# designdist: Design your own Dissimilarities

### Description

You can 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.

### Usage

 ```1 2 3``` ```designdist(x, method = "(A+B-2*J)/(A+B)", terms = c("binary", "quadratic", "minimum"), abcd = FALSE, alphagamma = FALSE, 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). 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).

### Value

`designdist` returns an object of class `dist`.

### Note

`designdist` does not use compiled code, and may be slow or use plenty of memory in large data sets. It is very easy to make errors when defining a function by hand. If an index is available in a function using compiled code, it is better to use the canned alternative.

Jari Oksanen

### References

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

`vegdist`, `betadiver`, `dist`, `raupcrick`.

### Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```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)) ## ... or in analysing beta diversity (without gradients) summary(dis) ```

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