abgdivparam {adiv} | R Documentation |

## Apportionment of Parametric Indices of Diversity

### Description

Function `abgdivparam`

calculates alpha, beta and gamma components of species diversity using parametric indices derived from Tsallis (HCDT) and Hill compositional indices. Alpha is for within-community diversity, beta for between-community diversity and gamma for the diversity of all combined communities.

### Usage

```
abgdivparam(comm, w = c("speciesab", "even"),
method = c("hillCJC", "hillR", "tsallis"), q = 2,
option = c("multiplicative", "additive", "proportional",
"C", "U", "V", "S", "Renyi"), tol = 1e-08)
## S3 method for class 'abgdivparam'
plot(x, legend = TRUE,
legendposi = "topright", type = "b",
col = if (is.numeric(x)) NULL else 1:nrow(x$div),
lty = if (is.numeric(x)) NULL else rep(1, nrow(x$div)),
pch = if (is.numeric(x)) NULL else 1:nrow(x$div),
ylim1 = range(x$div[c("Alpha", "Gamma"), ]), ylim2 = NULL, ...)
```

### Arguments

`comm` |
a data frame or a matrix typically with communities as rows, species as columns and an index of abundance as entries. |

`w` |
either a numeric vector giving weights for communities (same order as in comm), or a code: one of |

`method` |
a string with one of the following codes: |

`q` |
a vector with nonnegative value(s) for parameter |

`option` |
a string code: either |

`tol` |
numeric tolerance threshold: values between - |

`x` |
an object of class |

`legend` |
a logical. If TRUE a legend is given with the colour, the type of line (etc.) used to define the diversity curve of each diversity level (gamma, alpha, beta). |

`legendposi` |
a string that gives the position of the legend to be passed to function |

`type` |
a string to be passed to the graphic argument |

`col` |
vector of colours to be passed to the graphic argument |

`lty` |
vector of types of line (plain, broken etc.) to be passed to the graphic argument |

`pch` |
vector of types of point (open circle, close circle, square etc.) to be passed to the graphic argument |

`ylim1` |
a vector with two numerics indicating the range to be used to display alpha and gamma diversity. |

`ylim2` |
a vector with two numerics indicating the range to be used to display beta diversity. |

`...` |
other arguments can be added and passed to the functions |

### Details

Consider *j* a community (*j*=1,...,*m*), `a_{jk}`

the abundance of species *k* in community *j*. *q* is the parameter that increases with the importance given to abundant species compared to rare species in diversity.

The methods available are:
`tsallis`

(decomposition of Tsallis or HCDT entropy (Harvda and Charvat 1967; Daroczy 1970; Tsallis 1988) into alpha, beta, gamma components):

`^q\gamma_{Tsallis}=\left[1-\sum_k \left(\sum_{j=1}^m w_j \frac{a_{jk}}{\sum_k a_{jk}}\right)^q \right]/(q-1)`

`^q\alpha_{Tsallis}=\sum_{j=1}^m w_j \left[1-\sum_k \left( \frac{a_{jk}}{\sum_k a_{jk}}\right)^q\right]/(q-1)`

`hillR`

(Routledge decomposition of Hill diversity into alpha, beta, gamma components):

`^q\gamma_{Hill}=\left[\sum_k \left(\sum_{j=1}^m w_j \frac{a_{jk}}{\sum_k a_{jk}}\right)^q \right]^{1/(1-q)}`

`^q\alpha_{Hill-R}=\left[\sum_{j=1}^m w_j \sum_k \left( \frac{a_{jk}}{\sum_k a_{jk}}\right)^q\right]^{1/(1-q)}`

`hillCJC`

(Chiu et al. (2014) decomposition of species diversity into alpha, beta, gamma components, see Supplementary material Appendix 2 in Pavoine (2016) for a justification of the formulas):

`^q\gamma_{Hill}=\left[\sum_k \left(\sum_{j=1}^m w_j \frac{a_{jk}}{\sum_k a_{jk}}\right)^q \right]^{1/(1-q)}`

`^q\alpha_{Hill-CJC}=\frac{1}{m}\left[\sum_k \sum_{j=1}^m (w_j)^q \left( \frac{a_{jk}}{\sum_k a_{jk}}\right)^q\right]^{1/(1-q)}`

Then option `"additive"`

calculates `\beta`

diversity as `\gamma-\alpha`

.
Option `"proportional"`

calculates `\beta`

as `(\gamma-\alpha)/\gamma`

.
Option `"multiplicative"`

calculates `\beta`

diversity as `\gamma/\alpha`

.
Only for `method`

=`"hillCJC"`

, options `"C"`

, `"U"`

, `"V"`

, `"S"`

, use the multiplicative option and also calculate one of the transformations introduced by Chiu et al. (2014): indices `1-C_{qm}`

, `1-U_{qm}`

, `1-V_{qm}`

, and `1-S_{qm}`

, respectively. `"Renyi"`

calculates `\beta`

diversity as `ln(\gamma/\alpha)/ln(m)`

.

The weights of the sites (argument `w`

) can be `"even"`

(even weights), or `"speciesab"`

(proportional to the summed abundances of all species).

### Value

If only one value of `q`

is given, abgdivparam returns a vector with alpha, beta, and gamma diversities.
If more than one value of `q`

is given, it returns a list of two objects:

`q` |
the numeric vector of values for |

`div` |
a data frame with alpha, beta, gamma calculated for all values of |

Only if `method`

=`"hillCJC"`

and `option`

= `"C"`

, `"U"`

, `"V"`

, `"S"`

, or `"Renyi"`

, the index `1-C_{qm}`

(for `"C"`

), `1-U_{qm}`

(for `"U"`

), `1-V_{qm}`

(for `"V"`

), `1-S_{qm}`

(for `"S"`

) or the Renyi transformation (see above, for `"Renyi"`

) is also provided in the `div`

data frame under the name "transformed.beta".

The function `plot.abgdivparam`

returns a graphic.

### Author(s)

Sandrine Pavoine sandrine.pavoine@mnhn.fr

### References

Chiu, C.-H., Jost, L., Chao, A. (2014) Phylogenetic beta diversity, similarity, and differentiation measures based on Hill numbers. *Ecological Monographs*, **84**, 21–44.

Daroczy, Z. (1970) Generalized information functions. *Information and Control*, **16**, 36–51.

Havrda, M., Charvat F. (1967) Quantification method of classification processes: concept of structural alpha-
entropy. *Kybernetik*, **3**, 30–35

Hill, M.O. (1973) Diversity and evenness: a unifying notation and its consequences. *Ecology*, **54**, 427–432.

Pavoine, S. (2016) A guide through a family of phylogenetic dissimilarity measures among sites. *Oikos*, **125**, 1719–1732.

Rao, C.R. (1986) Rao's axiomatization of diversity measures. In: Kotz S, Johnson NL, editors. *Encyclopedia of Statistical Sciences*. New York: Wiley and Sons. pp. 614–617.

Routledge, R.D. (1979) Diversity indices: which ones are admissible? *Journal of Theoretical Biology*, **76**, 503–515.

### See Also

### Examples

```
data(batcomm)
abgdivparam(batcomm$ab)
plot(abgdivparam(batcomm$ab))
abgdivparam(batcomm$ab, q=0:4)
plot(abgdivparam(batcomm$ab, q=0:4))
```

*adiv*version 2.2.1 Index]