I present the Total Differential Value (TDV), an index designed for vegetation analysis based on the operational concept of differential species, as classically illustrated by Heinz Ellenberg and Dieter Mueller-Dombois. Given a phytosociological table and a grouping of its relevés, TDV is obtained by averaging the Differential Value (DiffVal) for each species in the table. DiffVal, grounded in combinatorial-discrete mathematics, quantifies the differential power of a species. The novelty of this approach lies in its distinction between two types of species absences: (i) absences from some relevés within a group and (ii) absences from all relevés representing a group. By leveraging the distribution of species absences among groups, this method effectively quantifies the differential power and distinguishes differential from non-differential species. I illustrate the computation of DiffVal and TDV and show that, because only differential species contribute to TDV, it reflects the strength of the differential species patterns in a classified table. TDV can be optimized (TDV-optimization), providing partitions of relevés. I demonstrate TDV-optimization using both an artificial and a well-known real-world data set. Key features of this method include its ability to identify patterns very closely resembling manual phytosociological tabulation and to detect reticulate patterns. TDV-optimization may lead to partitions where outlier or extreme relevés are isolated in groups; however, enforcing a minimum group size can highlight partitions with more balanced group sizes. An R package is now available, implementing DiffVal and TDV calculation as well as TDV-optimization.

Abbreviations: DiffVal = Differential Value; EllPar = Ellenberg’s partition into three groups; NDR = number of discrepant relevés; NSS = number of significant species; TDV = Total Differential Value.

biclustering, block structure, differential species, optimization, patterns, phytosociological tabulation, tabular classification, TDV-optimization, vegetation

Vegetation is a complex phenomenon driven by many deeply interacting factors. As bottom-up mechanistic approaches to vegetation are often hindered by this complexity, physiognomic-floristic patterns are sought as a means to simplify and handle this phenomenon (Armbruster 1995; Austin 2013).

Vegetation classification is the process of identifying, describing and interrelating vegetation units using vegetation relevés (see, e.g., Peet and Roberts 2013, p. 29). In specialized fields such as data science, machine learning, or pattern recognition, the term classification typically refers to the assignment of new individuals to existing classes, while clustering or partitioning refers to the process of identifying those classes (Bishop 2006; Witten 2009; Drummond 2011; Han et al. 2012; Bouveyron et al. 2019). In vegetation science, however, ‘classification’ is used more broadly to encompass both the establishment of classes and the assignment of objects to those classes.

Early in its history, vegetation classification was conducted solely by manipulating matrices of vegetation samples (relevés), as later described in detail by Ellenberg (1956) and translated to English by Mueller-Dombois and Ellenberg (1974). This manipulation forms the basis of Braun-Blanquet’s approach to vegetation, with the main purpose being the partitioning of relevés. In this process, the researcher seeks to divide the relevé data set into groups (subsets of relevés) that can be distinguished floristically (i.e., by subsets of the occurring species). The process is potentially highly subjective and particularly complex, even for moderately sized tables. The number of groups is expected to emerge naturally during the manipulation, adding further complexity. This technique is called tabulation (Poore 1955), tablework or table sorting (Češka and Roemer 1971), tabular sorting (Dale and Quadraccia 1973; Dale and Webb 1975), or tabular classification (Wildi 1989). The output of the technique is referred to as a differentiated table (Ellenberg 1956; Spatz 1972; Mueller-Dombois and Ellenberg 1974), or colloquially as a classified, ordered, rearranged or sorted table – simply put, a tabulation. After obtaining the groups, it is common to make fine adjustments to the species sorting, such as highlighting species considered characteristic of already known higher-ranked syntaxa, primarily to improve the final display of the table. Thus, part of this fine-tuning relies on information not contained in the relevé matrix (e.g., the current syntaxonomic system) and should be treated separately from clustering/partitioning, to which it is irrelevant.

As described in Ellenberg (1956), Whittaker (1962), and Mueller-Dombois and Ellenberg (1974), the first step of the tabulation is the recognition of species groups, known as differential species. These groups are identified based on their co-occurrence in a certain number of relevés, while being absent from others. Henceforth, I refer to these as patterns of differential species.

The concept of differential species dates back over 100 years (Braun-Blanquet 1925) and forms the foundation of the tabulation technique, as well as the delimitation of vegetation types (for a detailed analysis of the concept and its changes over the years, see Appendix 1). Mueller-Dombois and Ellenberg (1974, p. 178) describe “differential (or differentiating) species” as those found among the “species with restricted distributions or amplitudes in the relevés under comparison”. These authors operationalized the concept in its broadest sense, identifying species that are present in one or more groups but absent from others (such species cannot occur in all the emerging groups; they must be absent from at least one group to be considered differential). In this work, I adopt this definition, further exploring the notion of exclusiveness as presented and illustrated by Mueller-Dombois and Ellenberg (1974). It is worth noting that their definition differs slightly from the one given by Braun-Blanquet (1925, p. 134; 1932, p. 59), a distinction thoroughly examined in Appendix 1.

The concept of differential species is closely linked to two other key concepts from the Braun-Blanquet school: (i) characteristic species and (ii) fidelity. These concepts have also endured changes over the years. For a detailed discussion, see Appendix 1.

During the first half of the 20^th century, researchers discussed the feasibility of vegetation classification based on physiognomic-floristic patterns, emphasizing the need for objective sampling and classification methods (Goodall 1953; Whittaker 1962). A vast array of numerical methods for sampling vegetation and identifying patterns emerged in the latter half of the 20^th century (e.g., Whittaker 1962; Mueller-Dombois and Ellenberg 1974; Mucina and van der Maarel 1989) and continues into the 21^st century (e.g., Peet and Roberts 2013). Most methods aimed to replicate or approximate the tabulation technique, yet with diverse approaches and results. With such an abundance of methods, it seems unlikely that the current status quo can still benefit from further contributions.

