`Shannon.Rd`

Calculates the Shannon entropy of a probability vector.

Shannon(NorP, ...) bcShannon(Ns, Correction = "Best", CheckArguments = TRUE) # S3 method for ProbaVector Shannon(NorP, ..., CheckArguments = TRUE, Ps = NULL) # S3 method for AbdVector Shannon(NorP, Correction = "Best", Level = NULL, PCorrection = "Chao2015", Unveiling = "geom", RCorrection = "Rarefy", ..., CheckArguments = TRUE, Ns = NULL) # S3 method for integer Shannon(NorP, Correction = "Best", Level = NULL, PCorrection = "Chao2015", Unveiling = "geom", RCorrection = "Rarefy", ..., CheckArguments = TRUE, Ns = NULL) # S3 method for numeric Shannon(NorP, Correction = "Best", Level = NULL, PCorrection = "Chao2015", Unveiling = "geom", RCorrection = "Rarefy", ..., CheckArguments = TRUE, Ps = NULL, Ns = NULL)

Ps | A probability vector, summing to 1. |
---|---|

Ns | A numeric vector containing species abundances. |

NorP | A numeric vector, an integer vector, an abundance vector ( |

Correction | A string containing one of the possible asymptotic estimators: |

Level | The level of interpolation or extrapolation. It may be an a chosen sample size (an integer) or a sample coverage (a number between 0 and 1). Entropy extrapolation require its asymptotic estimation depending on the choice of |

PCorrection | A string containing one of the possible corrections to estimate a probability distribution in |

Unveiling | A string containing one of the possible unveiling methods to estimate the probabilities of the unobserved species in |

RCorrection | A string containing a correction recognized by |

... | Additional arguments. Unused. |

CheckArguments | Logical; if |

Bias correction requires the number of individuals to estimate sample `Coverage`

.

Correction techniques are from Miller (1955), Chao and Shen (2003), Grassberger (1988), Grassberger (2003), Schurmann (2003), Holste *et al.* (1998), Bonachela *et al.* (2008), Zhang (2012), Chao, Wang and Jost (2013).
More estimators can be found in the `entropy`

package.

Using `MetaCommunity`

mutual information, Chao, Wang and Jost (2013) calculate reduced-bias Shannon beta entropy (see the last example below) with better results than the Chao and Shen estimator, but community weights cannot be arbitrary: they must be proportional to the number of individuals.

The functions are designed to be used as simply as possible. `Shannon`

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 `bcShannon`

is called.

Entropy can be estimated at a specified level of interpolation or extrapolation, either a chosen sample size or sample coverage (Chao et al., 2014), rather than its asymptotic value.
Extrapolation relies on the estimation of the asymptotic entropy. If `Unveiling`

is "None", then the asymptotic estimation of entropy is made using the chosen `Correction`

, else the asymtpotic distribution of the community is derived and its estimated richness adjusted so that the entropy of a sample of this distribution of the size of the actual sample has the entropy of the actual sample.

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

Bonachela, J. A., Hinrichsen, H. and Munoz, M. A. (2008). Entropy estimates of small data sets. *Journal of Physics A: Mathematical and Theoretical* 41(202001): 1-9.

Chao, A. and Shen, T. J. (2003). Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample. *Environmental and Ecological Statistics* 10(4): 429-443.

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.

Chao, A., Gotelli, N. J., Hsieh, T. C., Sander, E. L., Ma, K. H., Colwell, R. K., Ellison, A. M (2014). Rarefaction and extrapolation with Hill numbers: A framework for sampling and estimation in species diversity studies. *Ecological Monographs*, 84(1): 45-67.

Grassberger, P. (1988). Finite sample corrections to entropy and dimension estimates. *Physics Letters A* 128(6-7): 369-373.

Grassberger, P. (2003). Entropy Estimates from Insufficient Samplings. *ArXiv Physics e-prints* 0307138.

Holste, D., Grosse, I. and Herzel, H. (1998). Bayes' estimators of generalized entropies. *Journal of Physics A: Mathematical and General* 31(11): 2551-2566.

Miller, G. (1955) Note on the bias of information estimates. In: Quastler, H., editor. *Information Theory in Psychology: Problems and Methods*: 95-100.

Shannon, C. E. (1948). A Mathematical Theory of Communication. *The Bell System Technical Journal* 27: 379-423, 623-656.

Schurmann, T. (2004). Bias analysis in entropy estimation. *Journal of Physics A: Mathematical and Theoretical* 37(27): L295-L301.

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

Zhang, Z. (2012). Entropy Estimation in Turing's Perspective. *Neural Computation* 24(5): 1368-1389.

Eric Marcon <Eric.Marcon@ecofog.gf>

`bcShannon`

, `Tsallis`

# 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 Shannon entropy Shannon(Ps)#> None #> 4.736023# Calculate the best estimator of Shannon entropy Shannon(Ns)#> UnveilJ #> 4.928035# Use metacommunity data to calculate reduced-bias Shannon beta as mutual information (bcShannon(Paracou618.MC$Ns) + bcShannon(colSums(Paracou618.MC$Nsi)) - bcShannon(Paracou618.MC$Nsi))#> UnveilJ #> 0.3439827