trait dataset without specifying site-species matrix fd_fric() computes FRic on the full trait dataset. You can compute FRic values for different sites by providing both a trait dataset and a site-species matrix to fd_fric():

fd_fric(traits_birds, site_sp_birds)
#>        site       FRic
#> 1  elev_250 171543.730
#> 2  elev_500 185612.548
#> 3 elev_1000 112600.176
#> 4 elev_1500  66142.748
#> 5 elev_2000  20065.764
#> 6 elev_2500  18301.176
#> 7 elev_3000  17530.651
#> 8 elev_3500   3708.735

Because the convex hull volume depends on the number and the units of the traits used, it is difficult to compare across datasets, that is why it has been suggested to standardize its value by the total volume comprising all species in the dataset (Villéger, Mason, and Mouillot 2008):

fd_fric(traits_birds, stand = TRUE)
#>   site FRic
#> 1   s1    1

The newly computed FRic values will then be comprised between 0 and 1. It is especially useful when comparing different sites:

fd_fric(traits_birds, site_sp_birds, stand = TRUE)
#>        site       FRic
#> 1  elev_250 0.74271733
#> 2  elev_500 0.80362981
#> 3 elev_1000 0.48751477
#> 4 elev_1500 0.28637225
#> 5 elev_2000 0.08687692
#> 6 elev_2500 0.07923694
#> 7 elev_3000 0.07590087
#> 8 elev_3500 0.01605737

Each row gives the standardized FRic values of each site.

Parallelization. The computation of this function can be parallelized thanks to the future package. Refer to the parallelization vignette to get more information about how to do so.

Memoization. By default, when loading fundiversity, the functions to compute convex hulls are memoised through the memoise package if it is installed. It means that repeated calls to fd_fric() with similar arguments won’t be recomputed each time but recovered from memory. To deactivate this behavior you can set the option fundiversity.memoise to FALSE by running the following line: options(fundiversity.memoise = FALSE). If you use it interactively it will only affect your current session. Add it to your script(s) or .Rprofile file to avoid toggling it each time.

Functional volume intersect (FRic_intersect) - fd_fric_intersect()

Sometimes you’re interested in the shared functional volumes between pairs of sites more than in the functional volumes of each site separately. fundiversity provides the fd_fric_intersect() function for this exact use case.

It follows the same interface as fd_fric() with similar named arguments:

fd_fric_intersect(traits_birds)
#>   first_site second_site FRic_intersect
#> 1         s1          s1       230967.7

fd_fric_intersect() computes the shared functional volumes between each pair of sites, including self-intersection which correspond to the functional volume of each site. Similarly to fd_fric() if no site-species data is provided, fd_fric_intersect() considers a site that contains all species from the trait dataset.

fd_fric_intersect(traits_birds, site_sp_birds[1:2,])
#>   first_site second_site FRic_intersect
#> 1   elev_250    elev_500       171532.6
#> 2   elev_250    elev_250       171543.7
#> 3   elev_500    elev_500       185612.5

The output is a data.frame where the two first columns (first_site and second_site) define the sites on which the intersection is computed, the third column (FRic_intersect) contains the volume of the intersection.

Similarly to fd_fric() the intersections volumes can be standardized:

fd_fric_intersect(traits_birds, site_sp_birds[1:2,], stand = TRUE)
#>   first_site second_site FRic_intersect
#> 1   elev_250    elev_500      0.7426689
#> 2   elev_250    elev_250      0.7427173
#> 3   elev_500    elev_500      0.8036298

Note that when standardizing the volumes, the behavior is similar to that of fd_fric() which means the function considers the total volume occupied by provided trait values, even if they are absent from all sites, this can lead to standardized self-intersection volumes lower than one.

Parallelization. The computation of this function can be parallelized thanks to the future package. Refer to the parallelization vignette to get more information about how to do so.