The results of numerical methods are often unsatisfactory for vegetation scientists (Roberts 2015), as they fail to replicate traditional tabular outputs (Mucina 1997). This may stem from the inappropriateness of certain algorithms for vegetation classification (Schmidtlein et al. 2010). Table 1 synthesizes the most relevant clustering/partitioning approaches used in vegetation analysis (including the one presented in this work). The table highlights the grouping approach (or model) and the space in which the groups are sought. The inherent characteristics of the model and space can influence the method’s suitability for the analysis objectives, as well as its consistency with theoretical models, expert knowledge or even ground truth. To address this, some key characteristics that can impact consistency are presented in Table 1 as questions. As the present work focuses on phytosociological tabulation as outlined in Mueller-Dombois and Ellenberg (1974), I highlight in green those answers that satisfy the tabulation mechanics and align with the concept of differential species, and I highlight in red those that do not. Four topics are addressed:

(4) Type of informative species

For those methods that recognize the relevance of subsets of species in the emergence of relevé groups, it is crucial to ensure the exclusivity of these species to one or more emerging groups, as this is a fundamental aspect of the tabulation technique. Note that several modern approaches to quantifying fidelity or indicator value use a concentration-based approach, which does not guarantee exclusivity to specific groups (see Appendix 1 for a detailed discussion of this issue). Relying on a concentration-based approach when identifying subsets of informative species may lead to different subsets from those obtained through phytosociological tabulation.

When the clustering/partitioning approach does not meet these four requirements of the phytosociological tabulation (italicized above), it risks misplacing relevés compared to the tabulation technique.

Madeira and Oliveira (2004) reviewed biclustering methods. I use their terminology to describe biclustering structures. The traditional tabulation technique aims to achieve exclusive-row and -column biclusters (see figure 1b in Podani and Feoli 1991; or figure 4 in Madeira and Oliveira 2004). Figure 1A depicts an exclusive-row and -column bicluster structure (dark grey and white areas), but non-exhaustive-in-the-rows (note the light grey area at the bottom, representing rows not attributed to any bicluster). The relaxation of the row’s exhaustiveness is significant in vegetation science, considering the high occurrence of dominant and companion species that may span all groups under comparison. Since the ideal scenario of an exclusive-row and -column diagonal of blocks is rarely observed in real-world data, there is an additional relaxation of column exclusiveness. This means that only exclusive-row biclusters are sought, where relevés can be associated with multiple dark grey blocks, but each species is still assigned to one block only. This adjusted structure is shown in Figure 1B.

Figure 1. 

Diagrams of bicluster structures (species in rows, relevés in columns). Dark and light grey blocks enclose the occurrences of species, while white areas contain only the absences of species. A) An exclusive-row and -column, non-exhaustive-in-the-rows structure. B) A relaxation of the previous structure, where only exclusive-rows, non-exhaustive-in-the-rows biclusters are depicted.

In manual tabulation, mutually exclusive groups of all columns (i.e., a partition of the relevés) are adjusted to the identified bicluster blocks to reveal exclusive species patterns among the final relevé groups. Figure 2A illustrates how a partition (represented by groups I, II and III) is overlaid on the bicluster structure from Figure 1B, showing how the groups are differentiated by the presence (or presence and absence) of certain species. The block shapes and groups in Figure 2A roughly depict the final tabulation from Mueller-Dombois and Ellenberg (1974) for the Arrhenatheretum data set.

Figure 2. 

Diagrams of bicluster structures and partitions of the columns/relevés as sought in the phytosociological tabulation. A) A partition of the columns/relevés (groups I, II and III) superimposed on the biclustering structure from Figure 1B. B) Illustration of some hypothetical species presences (1) and absences (0) over a simplified version of the biclustering structure shown on the left. Stochastic absences are shown in red, and differentiating absences are in green (see text for further explanation).

The bicluster structure in Figure 2B simplifies that of Figure 2A: the dark grey blocks, which enclose all presences of a species, are now aligned with groups I, II, and III, merging where appropriate into unique blocks. These blocks represent sets of differential species for a given group or combination of groups. For example, species 9 is a differential species for groups II and III because it is absent from group I. Nota bene 1: White areas (where only absences are recorded) ensure that a species is absent from at least one group – an essential property for differential species under the definition used here. Nota bene 2: In Figure 2B, all species within a block have at least one presence in each associated group. For clarity, if species 9 lacked its two presences in group II, it would instead be a differential species of group III and placed in the dark grey block below.

Recall that the bottom light grey area contains species left outside the biclusters. In phytosociological tabulation, some species absent from one or more relevé groups may remain in this area, though this can be considered subjective. In an automatic classification, however, each species is preferentially assigned to a unique block, ensuring that each block contains only species present in all its associated groups. Any later adjustments can be justified by the expert. Thus, in Figure 2B, the light grey area includes only species occurring at least once in all three groups (I, II, and III), exemplified by species 15.

Looking carefully at Figure 2B, it is possible to recognize two different kinds of absences. The first kind, represented in red, is inside the dark grey or the light grey areas, where it is possible to find absences together with occurrences of the species (remember that species associated with these areas show at least one presence in the associated groups). If present, such absences are generally seen as stochastic absences. The absences of the second kind (represented in green) occur inside the white areas, where there are no presences at all. As these absences allow the species to differentiate certain groups, I call them differentiating absences. The former can be attributed to stochastic processes that determine the absence of a species from a plot, even if such species is present in the species pool of the respective group(s). The latter can be attributed to a complete absence from the species pool of the respective group(s), e.g., due to biogeographical-related causes or strong environmental filtering.

The Total Differential Value (TDV), proposed in this work, seeks to answer the following question: “Can a numerical index be created to globally reflect the strength of differential species patterns in a differentiated table?” To answer this question, TDV leverages the distinction between stochastic and differentiating absences.

An ideal block of differential species is characterized by a high density of presences, coupled with a total absence from at least some of the remaining groups, preferably from all. An ideal tabulation would display large blocks of exclusive species, differentiating each group. However, in practice, the number of species between blocks varies, as does the density of occurrences of each species within the same block.

