# Hqz: Similarity-based entropy of a community In entropart: Entropy Partitioning to Measure Diversity

## Description

Calculates the entropy of order q of a probability vector according to a similarity matrix.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```Hqz(NorP, q = 1, Z = diag(length(NorP)), ...) bcHqz(Ns, q = 1, Z = diag(length(Ns)), Correction = "Best", SampleCoverage = NULL, CheckArguments = TRUE) ## S3 method for class 'ProbaVector' Hqz(NorP, q = 1, Z = diag(length(NorP)), ..., CheckArguments = TRUE, Ps = NULL) ## S3 method for class 'AbdVector' Hqz(NorP, q = 1, Z = diag(length(NorP)), Correction = "Best", ..., CheckArguments = TRUE, Ns = NULL) ## S3 method for class 'integer' Hqz(NorP, q = 1, Z = diag(length(NorP)), Correction = "Best", ..., CheckArguments = TRUE, Ns = NULL) ## S3 method for class 'numeric' Hqz(NorP, q = 1, Z = diag(length(NorP)), 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. Default is 1. `Z` A relatedness matrix, i.e. a square matrix whose terms are all positive, strictly positive on the diagonal. Generally, the matrix is a similarity matrix, i.e. the diagonal terms equal 1 and other terms are between 0 and 1. Default is the identity matrix to calculate neutral entropy. `Correction` A string containing one of the possible corrections: `"None"` (no correction), `"ChaoShen"`, `"MarconZhang"` or `"Best"`, the default value. The `"MarconZhang"` correction assumes a similarity matrix. `SampleCoverage` The sample coverage of `Ns` calculated elsewhere. Used to calculate the gamma diversity of meta-communities, see details. `...` Additional arguments. Unused. `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

Entropy is calculated following Leinster and Cobbold (2012) after Ricotta and Szeidl (2006): it is the entropy of order `q` of the community, using species ordinariness as the information function.

A similarity matrix is used (as for `Dqz`), not a distance matrix as in Ricotta and Szeidl (2006). See the example.

Bias correction requires the number of individuals. Use `bcHqz` and choose the `Correction`. Correction techniques are from Marcon et al. (2014).

Currently, the `"Best"` correction is the max value of `"ChaoShen"` and `"MarconZhang"`.

The functions are designed to be used as simply as possible. `Hqz` 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 `bcHqz` is called. Explicit calls to `bcHqz` (with bias correction) or to `Hqz.ProbaVector` (without correction) are possible to avoid ambiguity. The `.integer` and `.numeric` methods accept `Ps` or `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 `"ChaoShen"` correction to estimate quite well the entropy. `DivPart` and `GammaEntropy` functions use this tweak.

## Value

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

## Author(s)

Eric Marcon <[email protected]>

## References

Leinster, T. and Cobbold, C. (2012). Measuring diversity: the importance of species similarity. Ecology 93(3): 477-489.

Marcon, E., Zhang, Z. and Herault, B. (2014). The decomposition of similarity-based diversity and its bias correction. HAL hal-00989454(version 3).

Ricotta, C. and Szeidl, L. (2006). Towards a unifying approach to diversity measures: Bridging the gap between the Shannon entropy and Rao's quadratic index. Theoretical Population Biology 70(3): 237-243.

`Dqz`, `PhyloEntropy`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```# Load Paracou data (number of trees per species in two 1-ha plot of a tropical forest) data(Paracou618) # Prepare the similarity matrix DistanceMatrix <- as.matrix(EightSpTree\$Wdist^2/2) # Similarity can be 1 minus normalized distances between species Z <- 1 - DistanceMatrix/max(DistanceMatrix) # Calculate diversity of order 2 Ps <- EightSpAbundance/sum(EightSpAbundance) Hqz(Ps, 2, Z) # Equal to normalized Rao quadratic entropy when q=2 Rao(Ps, EightSpTree)/max(DistanceMatrix) # But different from PhyloEntropy for all other q, e.g. 1 Hqz(Ps, 1, Z) summary(PhyloEntropy(Ps, 1, EightSpTree)) ```