tdv {diffval}

The Total Differential Value of a phytosociological table

Description

Given a phytosociological table and a partition of its columns, this function calculates the respective Total Differential Value (TDV).

Usage

tdv(m_bin, p, output_type = "normal")

Arguments

m_bin

A matrix. A phytosociological table of 0s (absences) and 1s (presences), where rows correspond to taxa and columns correspond to relevés.

p

A vector of integer numbers with the partition of the relevés (i.e., a k-partition, consisting in a vector with values from 1 to k, with length equal to the number of columns of m_bin, ascribing each relevé to one of the k groups).

output_type

A character determining the amount of information returned by the function and also the amount of pre-validations. Possible values are "normal" (the default), "fast" and "full".

Details

The function accepts a phytosociological table (m_bin) and a k-partition of its columns (p), returning the corresponding TDV. TDV was proposed by Monteiro-Henriques and Bellu (2014). Monteiro-Henriques (2016) proposed TDV1, modifying TDV slightly with the objective of ensuring a value from 0 to 1. Yet, TDV is always within that range. In practice, both TDV and TDV1 have 0 as possible minimum value and 1 as possible maximum value, but TDV1 reduces further the contribution of differential taxa present in more than one group. TDV is then implemented here, for parsimony.

TDV is calculated using the DiffVal index for each (and all) of the taxa present in a tabulated phytosociological table M (also called sorted table). DiffVal index aims at characterizing how well a taxon works as a differential taxon in a such tabulated phytosociological table (for more information on differential taxa see Mueller-Dombois & Ellenberg, 1974).

An archetypal differential taxon of a certain group g of the partition p (a partition on the columns of M) is the one present in all relevés of group g, and absent from all the other groups of that partition. Therefore, DiffVal has two components, an inner one (\frac{a}{b}), which measures the presence of the taxon inside each of the groups, and an outer one (\frac{c}{d}), which measures the relevant absences of the taxon outside of each of the groups. Specifically, given a partition p with k groups, DiffVal is calculated for each taxon s as:

DiffVal_{s,p} = \frac{1}{e}\sum_{g=1}^k{\frac{a}{b}\frac{c}{d}}

where:

• a, is the total number of presences of taxon s within group g.

• b, is the total number of relevés of group g.

• c, is the total number of differentiating absences of taxon s, i.e., absences coming from the groups other than g from which the taxon s is completely absent.

• d, is the total number of relevés of all groups but g (i.e., the total number of relevés in the table - b).

• e, is the total number of groups in which the taxon s occurs at least once.

Therefore, for each taxon s and for each group g, the DiffVal index evaluates:

• \frac{a}{b}, i.e., the frequency of the presences of taxon s, relative to the size of group g; commonly called 'relative frequency.' \frac{a}{b} is only 1 if and only if taxon s occurs in all the relevés of group g.

• \frac{c}{d}, i.e., the frequency of the differentiating absences of taxon s outside group g, relative to the sum of sizes of all groups but g. Nota bene: absences in c are counted outside the group g but only in the groups from which taxon s is completely absent (these are the relevant absences, which produce differentiation among groups); in practice c corresponds to the sum of the sizes of all groups other than g that are empty. \frac{c}{d} is 1 if and only if the taxon s is absent from all groups but g.

Finally, \frac{1}{e} ensures that DiffVal is a value from 0 to 1.

The Total Differential Value (TDV or TotDiffVal) of a phytosociological table M tabulated/sorted by the partition p is:

TDV_{M,p} = \frac{1}{n}\sum_{i=1}^n{Diffval_{i,p}}

where:

• n, is the number of taxa in table M.

The division by the number of taxa present in M ensures that TDV remains in the [0,1] interval (as DiffVal is also in the same interval).

Value

If output_type = "normal" (the default) pre-validations are done and a list is returned, with the following components:

ifp

A matrix with the \frac{a}{b} values for each taxon in each group, for short called the 'inner frequency of presences'.

ofda

A matrix with the \frac{c}{d} values for each taxon in each group, for short called the 'outer frequency of differentiating absences'.

e

A vector with the e values for each taxon, i.e., the number of groups containing that taxon.

diffval

A matrix with the DiffVal for each taxon.

tdv

A numeric with the TDV of matrix ⁠m_bin,⁠ given the partition p.

If output_type = "full", some extra components are added to the output: afg, empty.size, gct (= e) and i.mul. These are intermediate matrices used in the computation of TDV.

If output_type = "fast", only TDV is returned and no pre-validations are done.

Author(s)

Tiago Monteiro-Henriques. E-mail: [e-mail].

References

Monteiro-Henriques T. & Bellu A. 2014. An optimization approach to the production of differentiated tables based on new differentiability measures. 23rd EVS European Vegetation Survey. Presented orally. Ljubljana, Slovenia.

Monteiro-Henriques T. 2016. A bunch of R functions to assist phytosociological tabulation. 25th Meeting of European Vegetation Survey. Presented in poster. Rome. Italy.

Mueller-Dombois D. & Ellenberg H. 1974. Aims and Methods of Vegetation Ecology. New York: John Wiley & Sons.

Examples

# Getting the Taxus baccata forests data set
data(taxus_bin)

# Creating a group partition, as the one presented in the original article of
# the data set
groups <- rep(c(1, 2, 3), c(3, 11, 19))

# Removing taxa occurring in only one relevé, in order to reproduce exactly
# the example in the original article of the data set
taxus_bin_wmt <- taxus_bin[rowSums(taxus_bin) > 1, ]

# Calculating TDV
result <- tdv(taxus_bin_wmt, groups)

# This is the TDV
result$tdv
# This is TDV1, reproducing exactly the value from the original article
sum(result$diffval / result$e) / nrow(taxus_bin_wmt)