Similarity-based entropy of a community

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

Usage

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.

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.

Examples

# 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)
#>      None 
#> 0.6033223 
# Equal to normalized Rao quadratic entropy when q=2
Rao(Ps, EightSpTree)/max(DistanceMatrix)
#>      None 
#> 0.6033223 
# But different from PhyloEntropy for all other q, e.g. 1
Hqz(Ps, 1, Z)
#>      None 
#> 0.9348275 
summary(PhyloEntropy(Ps, 1, EightSpTree))
#> alpha or gamma phylogenetic or functional entropy of order 1 of distribution -
#>  with correction: None
#> Phylogenetic or functional entropy was calculated according to the tree 
#> EightSpTree 
#> 
#> Entropy is normalized
#> 
#> Entropy equals: 1.034531