Calculates the HCDT, also known as Tsallis entropy of order q of a probability vector.
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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)
A probability vector, summing to 1.
A numeric vector containing species abundances.
A numeric vector, an integer vector, an abundance vector (
A number: the order of entropy. Some corrections allow only a positive number. Default is 1 for Shannon entropy.
A string containing one of the possible corrections:
The sample coverage of
Additional arguments. Unused.
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
bcTsallis and choose the
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
"Grassberger", Zhang and Grabchak (2014), Chao and Jost (2015) and Marcon (2015).
"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" 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
.numeric methods accept
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
"Grassberger" corrections to estimate quite well the entropy.
GammaEntropy functions use this tweak.
A named number equal to the calculated entropy. The name is that of the bias correction used.
Eric Marcon <[email protected]>
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.
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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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