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)