Memoization. By default, when loading fundiversity, the functions to compute convex hulls are memoised through the memoise package if it is installed. It means that repeated calls to fd_fric_intersect() with similar arguments won’t be recomputed each time but recovered from memory. To deactivate this behavior you can set the option fundiversity.memoise to FALSE by running the following line: options(fundiversity.memoise = FALSE). If you use it interactively it will only affect your current session. Add it to your script(s) or .Rprofile file to avoid toggling it each time.

Functional Divergence (FDiv) - fd_fdiv()

Functional Divergence (FDiv) represents how abundance is spread along the different traits (Villéger, Mason, and Mouillot 2008). When a species with extreme trait values has the highest abundance, then functional divergence is high.

Use the fd_fdiv() function to compute functional divergence:

# One-dimension FDiv
fd_fdiv(traits_birds[, 1, drop = FALSE])
#>   site      FDiv
#> 1   s1 0.7490732

# Multiple dimension FDiv
fd_fdiv(traits_birds)
#>   site      FDiv
#> 1   s1 0.7282172

When no site-species matrix is provided, FDiv is computed by default considering all the species together. If you provide a site-species matrix, then FDiv is computed across all sites:

fd_fdiv(traits_birds, site_sp_birds)
#>        site      FDiv
#> 1  elev_250 0.6847251
#> 2  elev_500 0.6937866
#> 3 elev_1000 0.7056772
#> 4 elev_1500 0.7269801
#> 5 elev_2000 0.7509511
#> 6 elev_2500 0.6985280
#> 7 elev_3000 0.6627204
#> 8 elev_3500 0.6422068

Similarly to FRic, if the included species differ between the site-species matrix and the trait dataset, fd_fdiv() will take the common subset of species.

The computation of this function can be parallelized thanks to the future package. Refer to the parallelization vignette to get more information about how to do so.

Functional Evenness (FEve) - fd_feve()

Functional Evenness (FEve) describes the regularity of the distribution of species (and their abundances) in trait space (Villéger, Mason, and Mouillot 2008). FEve is bounded between 0 and 1. FEve is close to 0 when most species (and abundances) are tightly packed in a portion of the trait space while it is close to 1 if species are regularly spread (with even abundances) along the trait space.

Use the fd_fdiv() function to compute functional divergence:

# One-dimension FEve
fd_feve(traits_birds[, 1, drop = FALSE])
#>   site      FEve
#> 1   s1 0.4454885

# Multiple dimension FEve
fd_feve(traits_birds)
#>   site      FEve
#> 1   s1 0.3743341

When no site-species matrix is provided, FEve is computed by default considering all the species together. If you provide a site-species matrix, then FEve is computed across all sites:

fd_feve(traits_birds, site_sp_birds)
#>        site      FEve
#> 1  elev_250 0.3841202
#> 2  elev_500 0.3846186
#> 3 elev_1000 0.3426688
#> 4 elev_1500 0.2965585
#> 5 elev_2000 0.3523994
#> 6 elev_2500 0.3552671
#> 7 elev_3000 0.3492529
#> 8 elev_3500 0.4222442

Similarly to FRic, if the included species differ between the site-species matrix and the trait dataset, fd_feve() will take the common subset of species.

The computation of this function can be parallelized thanks to the future package. Refer to the parallelization vignette to get more information about how to do so.

Memoization. By default, when loading fundiversity, the functions to compute convex hulls are memoised through the memoise package if it is installed. It means that repeated calls to fd_fdiv() with similar arguments won’t be entirely recomputed each time but recovered from memory. To deactivate this behavior you can set the option fundiversity.memoise to FALSE by running the following line: options(fundiversity.memoise = FALSE). If you use it interactively it will only affect your current session. Add it to your script(s) or .Rprofile file to avoid toggling it each time.

Functional Dispersion (FDis) - fd_fdis()

Functional Dispersion reflects changes in the abundance-weighted deviation of species trait values from the center of the functional space.