The usefulness of a single species in differentiating a specific group of relevés increases with (i) the frequency of the species’ presences in the group, and (ii) its exclusivity to the group, meaning it is absent from the other groups as much as possible. This feature is commonly sought in species indicator power measures (e.g., Dufrêne and Legendre 1997) and clustering sharpness measures (e.g., Podani and Feoli 1991). The maximum values of these measures occur when a species is present in all sites of one group, and only absences are recorded outside that group. The Differential Value (DiffVal), presented below, exhibits this same behaviour while introducing an innovative approach to considering absences outside the groups.

Given a phytosociological table T and a k-partition P of its relevés (i.e., a classification of all relevés into k groups, with each relevé assigned to only one group), the DiffVal expresses the ability of a species, say s, to differentiate the groups in which it occurs from the other groups. The DiffVal of species s, given partition P, is calculated by the following formula:

$DiffVa{l}_{s,P}=\frac{1}{e}\sum _{g=1}^k\frac{a_g}{b_g}\times \frac{c_{g’}}{d_{g’}}$ (1)

where a_g is the total number of presences of species s within group g; b_g is the total number of relevés in group g; c_g’ is the total number of differentiating absences of species s in groups other than g; d_g’’ is the total number of relevés in all groups excluding group g; and e is the total number of groups in which species s occurs at least once.

DiffVal is, therefore, a summation of k summands, each one representing the partial differential value of the species s to each one of the k groups. Note that a_g/b_g is the frequency of the presences (of species s inside group g), relative to the group g size (i.e., the number of relevés in g). This is the constancy of species s to group g (Mueller-Dombois and Ellenberg 1974). The ratio will equal its maximum value, 1, if, and only if, species s occurs in all the relevés of group g. Additionally, note that c_g’/d_g’ is the frequency of the differentiating absences (of species s outside group g), relative to the sum of the sizes of all groups but g. That is, absences in c_g’ are counted outside the group g but only in the groups from which species s is completely absent (as these are the relevant absences, which produce differentiation among groups). In practice c_g’ corresponds to the sum of the sizes of all groups other than g that are empty. Thus, c_g’/d_g’ will equal its maximum, 1, if, and only if, species s is completely absent from all groups but g. Finally, 1/e ensures that DiffVal is a value from 0 to 1. Note that DiffVal is simply based on counting the presences and absences of a species, and the size of the given groups (i.e., it is grounded in combinatorial-discrete mathematics) and was devised to measure the differential power of a species.

The TDV is the mean of the DiffVal outcomes for species in table T. It is a global measure (ranging from 0 to 1) of how well the k groups can be distinguished from one another using differential species. Thus, the TDV of a partition P over the phytosociological table T is given by:

$TD{V}_{P,T}=\frac{1}{n}\sum _{s=1}^{n}DiffVa{l}_{s,P}$ (2)

where n is the total number of species in table T and DiffVal_s,P is the differential value of species s, given partition P.

Table 2 exemplifies the calculation of DiffVal and TDV on a hypothetical matrix of seven species and nine relevés partitioned into three groups. For computation purposes it is worth mentioning that when a species is not present in a group, the respective DiffVal summand is always zero (since the relative frequency a_g/b_g equals zero when a_g = 0), meaning that the summands only need be computed for the groups containing the species. Similarly, when the species occurs in all groups, all summands are zero (since c_g'/d_g' equals zero for all summands where c_g' = 0). In such cases, performing the calculation is unnecessary, as DiffVal will also be zero. For illustrative purposes, all calculations are shown in Table 2.

Note that an interesting property emerges from the way the distinction between stochastic and differentiating absences is incorporated into DiffVal. Specifically, the distribution of differentiating absences among the groups under comparison allows for the establishment of exclusiveness degrees (i.e., species exclusive to one group, two groups, three groups, etc.), quantifying the differential power of the species and ultimately distinguishing differential from non-differential species. The latter species, which occur in all vegetation groups of the phytosociological table, do not contribute to TDV, as their respective DiffVal is zero.

The number of possible partitions of a set with n elements is well-studied in combinatorics and corresponds to the n^th Bell number (Bell 1934), usually denoted B_n. For example, 10 relevés can be partitioned in B₁₀ = 115,975 different ways. If we constrain the number of groups, say to k, the possible partitions with k groups (of the same n elements) correspond to the Stirling number of the second kind (Boyadzhiev 2012), usually denoted S (n,k). For instance, the same 10 relevés can be partitioned into three groups in S (10,3) = 9,330 different ways, and into five groups in S (10,5) = 42,525 different ways. The “Stirling2” function from the R package multicool (Curran et al. 2024) implements these calculations.

The TDV metric may provide a way to assist tabulation. Specifically, if we accept that DiffVal is a good measure of the differential value of a species, and if we accept that TDV is a good measure of how well a partition is characterized in terms of the strength of the differential species patterns it contains, then optimizing TDV would allow us to find tabulations close to the ones sought in Braun-Blanquet’s school.

As Bell numbers and Stirling numbers of the second kind increase rapidly, calculating TDV for all possible partitions of the relevés in a phytosociological table is often impracticable. In other words, complete enumeration is not a viable option, as is the case with other criteria (see, e.g., Podani and Feoli 1991; Peet and Roberts 2013). However, appropriate optimization techniques (e.g., local search-based metaheuristics, also known as non-hierarchical iterative reallocation algorithms) can be used to maximize TDV, as will be shown hereafter.

R package diffval (Monteiro-Henriques and Cerdeira 2023) includes functions dedicated to the optimization of TDV. In the following subchapters, I will illustrate TDV-optimization using variations of the hill climbing algorithm (a local search technique; see, e.g., Basseur and Goëffon 2015), implemented in function “optim_tdv_hill_climb”. The function’s documentation provides a detailed description of the optimization procedure. Monteiro-Henriques and Cerdeira (2023) also provide functions for a mixed-integer linear programming approach (for the two-group case) and a simulated annealing algorithm (for two or more groups). Additionally, these authors are working on a mathematical proof showing that finding a partition that maximizes TDV is NP-hard (a class of problems in computational complexity theory) for two or more groups.

