# diversity-method: Heterogeneity and evenness In nfrerebeau/tabula: Analysis, Seriation and Visualization of Archaeological Count Data

## Description

`diversity` returns a diversity or dominance index. `evenness` returns an evenness measure.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```diversity(object, ...) evenness(object, ...) ## S4 method for signature 'CountMatrix' diversity(object, method = c("berger", "brillouin", "mcintosh", "shannon", "simpson"), simplify = FALSE, ...) ## S4 method for signature 'CountMatrix' evenness(object, method = c("brillouin", "mcintosh", "shannon", "simpson"), simplify = FALSE, ...) ```

## Arguments

 `object` A m x p matrix of count data. `...` Further arguments passed to other methods. `method` A `character` string or vector of strings specifiying the index to be computed (see details). Any unambiguous substring can be given. `simplify` A `logical` scalar: should the result be simplified to a matrix? The default value, `FALSE`, returns a list.

## Details

Diversity measurement assumes that all individuals in a specific taxa are equivalent and that all types are equally different from each other (Peet 1974). A measure of diversity can be achieved by using indices built on the relative abundance of taxa. These indices (sometimes referred to as non-parametric indices) benefit from not making assumptions about the underlying distribution of taxa abundance: they only take evenness and richness into account. Peet (1974) refers to them as indices of heterogeneity.

Diversity indices focus on one aspect of the taxa abundance and emphasize either richness (weighting towards uncommon taxa) or dominance (weighting towards abundant taxa; Magurran 1988).

Evenness is a measure of how evenly individuals are distributed across the sample.

The following heterogeneity index and corresponding evenness measures are available (see Magurran 1988 for details):

berger

Berger-Parker dominance index. The Berger-Parker index expresses the proportional importance of the most abundant type.

brillouin

Brillouin diversity index. The Brillouin index describes a known collection: it does not assume random sampling in an infinite population. Pielou (1975) and Laxton (1978) argues for the use of the Brillouin index in all circumstances, especially in preference to the Shannon index.

mcintosh

McIntosh dominance index. The McIntosh index expresses the heterogeneity of a sample in geometric terms. It describes the sample as a point of a S-dimensional hypervolume and uses the Euclidean distance of this point from the origin.

shannon

Shannon-Wiener diversity index. The Shannon index assumes that individuals are randomly sampled from an infinite population and that all taxa are represented in the sample (it does not reflect the sample size). The main source of error arises from the failure to include all taxa in the sample: this error increases as the proportion of species discovered in the sample declines (Peet 1974, Magurran 1988). The maximum likelihood estimator (MLE) is used for the relative abundance, this is known to be negatively biased.

simpson

Simpson dominance index for finite sample. The Simpson index expresses the probability that two individuals randomly picked from a finite sample belong to two different types. It can be interpreted as the weighted mean of the proportional abundances.

The `berger`, `mcintosh` and `simpson` methods return a dominance index, not the reciprocal or inverse form usually adopted, so that an increase in the value of the index accompanies a decrease in diversity.

## Value

If `simplify` is `FALSE`, then `diversity` and `evenness` return a list (default), else return a matrix.

## Note

Ramanujan approximation is used for x! computation if x > 170.

N. Frerebeau

## References

Berger, W. H. & Parker, F. L. (1970). Diversity of Planktonic Foraminifera in Deep-Sea Sediments. Science, 168(3937), 1345-1347. DOI: 10.1126/science.168.3937.1345.

Brillouin, L. (1956). Science and information theory. New York: Academic Press.

Laxton, R. R. (1978). The measure of diversity. Journal of Theoretical Biology, 70(1), 51-67. DOI: 10.1016/0022-5193(78)90302-8.

Magurran, A. E. (1988). Ecological Diversity and its Measurement. Princeton, NJ: Princeton University Press. DOI:10.1007/978-94-015-7358-0.

McIntosh, R. P. (1967). An Index of Diversity and the Relation of Certain Concepts to Diversity. Ecology, 48(3), 392-404. DOI: 10.2307/1932674.

Peet, R. K. (1974). The Measurement of Species Diversity. Annual Review of Ecology and Systematics, 5(1), 285-307. DOI: 10.1146/annurev.es.05.110174.001441.

Pielou, E. C. (1975). Ecological Diversity. New York: Wiley. DOI: 10.4319/lo.1977.22.1.0174b

Shannon, C. E. (1948). A Mathematical Theory of Communication. The Bell System Technical Journal, 27, 379-423. DOI: 10.1002/j.1538-7305.1948.tb01338.x.

Simpson, E. H. (1949). Measurement of Diversity. Nature, 163(4148), 688-688. DOI: 10.1038/163688a0.

`turnover` `similarity`
Other alpha-diversity: `richness`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56``` ```# Shannon diversity index # Data from Magurran 1988, p. 145-149 birds <- CountMatrix( data = c(35, 26, 25, 21, 16, 11, 6, 5, 3, 3, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 30, 30, 3, 65, 20, 11, 0, 4, 2, 14, 0, 3, 9, 0, 0, 5, 0, 0, 0, 0, 1, 1), nrow = 2, byrow = TRUE, dimnames = list(c("oakwood", "spruce"), NULL)) diversity(birds, "shannon") # 2.40 2.06 evenness(birds, "shannon") # 0.80 0.78 # Brillouin diversity index # Data from Magurran 1988, p. 150-151 moths <- CountMatrix(data = c(17, 15, 11, 4, 4, 3, 3, 3, 2, 2, 1, 1, 1), nrow = 1, byrow = TRUE) diversity(moths, "brillouin") # 1.88 evenness(moths, "brillouin") # 0.83 # Simpson dominance index # Data from Magurran 1988, p. 152-153 trees <- CountMatrix( data = c(752, 276, 194, 126, 121, 97, 95, 83, 72, 44, 39, 16, 15, 13, 9, 9, 9, 8, 7, 4, 2, 2, 1, 1, 1), nrow = 1, byrow = TRUE ) diversity(trees, "simpson") # 1.19 evenness(trees, "simpson") # 0.21 # McIntosh dominance index # Data from Magurran 1988, p. 154-155 invertebrates <- CountMatrix( data = c(254, 153, 90, 69, 68, 58, 51, 45, 40, 39, 25, 23, 19, 18, 16, 14, 14, 11, 11, 11, 11, 10, 6, 6, 6, 6, 5, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1), nrow = 1, byrow = TRUE ) diversity(invertebrates, "mcintosh") # 0.71 evenness(invertebrates, "mcintosh") # 0.82 # Berger-Parker dominance index # Data from Magurran 1988, p. 156-157 fishes <- CountMatrix( data = c(394, 3487, 275, 683, 22, 1, 0, 1, 6, 8, 1, 1, 2, 1642, 5681, 196, 1348, 12, 0, 1, 48, 21, 1, 5, 7, 3, 90, 320, 180, 46, 2, 0, 0, 1, 0, 0, 2, 1, 5, 126, 17, 115, 436, 27, 0, 0, 3, 1, 0, 0, 1, 0, 32, 0, 0, 5, 0, 0, 0, 0, 13, 9, 0, 0, 4), nrow = 5, byrow = TRUE, dimnames = list(c("station 1", "station 2", "station 3", "station 4", "station 5"), NULL) ) diversity(fishes, "berger") # 0.71 0.63 0.50 0.60 0.51 ```