# Tsallis (HCDT) Entropy of a community

### 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, Correction = "Best", CheckArguments = TRUE,
Ps = NULL, Ns = NULL)
bcTsallis(Ns, q = 1, Correction = "Best", CheckArguments = TRUE)
## S3 method for class 'ProbaVector'
Tsallis(NorP, q = 1, Correction = "Best", CheckArguments = TRUE,
Ps = NULL, Ns = NULL)
## S3 method for class 'AbdVector'
Tsallis(NorP, q = 1, Correction = "Best", CheckArguments = TRUE,
Ps = NULL, Ns = NULL)
## S3 method for class 'integer'
Tsallis(NorP, q = 1, Correction = "Best", CheckArguments = TRUE,
Ps = NULL, 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 ( |

`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: |

`CheckArguments` |
Logical; if |

### 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.

### Value

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

### Author(s)

Eric Marcon <Eric.Marcon@ecofog.gf>

### 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. (2014). Entropic Representation and Estimation of Diversity Indices. *arXiv* 1403.3031(v. 1): 1-12.

### 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)
``` |

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