In the diffval package, users can also find the “tabulation” function, which, given a phytosociological table and a partition of its relevés, rearranges the table’s rows and columns to display exclusive species at the top. The reordering follows these steps: (i) Species are first ordered by the increasing number of groups in which they occur (i.e., by exclusiveness level/degree). (ii) Within each exclusiveness level, species are further ordered lexicographically based on the groups to which they belong. (iii) Finally, within each level defined by the previous two steps (also known as shortlex order), species are reordered by the decreasing sum of their relative frequencies across the groups in which they occur.

The columns are also reordered according to the increasing order of the assigned group membership numbers. Optionally, the rearranged table can be visualized graphically (an example is provided below).

It is important to note that the tabular rearrangement performed by the “tabulation” function is based on simple combinatorial rules (primarily the shortlex order). Its effectiveness in generating a meaningful phytosociological tabulation depends entirely on the data and the given partition.

I illustrate TDV-optimization using an artificial data set (Example 1), and a real-world data set (Example 2). I contrast TDV-optimization with ten clustering/partitioning methods (see Table 3) using seven statistics (abbreviations in bold): (1) PARTANA, the ratio of within-cluster similarity to among-cluster similarity (Roberts 2015), (2) the average silhouette width (ASW, Rousseeuw 1987), (3) the minimum silhouette width (MSW, Lengyel et al. 2021), (4) OptimClass 1, the total count of faithful species across all clusters of the partition, at significance levels of 0.05, 0.01, 10^-3 and 10^-6 (Tichý et al. 2010), (5) the sum of significant indicator values (IndVal) assigned to the cluster where each species has its highest indicator value, at α = 0.05 (ssIndVal, see Dufrêne and Legendre 1997; Roberts 2015), (6) the number of significant indicator species that contribute to ssIndVal (nsIndic, see Dufrêne and Legendre 1997), and (7) The average ISAMIC score (avISAMIC, see Roberts 2015). For both examples, I provide additional statistics such as the number of discrepant relevés in comparison to known partitions, the number of detected differential species, and, of course, the TDV.

IndVal p-values are calculated using permutations. To obtain stable estimates, ssIndval and nsIndic are averaged over 100 runs of the “indval” function from the labdsv package (Roberts 2023), with 9999 randomizations per run.

For all methods that required dissimilarity matrices, the Bray-Curtis index (Bray and Curtis 1957) was used, calculated using the vegan (Oksanen et al. 2024) and labdsv (Roberts 2023) packages, as it is one of the most widely used metrics in vegetation science. I focused on three groups, as the artificial data set in Example 1 follows a three-group structure, and the real-world data set in Example 2 is traditionally classified into three groups.

OptimClass 1 was calculated using Fisher’s exact test for the right-tailed hypothesis (Tichý et al. 2010), ensuring that each species was counted only once, even when the test was significant for two groups. Note that these methodological choices are implemented differently in the function “optimclass” from the optpart package (Roberts 2020). Suppl. material 1 contains the functions used to calculate OptimClass, adapted from Roberts (2020).

Example 1: An artificial data set

Reticulated patterns are misaligned subsets of species that produce multiple, sound groupings of relevés (see also Dale 1988). Holzner et al. (1978) provided an extreme example of reticulate patterns of differential species, highlighting the challenges of treating these collectively. As a solution, the authors proposed working with subsets of species to obtain appropriate clusters. Since DiffVal does not assign differential value to species present in all groups, I hypothesize that TDV-optimization, using a local search-based metaheuristic such as random restart hill climbing, can focus on different subsets of the relevé matrix. This approach may help detect reticulate patterns among partitions with higher TDV.

To test this, I created an artificial data set (107 species × 64 relevés) containing two distinct patterns of differential species distributed across three groups. The first pattern included three blocks of 16, 18, and 30 relevés, with 6, 8 and 12 differential species, respectively. The second pattern comprised three additional blocks of 14, 22, and 28 relevés, with 5, 11, and 9 differential species, respectively. These two patterns were randomly interwoven, ensuring that the blocks composing each pattern remained misaligned. The remaining 56 species were considered common-to-rare, without any associated pattern. Species presences within each block were randomly generated with varying theoretical relative frequencies, decreasing exponentially from 0.9 to around 0.1. A similar process was applied to the 56 common-to-rare species, but their presences were distributed across all 64 relevés, with frequencies decreasing exponentially from 1.0 to around 0.05.

The first pattern of differential species defined three groups of relevés, labelled 1, 2, and 3 (hereafter pattern ‘123’), while the second pattern defined three groups labelled a, b, and c (pattern ‘abc’). Each relevé was assigned both a number and a letter based on the differential species it contained (Figure 3). Figure 3A makes pattern ‘123’ apparent by sorting relevés according to their numerical labels, whereas Figure 3B reveals pattern ‘abc’ by sorting them by their letter labels. Species occurrences in each block of differential species are indicated by different colours, while the 56 common-to-rare species are shown in grey. The code to generate the artificial data set is provided in Suppl. material 2.

I submitted this data set, with relevés in random order, to the classification procedures listed in Table 3. Recall that there are S (64,3) possible ways to partition the 64 relevés into three groups – an astonishing 7.6×10¹⁸ different solutions to this specific classification problem.