You can compute Functional Dispersion (FDis) using the fd_fdis() function by providing a trait dataset:

fd_fdis(traits_birds)
#>   site     FDis
#> 1   s1 146.2072

If you don’t provide a site-species matrix, fd_fdis() considers all species provided in the trait dataset present at equal abundances in the same site. You can also provide a site-species matrix to compute FDis at different sites:

fd_fdis(traits_birds, site_sp_birds)
#>        site      FDis
#> 1  elev_250 165.58561
#> 2  elev_500 168.34567
#> 3 elev_1000 173.92101
#> 4 elev_1500 156.19006
#> 5 elev_2000  86.39285
#> 6 elev_2500  88.28895
#> 7 elev_3000  97.11743
#> 8 elev_3500  80.70000

The computation of this function can be parallelized thanks to the future package. Refer to the parallelization vignette to get more information about how to do so.

Rao’s Quadratic Entropy (Q) - fd_raoq()

Rao’s Quadratic entropy assesses the multi-dimensional divergence in trait space (Rao 1982). It is the abundance-weighted variance of the trait dissimilarities between all species pairs.

You can compute Rao’s Quadratic entropy (Q) using the fd_raoq() function by providing a trait dataset:

fd_raoq(traits_birds)
#>   site        Q
#> 1   s1 170.0519

If you don’t provide a site-species matrix, fd_raoq() considers all species provided in the trait dataset present at equal abundances in the same site. You can also provide a site-species matrix to compute Q at different sites:

fd_raoq(traits_birds, site_sp_birds)
#>        site         Q
#> 1  elev_250 194.78095
#> 2  elev_500 197.08184
#> 3 elev_1000 200.52231
#> 4 elev_1500 178.24801
#> 5 elev_2000  97.32416
#> 6 elev_2500 102.22461
#> 7 elev_3000 113.22049
#> 8 elev_3500  87.04750

Because the computation of Rao’s quadratic entropy requires dissimilarities between all pair of species in the dataset, if you provide a trait dataset fd_raoq(), the function will compute the Euclidean distance between all pairs of species. If you wish to directly provide species dissimilarities, you can do so through the dist_matrix argument:

# Compute dissimilarity between species with the Manhattan distance
trait_dissim <- dist(traits_birds, method = "manhattan")

fd_raoq(dist_matrix = trait_dissim)
#>   site       Q
#> 1   s1 190.589

fd_raoq(sp_com = site_sp_birds, dist_matrix = as.matrix(trait_dissim))
#>        site        Q
#> 1  elev_250 218.3636
#> 2  elev_500 220.8257
#> 3 elev_1000 220.0048
#> 4 elev_1500 196.6785
#> 5 elev_2000 112.6211
#> 6 elev_2500 117.6497
#> 7 elev_3000 127.5911
#> 8 elev_3500 104.8981

NB: if you want to provide both a site-species matrix and a trait dissimilarity matrix please specify explicitly the arguments names.

Large site-species data / sparse matrices

Sparse matrices are memory efficient ways of storing matrix object that contains many zeros. fundiversity is fully compatible with sparse matrices through the Matrix package. They can be used to encode site-species information or distance matrices.

Provide Matrix objects as inputs of the indices function fundiversity, they will transparently use them for efficient computation.

# Convert site-species matrix to sparse matrix
sparse_site_sp <- Matrix::Matrix(site_sp_birds, sparse = TRUE)

fd_raoq(traits_birds, site_sp_birds)
#>        site         Q
#> 1  elev_250 194.78095
#> 2  elev_500 197.08184
#> 3 elev_1000 200.52231
#> 4 elev_1500 178.24801
#> 5 elev_2000  97.32416
#> 6 elev_2500 102.22461
#> 7 elev_3000 113.22049
#> 8 elev_3500  87.04750

Standardizing trait data

fundiversity does not perform any transformation on the input trait or dissimilarity data. In fd_raoq() if you provide only continuous trait data then the function will attempt computing Euclidean distance between the species.

