Traditional phytosociological work was based on the subjective delimitation of vegetation units, made either already during the field reconnaissance and sampling or in the process of manual sorting of releves and species within tables. The need for more formal, transparent, efficient, and repeatable classification procedures led to the introduction of numerical classification methods in phytosociology since the 1960s. They can be either agglomerative or divisive. Agglomerative methods start with linking individual releves based on the similarity of their species composition, forming releve clusters and subsequently linking these clusters to form a hierarchical classification, usually presented as a dendrogram. Divisive methods start with dividing the set of releves into subsets, which are further divided into subsets on a lower hierarchical level, thus eventually proceeding to the single releves. The most popular divisive method is two-way indicator species analysis (TWINSPAN), which uses the ordination method of correspondence analysis to divide the releves into subsets. Simultaneously with the classification of releves, TWINSPAN classifies species, and produces an ordered species-by-releve table similar to that used in traditional phytosociology (see the section entitled 'Phytosociological tables'). The classifications of the same data sets produced by agglomarative clustering and TWINSPAN usually roughly correspond but differ in details. Agglomerative clustering is the method of choice when cluster homogeneity is the principal goal, while TWINSPAN better reflects the main gradients in species composition of the input data set. An important choice in any numerical procedure is the transformation of cover-abundance data, which determines to what degree species cover-abundance will be accounted for in the analysis.
In addition to numerical classification, phytosociology frequently uses various ordination methods, such as correspondence analysis (CA), detrended correspondence analysis (DCA), or principal components analysis (PCA). Sometimes, ordination and classification are perceived as antagonististic approaches, representing the Gleasonian continuum concept and the Clementsian concept of superorganism, respectively. However, phytosociologists never engaged in that ideological debate, and nowadays both approaches seem to be reconciled: classification studies often use ordination to visualize the position of vegetation units along gradients, and ordination patterns are used to propose the delimitation of releve groups for certain purposes.
Applicability of an established classification crucially depends on finding those species that are typical of releve groups (vegetation units) and make them recognizable by simple floristic criteria. Such species may include the most frequent species, dominant species, or diagnostic species. The former two groups of species can be easily defined by setting some threshold of constancy or cover-abundance values that a species must exceed to be considered as frequent (constant) or dominant, respectively. Diagnostic species are determined based on the concept of fidelity, which quantifies the degree of concentration of a species' occurrence or abundance in the releves of the target vegetation unit. If a species occurs mainly in the releves of the target vegetation unit while it is largely absent elsewhere, it is considered as faithful to this vegetation unit. Fidelity can be quantified by various statistical measures. If it is based on species presence/absence, various measures of association between categorical variables can be used, for example, chi-square, G statistic, or phi coefficient of association. Some fidelity measures have also been proposed to deal with cover-abundances, for example, the Dufrene-Legendre indicator value. The properties of different fidelity measures vary slightly, for example, with respect to the weight given to rare or common species. Statistical significance of fidelity can be either derived directly from the values of some of these measures or determined by a separate procedure such as permutation test. Apart from the selection of the appropriate fidelity measure, fidelity can be measured in two different ways. First, species occurrence in the target group of releves can be compared with all the releves in the data set that do not belong to the target group, irrespective of the divisions of the rest of the data set. Second, species frequency in that group of releves where it is most common is compared with its frequency in the group where this species is the second most common. In both cases, some arbitrary threshold fidelity value is selected and species that exceed this value are considered as diagnostic. The first approach is not affected by the divisions of the data set outside the target vegetation unit, thus yielding a more general result, whereas the latter approach is only valid in the context of a given table or classification, but it provides a clearer separation of vegetation units through diagnostic species within this table or classification. The results of both approaches depend on the geographical extent, sampling design, and delimitation of the available set of releves, the 'universe of investigation'.
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