Optimization procedures (OPTPART, OPTSIL, and TDV-optimization) were run multiple times with random initial partitions, retaining local optima for further analysis. OPTPART was run 50,000 times with default settings: a maximum of 100 iterations per run and a minimum PARTANA ratio increment of 0.001 to continue iterating. OPTSIL was run 20,000 times with default values for the maximum of 100 iterations per run. TDV-optimization was run 5,000 times using the “optim_tdv_hill_climb” function (Monteiro-Henriques and Cerdeira 2023). Each run began with 1,500 iterations of stochastic hill climbing, followed by up to 30 iterations of greedy hill climbing (see Basseur and Goëffon 2015). These iteration numbers were chosen based on initial tests, which confirmed that TDV plateaued in single runs. The 500 highest-TDV partitions were retained for further analysis. The minimum group size (min_g_size) was set to the default value of 1.

For each optimization procedure, I analysed the partition with the highest observed value of the optimized criterion and searched the local optima for partitions closest to patterns ‘123’ and ‘abc’. For the OPTSIL+REMOS method, I also analysed the solution with the highest ASW and searched the local optima for partitions as close as possible to these patterns. Partition discrepancy (or distance) was measured as the minimum number of relevés that must be reassigned to obtain one partition from another (see MINDMT in Podani 1986). Hereafter, I refer to this metric as “number of discrepant relevés” (NDR) followed by the target partition (e.g., NDR’123’ and NDR’abc’). Suppl. material 1 includes the function for calculating discrepant relevés between two partitions. Suppl. material 3 displays the graphical output of the “tabulation” function for each partition, sorting species by their degree of exclusiveness. Each output provides additional information on OptimClass 1 and IndVal significance, highlighting the species in patterns ‘123’ and ‘abc’. Alongside the tabulations, biplots of nonmetric multidimensional scaling (NMDS) and correspondence analysis (CA) are provided (Oksanen et al. 2024), highlighting the partition groups.

Table 4 shows the statistics obtained for the partitions returned by each method. In addition to the previously mentioned statistics, Table 4 includes:

1. The number of significant species among the 56 common-to-rare species (NSS-c-to-r).
2. The number of significant species among the 26 species composing the pattern ‘123’ (NSS’123’).
3. The NSS’123’ that are exclusive to one or two groups (NSSExc’123’).
4. The number of significant species among the 25 species composing the pattern ‘abc’ (NSS’abc’).
5. The NSS’abc’ that are exclusive to one or two groups (NSSExc’abc’).
6. The number of discrepant relevés relative to pattern ‘123’ (NDR’123’).
7. The number of discrepant relevés relative to pattern ‘abc’ (NDR’abc’).

Significance for (1) to (5) was determined using Fisher’s exact test, exactly as in OptimClass 1, with a significance level (α) of 0.05. The sum of NSS-c-to-r, NSS’123’, and NSS’abc’ equals OptimClass 1 (α = 0.05).

NSS-c-to-r counts how many of these 56 species showed a statistically significant concentration in at least one group. At the commonly accepted significance level of α = 0.05, an average of 2.8 species are expected to be falsely detected as concentrated in at least one group. Table 4 shows that some methods (k-means, ISOPAM, Flexible β, OPTPART, Ward and PAM) produced particularly high NSS-c-to-r values (8 to 14). These high false positive rates (14 to 25%) stem from the fact that group formation was not independent of the distribution of these 56 species (see Discussion). These methods also yielded the highest OptimClass 1 values (α = 0.05), ranging from 34 to 45. The solution for such an overabundance of significant species is usually to subjectively decrease α (Tichý et al. 2010; Lötter et al. 2013; Roberts 2015; Lengyel et al. 2021). However, this greatly reduces the test’s statistical power. OptimClass 1 declines sharply for α values lower than 0.05, especially for these six methods. Values range from 9 to 14, for α = 10^-3, and from 2 to 6, for α = 10^-6. Recall pattern ‘123’ consisted of 26 differential species, while pattern ‘abc’ contained 25. Suppl. material 4 provides an analysis of the statistical power of Fisher’s exact test, adjusted to the scale of Example 1, illustrating the extent to which lowering α reduces differential/concentrated species detectability. Suppl. material 5 includes a graphical analysis of how OptimClass 1 varies with α across clustering/partitioning methods.

Despite the high false positive rates among the common-to-rare species, k-means and OPTPART detected 32 of the 51 differential species as significant (NSSExc’123’ + NSSExc’abc’). ISOPAM found 31. These are notably high values, given that the corresponding partitions deviate substantially from patterns ‘123’ and ‘abc’ and represent a mixture of some of the artificially created groups.

For partitions J and K, which are closer to partitions ‘123’ and ‘abc’, the figures are necessarily different, showing lower values of NSS-c-t-r. Unsurprisingly, partitions ‘123’ and ‘abc’ (found by OPTSIL, OPTSIL+REMOS and TDV-optimization) have OptimClass 1 values (α = 0.05) that are closer to the actual number of differential species composing each pattern. These numbers reflect high statistical power, ranging from 88 to 96% (see the NSS of the respective pattern), and acceptable false positive rates, ranging from 4 to 5% (see NSS-c-to-r and the NSS of the other pattern).

It is worth mentioning that: (i) Mod-TWINSPAN and DIANA had relatively low NSS-c-to-r values; (ii) PAM and ISOPAM showed the highest values of ssIndVal and nsIndic; (iii) Flexible β and DIANA had relatively high values of avISAMIC; and (iv) k-means showed a high value for the PARTANA ratio.

The number of significant indicators (nsIndic) was consistently between OptimClass 1 values for α = 0.05 and those for α = 0.01.

For the optimization procedures, Table 4 shows that, as expected, each method performs well according to its criterion. OPTPART found the partition with the highest PARTANA ratio, OPTSIL the partition with the highest ASW, and TDV-optimization the partition with the highest TDV.

OPTSIL found partition ‘123’ (among the set of local optima) and got extremely close to partition ‘abc’, with just two discrepant relevés (partition K, which also has the highest ASW). It is not possible for OPTISIL to find partition ‘abc’ exactly, as its ASW is lower than the ASW of partition K, and applying OPTSIL directly to partition ‘abc’ leads to convergence with partition K. The increase in ASW can only be explained by the noise that other (randomly distributed) species introduce to the ASW.

