# Tsallis: Tsallis (HCDT) Entropy of a community In entropart: Entropy Partitioning to Measure Diversity

## Description

Calculates the HCDT, also known as Tsallis entropy of order q of a probability vector.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```Tsallis(NorP, q = 1, ...) bcTsallis(Ns, q = 1, Correction = "Best", SampleCoverage = NULL, CheckArguments = TRUE) ## S3 method for class 'ProbaVector' Tsallis(NorP, q = 1, ..., CheckArguments = TRUE, Ps = NULL) ## S3 method for class 'AbdVector' Tsallis(NorP, q = 1, Correction = "Best", ..., CheckArguments = TRUE, Ns = NULL) ## S3 method for class 'integer' Tsallis(NorP, q = 1, Correction = "Best", ..., CheckArguments = TRUE, Ns = NULL) ## S3 method for class 'numeric' Tsallis(NorP, q = 1, Correction = "Best", ..., CheckArguments = TRUE, Ps = NULL, Ns = NULL) ```

## Arguments

 `Ps` A probability vector, summing to 1. `Ns` A numeric vector containing species abundances. `NorP` A numeric vector, an integer vector, an abundance vector (`AbdVector`) or a probability vector (`ProbaVector`). Contains either abundances or probabilities. `q` A number: the order of entropy. Some corrections allow only a positive number. Default is 1 for Shannon entropy. `Correction` A string containing one of the possible corrections: `"None"` (no correction), `"ChaoShen"`, `"GenCov"`, `"Grassberger"`, `"Holste"`, `"Bonachela"`, `"ZhangGrabchak"`, or `"ChaoWangJost"`, `"Marcon"`, `"UnveilC"`, `"UnveiliC"`, `"UnveilJ"` or `"Best"`, the default value. Currently, `"Best"` is `"ChaoWangJost"`. `SampleCoverage` The sample coverage of `Ns` calculated elsewhere. Used to calculate the gamma diversity of meta-communities, see details. `...` Additional arguments. Unused. `CheckArguments` Logical; if `TRUE`, the function arguments are verified. Should be set to `FALSE` to save time when the arguments have been checked elsewhere.

## Details

Tsallis (Havrda and Charvat, 1967; Daroczy, 1970; Tsallis, 1988) generalized entropy is a generalized measure of diversity (Jost, 2006).

Bias correction requires the number of individuals to estimate sample `Coverage`. Use `bcTsallis` and choose the `Correction`. Correction techniques are from Chao and Shen (2003), Grassberger (1988), Holste et al. (1998), Bonachela et al. (2008), (Marcon et al., 2014), which is actually the max value of `"ChaoShen"` and `"Grassberger"`, Zhang and Grabchak (2014), Chao and Jost (2015) and Marcon (2015).

Currently, the `"Best"` correction is `"ChaoWangJost"` (Chao, Wang and Jost, 2013 for q=1; Chao and Jost, 2015). This estimator contains an unbiased part concerning observed species, equal to that of Zhang and Grabchak (2014), and a (biased) estimator of the remaining bias based on the estimation of the species-accumulation curve. It is very efficient but very slow if the number of individuals is more than a few hundreds.

The unveiled estimators rely on Chao et al. (2015), completed by Marcon (2015). The actual probabilities of observed species are estimated and completed by a geometric distribution of the probabilities of unobserved species. The number of unobserved species is estimated by the Chao1 estimator (`"UnveilC"`), following Chao et al. (2015), or by the iChao1 (`"UnveiliC"`) or the jacknife (`"UnveilJ"`). The `"UnveilJ"` correction often has a lower bias but a greater variance (Marcon, 2015).

The functions are designed to be used as simply as possible. `Tsallis` is a generic method. If its first argument is an abundance vector, an integer vector or a numeric vector which does not sum to 1, the bias corrected function `bcTsallis` is called. Explicit calls to `bcTsallis` (with bias correction) or to `Tsallis.ProbaVector` (without correction) are possible to avoid ambiguity. The `.integer` and `.numeric` methods accept `Ps` or `Ns` arguments instead of `NorP` for backward compatibility.

The size of a metacommunity (see `MetaCommunity`) is unknown so it has to be set according to a rule which does not ensure that its abundances are integer values. Then, classical bias-correction methods do not apply. Providing the `SampleCoverage` argument allows applying the `"ChaoShen"` and `"Grassberger"` corrections to estimate quite well the entropy. `DivPart` and `GammaEntropy` functions use this tweak.

## Value

A named number equal to the calculated entropy. The name is that of the bias correction used.

## Author(s)

Eric Marcon <[email protected]>

## References

Chao, A. and Jost, L. (2015) Estimating diversity and entropy profiles via discovery rates of new species. Methods in Ecology and Evolution 6(8): 873-882.

Chao, A., Hsieh, T. C., Chazdon, R. L., Colwell, R. K., Gotelli, N. J. (2015) Unveiling the Species-Rank Abundance Distribution by Generalizing Good-Turing Sample Coverage Theory. Ecology 96(5): 1189-1201.

Chao, A., Wang, Y. T. and Jost, L. (2013). Entropy and the species accumulation curve: a novel entropy estimator via discovery rates of new species. Methods in Ecology and Evolution 4(11):1091-1100.

Havrda, J. and Charvat, F. (1967). Quantification method of classification processes. Concept of structural a-entropy. Kybernetika 3(1): 30-35.

Daroczy, Z. (1970). Generalized information functions. Information and Control 16(1): 36-51.

Jost, L. (2006). Entropy and diversity. Oikos 113(2): 363-375.

Marcon, E. (2015) Practical Estimation of Diversity from Abundance Data. HAL 01212435: 1-27.

Marcon, E., Scotti, I., Herault, B., Rossi, V. and Lang, G. (2014). Generalization of the partitioning of Shannon diversity. PLOS One 9(3): e90289.

Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics 52(1): 479-487.

Zhang, Z., and Grabchak, M. (2016). Entropic Representation and Estimation of Diversity Indices. Journal of Nonparametric Statistics, 28(3): 563-575.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```# Load Paracou data (number of trees per species in two 1-ha plot of a tropical forest) data(Paracou618) # Ns is the total number of trees per species Ns <- as.AbdVector(Paracou618.MC\$Ns) # Species probabilities Ps <- as.ProbaVector(Paracou618.MC\$Ns) # Whittaker plot plot(Ns) # Calculate entropy of order 1, i.e. Shannon's entropy Tsallis(Ps, 1) # Calculate it with estimation bias correction Tsallis(Ns, 1) ```

entropart documentation built on Feb. 6, 2018, 1:04 a.m.