In order to get comparable functional diversity indices you can standardize the trait data. One option would be to consider the scale() function to scale each continuous trait with a mean of zero and a standard deviation of one (z-score). Each trait will then have the same importance when computing functional diversity indices:

traits_birds_sc <- scale(traits_birds)
summary(traits_birds_sc)
#>  Bill.width..mm.   Bill.length..mm.    Kipp.s.index      Bodymass..g.     
#>  Min.   :-1.1854   Min.   :-0.78608   Min.   :-1.8545   Min.   :-0.49301  
#>  1st Qu.:-0.7330   1st Qu.:-0.52857   1st Qu.:-0.7432   1st Qu.:-0.44701  
#>  Median :-0.2974   Median :-0.28257   Median :-0.2492   Median :-0.34440  
#>  Mean   : 0.0000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.00000  
#>  3rd Qu.: 0.4157   3rd Qu.: 0.09482   3rd Qu.: 0.3682   3rd Qu.:-0.03999  
#>  Max.   : 3.9738   Max.   : 7.20531   Max.   : 3.3317   Max.   : 5.79396

# Unscaled
fd_fric(traits_birds)
#>   site     FRic
#> 1   s1 230967.7

# Scaled
fd_fric(traits_birds_sc)
#>   site    FRic
#> 1   s1 88.9286

Another solution to make trait comparable is to scale them between 0 and 1 by scaling each trait by its maximum and minimum values:

min_values <- as.numeric(lapply(as.data.frame(traits_birds), min))
max_values <- as.numeric(lapply(as.data.frame(traits_birds), max))

traits_birds_minmax <- apply(traits_birds, 1, function(x) {
  (x - min_values)/(max_values - min_values)
})
traits_birds_minmax <- t(traits_birds_minmax)
summary(traits_birds_minmax)
#>  Bill.width..mm.   Bill.length..mm.   Kipp.s.index     Bodymass..g.     
#>  Min.   :0.00000   Min.   :0.00000   Min.   :0.0000   Min.   :0.000000  
#>  1st Qu.:0.08769   1st Qu.:0.03222   1st Qu.:0.2143   1st Qu.:0.007317  
#>  Median :0.17212   Median :0.06301   Median :0.3095   Median :0.023637  
#>  Mean   :0.22977   Mean   :0.09837   Mean   :0.3576   Mean   :0.078418  
#>  3rd Qu.:0.31034   3rd Qu.:0.11023   3rd Qu.:0.4286   3rd Qu.:0.072057  
#>  Max.   :1.00000   Max.   :1.00000   Max.   :1.0000   Max.   :1.000000

There are several other options available to standardize trait values, reviewed in Leps et al. (2006).

If not all the traits you use are continuous, refer to the next section, which suggests ways of computing functional diversity indices with non-continuous traits.

Non-continuous traits?

Do not panic. You can still compute the above-mentioned functional diversity indices. However, as all indices need continuous descriptors for all considered species, you need to transform the non-continuous trait data into a continuous form. The general idea is to obtain from the trait table a table of quantitative descriptions by defining specific dissimilarity and projecting species dissimilarities onto quantitative space using Principal Coordinates Analysis (PCoA). The framework is fully described in Maire et al. (2015).

To compute dissimilarity with non-continuous traits you can user Gower’s distance (Gower 1971) or its following adaptations (Pavoine et al. 2009; Podani 1999). You can use the following functions: cluster::daisy(), FD::gowdis(), ade4::dist.ktab(), or vegan::vegdist().

Then you can project these dissimilarities with Principal Coordinates using ape::pcoa() for example. You can then select the first dimensions that explains the most variance and use theses as the input “traits” to compute functional diversity indices.

Missing values in traits?

