Tsallis (HCDT) Entropy of a community

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Description

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

Usage

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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 (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".

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.

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

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  # 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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