In contrast, OPTISIL+REMOS improved the MSW of partition K, converging to partition ‘abc’ (which is a local optimum of OPTISIL+REMOS). However, since the MSW is negative for partition ‘123’, it becomes impossible for OPTSIL+REMOS to find it. Applying the REMOS algorithm directly to partition ‘123’ changes 13 relevés from their original groups, leading to partition L, which has the highest ASW found by the REMOS procedure and is the closest to pattern ‘123’.

SillyPutty (Bombina et al. 2024) has recently been proposed as a novel clustering method, but it refines partitions by reallocating objects with negative silhouettes in the same way as the REMOS1 algorithm (Lengyel et al. 2021). I tested the “RandomSillyPutty” function (50,000 runs starting from random partitions), and the results are identical to those shown in Table 4 for OPTSIL+REMOS. Suppl. material 6 contains an analysis of how discrepant the local optima yielded by the optimization procedures (including SillyPutty) were from the patterns ‘123’ and ‘abc’.

TDV-optimization was the only method capable of converging to local optima that identified both patterns (‘123’ and ‘abc’). However, the highest TDV corresponds to a partition (M) where two groups consist of single relevés. Partition M also exhibits the highest avISAMIC. Nevertheless, when TDV-optimization is applied with the parameter min_g_size = 4, partitions ‘123’ and ‘abc’ are ranked first and second among the returned solutions (see Discussion).

Figure 3. 

Artificial data set displaying its two reticulated patterns of differential species. Species presences are represented as small rectangles (coloured or grey). A) Relevés are sorted by the numbers in their labels, revealing pattern ‘123’ with 26 differential species. B) Relevés are sorted by the letters in their labels, revealing pattern ‘abc’ with 25 differential species.

Table 4.

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Statistics for the partitions returned by the tested methods for the Example 1 data set. The best scores in each row are in bold, and the maximum (or minimum) values are in bold italics.

	Agglomerative clustering		k-means	Mod-TWINSPAN	PAM	DIANA	ISOPAM	OPTPART selected partitions:			OPTSIL selected partitions:		OPTSIL+ REMOS selected partitions:		TDV-optimization selected partitions:
	Ward	Flexible β						maximum found	local optimum closest to ‘123’	local optimum closest to ‘abc’	maximum found and closest to ‘abc’	‘123’ is among local optima	maximum found and closest to ‘123’	‘abc’ is among local optima	maximum found	‘123’ is among local optima	‘abc’ is among local optima
Partition id:	A	B	C	D	E	F	G	H	I	J	K	‘123’	L	‘abc’	M	‘123’	‘abc’
PARTANA	1.207	1.183	1.258	1.189	1.187	1.173	1.258	1.265	1.240	1.255	1.249	1.228	1.256	1.243	1.246	1.228	1.243
ASW	0.0814	0.0714	0.1018	0.0729	0.0723	0.0496	0.1024	0.0989	0.0946	0.1010	0.1042	0.1036	0.1039	0.1014	0.0491	0.1036	0.1014
MSW (×10)	-0.965	-1.545	-0.087	-0.758	-0.925	-1.379	0.012	-0.336	-0.51	-0.461	-0.318	-0.229	0.002	0.022	-2.461	-0.229	0.022
OptimClass 1 (α = 0.05)1	43 (25)	34 (24)	42 (33)	33 (27)	43 (24)	29 (22)	43 (32)	45 (35)	41 (24)	31 (23)	29 (25)	26 (24)	39 (24)	28 (24)	5 (5)	26 (24)	28 (24)
OptimClass 1 (α = 0.01) 1	25 (17)	16 (15)	18 (17)	19 (16)	23 (15)	16 (15)	22 (19)	20 (19)	23 (17)	17 (15)	16 (16)	18 (18)	22 (18)	17 (17)	0 (0)	18 (18)	17 (17)
OptimClass 1 (α = 10-3) 1	10 (6)	12 (11)	11 (10)	11 (10)	11 (6)	10 (10)	11 (10)	9 (9)	14 (11)	12 (11)	12 (12)	14 (14)	16 (13)	14 (14)	0 (0)	14 (14)	14 (14)
OptimClass 1 (α = 10-6) 1	3 (3)	3 (3)	5 (4)	3 (3)	2 (1)	4 (4)	6 (5)	4 (4)	4 (3)	7 (6)	8 (8)	9 (9)	5 (4)	9 (9)	0 (0)	9 (9)	9 (9)
ssIndval (α = 0.05)	10.66	10.41	10.64	9.82	12.08	7.94	12.00	9.99	10.88	9.76	9.74	9.61	11.89	10.01	1.87	9.61	10.02
nsIndic (α = 0.05)	31.93	25.00	30.53	29.39	36.53	20.06	36.37	27.90	30.87	22.32	24.22	21.00	34.27	23.81	2.00	21.00	23.87
avISAMIC	0.6654	0.6846	0.6695	0.6717	0.6709	0.6794	0.6710	0.6636	0.6705	0.6657	0.6640	0.6771	0.6731	0.6670	0.8797	0.6771	0.6670
NSS-c-to-r (α = 0.05)	11	10	8	4	14	5	9	10	11	3	5	2	7	3	4	2	3
	(20%)	(18%)	(14%)	(7%)	(25%)	(9%)	(16%)	(18%)	(20%)	(5%)	(9%)	(4%)	(12%)	(5%)	(7%)	(4%)	(5%)
NSS’123’ (α = 0.05)	17	21	17	16	18	19	18	15	20	6	0	23	19	1	0	23	1
	(65%)	(81%)	(65%)	(62%)	(69%)	(73%)	(69%)	(58%)	(77%)	(23%)	(0%)	(88%)	(73%)	(4%)	(0%)	(88%)	(4%)
NSSExc’123’ (α = 0.05)	10	20	17	13	12	19	18	14	19	3	0	23	19	0	0	23	0
	(38%)	(77%)	(65%)	(50%)	(46%)	(73%)	(69%)	(54%)	(73%)	(12%)	(0%)	(88%)	(73%)	(0%)	(0%)	(88%)	(0%)
NSS’abc’ (α = 0.05)	15	3	17	13	11	5	16	20	10	22	24	1	13	24	1	1	24
	(60%)	(12%)	(68%)	(52%)	(44%)	(20%)	(64%)	(80%)	(40%)	(88%)	(96%)	(4%)	(52%)	(96%)	(4%)	(4%)	(96%)
NSSExc’abc’ (α = 0.05)	12	1	15	12	8	2	13	18	4	20	24	1	5	24	1	1	24
	(48%)	(4%)	(60%)	(48%)	(32%)	(8%)	(52%)	(72%)	(16%)	(80%)	(96%)	(4%)	(20%)	(96%)	(4%)	(4%)	(96%)
NDR’123’	23	13	18	18	21	16	17	21	12	34	36	0	13	36	33	0	36
NDR’abc’	22	39	22	26	25	37	22	20	29	7	2	36	25	0	34	36	0
TDV	0.0515	0.0503	0.0615	0.0601	0.0433	0.0678	0.0586	0.0700	0.0591	0.0700	0.0904	0.1010	0.0511	0.1024	0.1622	0.1010	0.1024

