# Simpson entropy of a community

### Description

Calculates the Simpson entropy of a probability vector.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
Simpson(NorP, Correction="Lande", CheckArguments = TRUE,
Ps = NULL, Ns = NULL)
bcSimpson(Ns, Correction = "Lande", CheckArguments = TRUE)
## S3 method for class 'ProbaVector'
Simpson(NorP, Correction="Lande", CheckArguments = TRUE,
Ps = NULL, Ns = NULL)
## S3 method for class 'AbdVector'
Simpson(NorP, Correction="Lande", CheckArguments = TRUE,
Ps = NULL, Ns = NULL)
## S3 method for class 'integer'
Simpson(NorP, Correction="Lande", CheckArguments = TRUE,
Ps = NULL, Ns = NULL)
## S3 method for class 'numeric'
Simpson(NorP, Correction="Lande", 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 ( |

`Correction` |
A string containing one of the possible corrections accepted by |

`CheckArguments` |
Logical; if |

### Details

Lande's correction has been derived (Lande, 1996; Good, 1953) especially for Simpson entropy, while other corrections are for generalized Tsallis entropy. It is identical to the unbiased estimator proposed by Simpson, although arguments were different. It equals the plug-in etimator multiplied by n/(n-1) where n is the total number of individuals.

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

. Use `bcSimpson`

and choose the `Correction`

.

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

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 `bcSimpson`

is called. Explicit calls to `bcSimpson`

(with bias correction) or to `Simpson.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

Good, I. J. (1953). On the Population Frequency of Species and the Estimation of Population Parameters. *Biometrika* 40(3/4): 237-264.

Lande, R. (1996). Statistics and partitioning of species diversity, and similarity among multiple communities. *Oikos* 76: 5-13.

Simpson, E. H. (1949). Measurement of diversity. *Nature* 163(4148): 688.

### See Also

`Tsallis`

, `bcSimpson`

### Examples

1 2 3 4 5 6 7 8 9 | ```
# 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)
# Whittaker plot
plot(Ns)
# Calculate an unbiased estimator of Simpson's index of diversity
Simpson(Ns)
``` |