# Similarity-based beta entropy of a community

### Description

Calculates the similarity-based beta entropy of order *q* of a community belonging to a metacommunity.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
HqzBeta(NorP, NorPexp = NULL, q = 1, Z = diag(length(Ps)), Correction = "Best",
CheckArguments = TRUE,
Ps = NULL, Ns = NULL, Pexp = NULL, Nexp = NULL)
bcHqzBeta(Ns, Nexp = NULL, q = 1, Z = diag(length(Ns)), Correction = "Best",
CheckArguments = TRUE)
## S3 method for class 'ProbaVector'
HqzBeta(NorP, NorPexp = NULL, q = 1, Z = diag(length(Ps)), Correction = "Best",
CheckArguments = TRUE, Ps = NULL, Ns = NULL, Pexp = NULL, Nexp = NULL)
## S3 method for class 'AbdVector'
HqzBeta(NorP, NorPexp = NULL, q = 1, Z = diag(length(Ps)), Correction = "Best",
CheckArguments = TRUE, Ps = NULL, Ns = NULL, Pexp = NULL, Nexp = NULL)
## S3 method for class 'integer'
HqzBeta(NorP, NorPexp = NULL, q = 1, Z = diag(length(Ps)), Correction = "Best",
CheckArguments = TRUE, Ps = NULL, Ns = NULL, Pexp = NULL, Nexp = NULL)
## S3 method for class 'numeric'
HqzBeta(NorP, NorPexp = NULL, q = 1, Z = diag(length(Ps)), Correction = "Best",
CheckArguments = TRUE, Ps = NULL, Ns = NULL, Pexp = NULL, Nexp = NULL)
``` |

### Arguments

`Ps` |
The probability vector of species of the community. |

`Pexp` |
The probability vector of species of the metacommunity. |

`Ns` |
A numeric vector containing species abundances of the community. |

`Nexp` |
A numeric vector containing species abundances of the metacommunity. |

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

`NorPexp` |
A numeric vector, an integer vector, an abundance vector ( |

`q` |
A number, the order of diversity. Default is 1. |

`Z` |
A relatedness matrix, |

`Correction` |
A string containing one of the possible corrections: currently, no correction is available so |

`CheckArguments` |
Logical; if |

### Details

The derivation of similarity-based beta entropy can be found in Marcon *et al.* (2014).

Bias correction requires the number of individuals.

Note that beta entropy value is related to alpha entropy (if *q* is not 1) and cannot be compared accross communities (Jost, 2007). Beta entropy of a community is not meaningful in general, do rather calculate the `BetaDiversity`

of the metacommunity.

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

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 `bcHqzBeta`

is called. Explicit calls to `bcHqzBeta`

(with bias correction) or to `HqzBeta.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

Jost (2007), Partitioning diversity into independent alpha and beta components. *Ecology* 88(10): 2427-2439.

Marcon, E., Zhang, Z. and Herault, B. (2014). The decomposition of similarity-based diversity and its bias correction. *HAL* hal-00989454(version 1).

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 | ```
# Load Paracou data (number of trees per species in two 1-ha plot of a tropical forest)
data(Paracou618)
# Ps is the vector of probabilities
Ps <- as.ProbaVector(Paracou618.MC$Ps)
# Probability distribution of the first plot
Ps1 <- as.ProbaVector(Paracou618.MC$Psi[, 1])
# Prepare the similarity matrix
DistanceMatrix <- as.matrix(Paracou618.dist)
# Similarity can be 1 minus normalized distances between species
Z <- 1 - DistanceMatrix/max(DistanceMatrix)
# Divergence of order 2 between plot 1 and the whole forest
HqzBeta(Ps1, Ps, q=2, Z)
``` |

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