Sometimes, some of the trait values can be missing for some species in your dataset. Because fundiversity does not want to make assumptions without telling you, by default it drops the species data for which the trait is missing.

If you want to use data with missing values you can use dissimilarity metrics that accept missing trait values such as some of the methods specified in vegan::vegdist().

Another solution, would be to impute the missing trait value to fill it. Many imputation methods exists and trait imputation is out of the scope of fundiversity but you can find some details on how to proceed in the review by Penone et al. (2014).

Functions summary table

Function Name Index Name Parallelizable1 Memoizable2
fd_fric() FRic
fd_fric_intersect() FRic_intersect
fd_fdiv() FDiv
fd_feve() FEve
fd_fdis() FDis
fd_raoq() Rao’s Q

References

Gower, John C. 1971. “A General Coefficient of Similarity and Some of Its Properties.” Biometrics, 857–71.

Leps, Jan, Francesco Bello, Sandra Lavorel, and Sandra Berman. 2006. “Quantifying and Interpreting Functional Diversity of Natural Communities: Practical Considerations Matter.” Preslia 78 (November): 481–501.

Maire, Eva, Gaël Grenouillet, Sébastien Brosse, and Sébastien Villéger. 2015. “How Many Dimensions Are Needed to Accurately Assess Functional Diversity? A Pragmatic Approach for Assessing the Quality of Functional Spaces.” Global Ecology and Biogeography 24 (6): 728–40. https://doi.org/10.1111/geb.12299.

Nowak, Larissa, W. Daniel Kissling, Irene M. A. Bender, D. Matthias Dehling, Till Töpfer, Katrin Böhning-Gaese, and Matthias Schleuning. 2019. “Data from: Projecting Consequences of Global Warming for the Functional Diversity of Fleshy-Fruited Plants and Frugivorous Birds Along a Tropical Elevational Gradient.” Data Dryad Digital Repository. https://doi.org/10.5061/DRYAD.C0N737B.

Nowak, Larissa, W. Daniel Kissling, Irene M. A. Bender, D. Matthias Dehling, Till Töpfer, Katrin Böhning-Gaese, and Matthias Schleuning. 2019. “Projecting Consequences of Global Warming for the Functional Diversity of Fleshy-Fruited Plants and Frugivorous Birds Along a Tropical Elevational Gradient.” Edited by Kenneth Feeley. Diversity and Distributions 25 (9): 1362–74. https://doi.org/10.1111/ddi.12946.

Pavoine, Sandrine, Jeanne Vallet, Anne-Béatrice Dufour, Sophie Gachet, and Hervé Daniel. 2009. “On the Challenge of Treating Various Types of Variables: Application for Improving the Measurement of Functional Diversity.” Oikos 118 (3): 391–402. https://doi.org/10.1111/j.1600-0706.2008.16668.x.

Penone, Caterina, Ana D. Davidson, Kevin T. Shoemaker, Moreno Di Marco, Carlo Rondinini, Thomas M. Brooks, Bruce E. Young, Catherine H. Graham, and Gabriel C. Costa. 2014. “Imputation of Missing Data in Life-History Trait Datasets: Which Approach Performs the Best?” Methods in Ecology and Evolution 5 (9): 961–70. https://doi.org/10.1111/2041-210X.12232.

Podani, János. 1999. “Extending Gower’s General Coefficient of Similarity to Ordinal Characters.” Taxon, 331–40.

Rao, C. Radhakrishna. 1982. “Diversity and Dissimilarity Coefficients: A Unified Approach.” Theoretical Population Biology 21 (1): 24–43. https://doi.org/10.1016/0040-5809(82)90004-1.

Villéger, Sébastien, Norman W. H. Mason, and David Mouillot. 2008. “New Multidimensional Functional Diversity Indices for a Multifaceted Framework in Functional Ecology.” Ecology 89 (8): 2290–2301. https://doi.org/10.1890/07-1206.1.