¹ For OptimClass 1, I provide in parentheses the number of significant species that are exclusive to one or two groups.

Example 2: The Arrhenatheretum data set

Ellenberg (1956) used the Arrhenatheretum data set to meticulously illustrate the tabulation technique. That illustration was later translated into English by Mueller-Dombois and Ellenberg (1974) and has since become a classical data set for testing the performance of methodologies aimed at reproducing the technique (e.g., van der Maarel et al. 1978; van der Maarel 1979; Gauch 1982; Kuhn 1983; Popma et al. 1983; Wildi 1989; Bruelheide and Flintrop 1994). Table 5 lists some of these works and their performance compared to Ellenberg’s partition into three groups (hereafter EllPar). The groups found by Ellenberg, as presented in Mueller-Dombois and Ellenberg (1974), are not necessarily the ground truth, but is the best solution found by the author that complies with tabulation requirements. In previous works, different authors tested various aspects of the original data set (e.g., cover vs. presence/absence), and it is not always clear whether the complete data set was used or only the portion that was analysed by Ellenberg (1956). Therefore, for a thorough comparison, I tested the performance of the clustering/partitioning methods listed in Table 3, calculating the same statistics used in Example 1. I used the original data set, trimmed to species with constancy between 10 and 60% as done by Ellenberg, and converted values to presence-absence.

Table 5.

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Research works aiming to classify the Arrhenatheretum data set and their performance compared to the three-group partition proposed by Ellenberg (1956).

Reference	Method	No. of discrepant relevés compared to Ellenberg’s partition
van der Maarel et al. (1978)	TABORD	7
van der Maarel (1979)	Agglomerative clustering (Euclidean distance, Ward’s method)	3
Gauch (1982)	TWINSPAN	7
Gauch (1982)	COMPCLUS (non-hierarchical composite clustering)	2
Kuhn (1983) 1	VEGTAB, using the minimum spanning tree1	6
Popma et al. (1983)	FLEXCLUS, modified single linkage clustering + centroid clustering	8
Wildi (1989)	Agglomerative clustering (complete linkage)	3

¹ This approach does not return a partition of the relevés but a sequence to sort the relevés in the table. The sequence was divided manually into groups by the author and considering the groups given by Ellenberg, for comparison.

For the Arrhenatheretum data set, OPTPART was run 5,000 times with default values for the maximum number of iterations per run (100) and for the minimum increment in the PARTANA ratio required to continue iterating (0.001). OPTSIL was run 5,000 times with default values for the maximum number of iterations per run (100). TDV-optimization was run 500 times using the function “optim_tdv_hill_climb”. Each run began with 300 iterations of stochastic hill climbing, followed by up to 30 iterations of greedy hill climbing. These iteration numbers were chosen based on initial tests, which confirmed that TDV plateaued in individual runs. The 100 highest-TDV partitions were retained for subsequent analysis. The minimum group size (min_g_size) was set to the default value of 1.

OPTPART consistently converged to a two-group solution with a PARTANA ratio of 2.183. In the OPTPART algorithm, “no minimum cluster size is enforced” (Roberts 2020). The fact that it converged into two groups is “an indication that a partition with a fewer number of clusters achieves a better within/among similarity ratio than does a larger number” (Roberts 2020). This is consistent with the idea expressed by van der Maarel et al. (1978) and van der Maarel (1979) that a two-group partition could be appropriate to classify the Arrhenatheretum data set. A solution with a higher PARTANA ratio may exist for a three-group partition, but as no other partitions were returned by OPTPART (not even as local optima), this method is not included in the upcoming comparisons.

Table 6 shows the statistics calculated for the partitions of the Arrhenatheretum data set, as obtained by each method. The specific statistics in this example are:

1. The number of species not considered differential by Ellenberg, but that are exclusive to one or two groups (nEll-not-diff-but-excl).
2. The number of species considered differential by Ellenberg that are exclusive to one or two groups (nEll-diff-found).
3. The number of discrepant relevés, considering Ellenberg’s partition (NDR’EllPar’).

Suppl. material 7 displays the graphical output of the “tabulation” function for each partition. Each output provides additional information on OptimClass 1 and IndVal significance, highlighting the species considered differential by Ellenberg (1956).

OPTSIL found the solution with the highest ASW, followed by OPTSIL+REMOS. TDV-optimization yielded the highest values of TDV and avISAMIC. The second highest value of TDV was obtained by Mod-TWINSPAN. TDV-optimization also found the solution with the highest PARTANA ratio, again followed by Mod-TWINSPAN (but recall that OPTPART was not included in the comparison).

Ellenberg (1956) highlighted 26 differential species in the data set. OptimClass 1 for α = 0.05 presented values close to the Ellenberg figure. As observed in Example 1, lowering α involves a sharp loss of statistical power and erratic behaviour in identifying to the best-performing method.

In general, all methods identified a relatively high proportion of the differential species defined by Ellenberg (nEll-diff-found), with Ward, k-means and OPTSIL performing relatively worse.

The local optimum closest to EllPar found by the OPTSIL algorithm is the same partition produced by k-means (partition c). Partition c has the highest number of species exclusive to one or two groups (nEll-not-diff-but-excl + nEll-diff-found), but the respective TDV is relatively low. Recall that DiffVal weighs the constancy in groups and the degree of exclusiveness. Partition c has a relatively low number of species exclusive to just one group.

The local optimum closest to EllPar found by the OPTSIL+REMOS procedure is the same partition produced by Flexible β (partition b), differing from EllPar by only three discrepant relevés.

ISOPAM produced an interesting partition that, despite having eight discrepant relevés (NDR’EllPar’), closely resembled EllPar in terms of differential species. The Bromus erectus group remains the same as in EllPar, with the difference lying in how the remaining relevés are split. The tabulation shown in Suppl. material 7: figure S7 reveals that ISOPAM’s split designates Pastinaca sativa as a differential species for one group and Silaus pratensis for the other, though both exhibit low constancy within their respective groups. In contrast, EllPar (Suppl. material 7: figure S14) identifies Deschampsia caespitosa, Angelica sylvestris, Carex acutiformis, Filipendula ulmaria, and Pimpinella magna as differentials for one group, while the other group lacks its own differential species. The differential species block is moderately dense, but every relevé within it contains at least one species from the block.

The statistic nEll-not-diff-but-excl counts the number of species not considered differential by Ellenberg but found to be exclusive to one or two groups (and thus are potential differentials if the partition is to be accepted). High values of this measure indicate that species considered pervasive in Ellenberg’s tabulation contribute to differentiate the groups of the partition under analysis. The baseline for this measure is not 0, as Ellenberg did not highlight as differential some species that were exclusive to two groups. As a result, nEll-not-diff-but-excl for Ellenberg’s own partition is 5. While the decision that was made not to consider these species as differential may have been subjective (or based on expert knowledge), all had a low number of presences in the matrix: four for Lotus corniculatus and Galium boreale, and three for Silene inflata, Silaus pratensis and Pastinaca sativa. The automatic tabulation performed by the “tabulation” function, which is primarily based on the shortlex order, assign these species to the respective differential block (see Suppl. material 7).

Figure 4 presents the output of the “tabulation” function for partition k, which ranks ninth among the solutions with the highest TDV and is the closest to EllPar (NDR’EllPar’ = 1). The only discrepant relevé is R5. This change is understandable, as Holcus lanatus becomes an exclusive species of a single group without the loss of any other differential species or changes in their degree of exclusiveness. The “tabulation” function outputs for the top ten solutions with the highest TDV are provided in Suppl. material 8.

Suppl. material 8: figure S1 illustrates the solution with the highest TDV (0.2511). For the most part, the top eight solutions were largely similar, generally isolating one or two relevés while maintaining two primary groups: the Bromus erectus group and the Cirsium oleraceum + Geum rivale group. In some cases, the isolated relevés were merely a subset of the Bromus erectus group, suggesting that a two-group partition may be appropriate (see Suppl. material 8: figures S2, S4–S8). The TDV-optimization result for two groups (using the same parametrization) is provided in Suppl. material 9.

Several of the top solutions from TDV-optimization (see Suppl. material 8) closely resembled partitions obtained by other methods. The first TDV-optimization solution (partition j) differed by only one relevé (R16) to partition d (Mod-TWINSPAN) and by two relevés (R16 and R12) to partition b (Flexible β and the OPTSIL+REMOS local optimum). Partition i (OPTSIL+REMOS maximum) had a single discrepant relevé (R16) compared to the third TDV-optimization solution. Partition h (OPTSIL maximum) was identical to the eighth TDV-optimization solution.

Table 6.

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Statistics for the partitions returned by the tested methods for the Arrhenatheretum data set. The best scores in each row are in bold, and the maximum (or minimum) values are in bold italics.

	Agglomerative clustering		k-means	Mod-TWINSPAN	PAM	DIANA	ISOPAM	OPTSIL selected partitions:		OPTSIL+ REMOS selected partitions:		TDV-optimization selected partitions:		Ellenberg’s partition
	Ward	Flexible β						maximum found	local optimum closest to EllPar	maximum found	local optimum closest to EllPar	maximum found	local optimum closest to EllPar
Partition id:	a	b	c	d	e	f	g	h	c	i	b	j	k	EllPar
PARTANA	1.693	1.978	1.750	2.015	1.724	1.819	1.709	1.978	1.750	1.936	1.978	2.094	1.854	1.803
ASW	0.1556	0.1695	0.1824	0.1489	0.1679	0.1883	0.1644	0.2280	0.1824	0.2079	0.1695	0.1622	0.1697	0.1627
MSW (×10)	-1.302	0.332	-0.346	-0.637	-0.641	0.157	-0.658	-0.728	-0.346	0.020	0.332	-1.435	-1.299	-1.354
OptimClass 1 (α = 0.05) 1	21 (18)	20 (19)	23 (19)	21 (19)	22 (18)	25 (19)	25 (21)	15 (14)	23 (19)	22 (20)	20 (19)	21 (21)	22 (21)	23 (21)
OptimClass 1 (α = 0.01) 1	12 (10)	10 (10)	13 (11)	12 (12)	14 (12)	17 (14)	14 (13)	9 (9)	13 (11)	12 (12)	10 (10)	11 (11)	11 (11)	12 (12)
OptimClass 1 (α = 10-3) 1	2 (2)	3 (3)	2 (2)	3 (3)	2 (2)	3 (3)	2 (2)	5 (5)	2 (2)	4 (4)	3 (3)	4 (4)	3 (3)	4 (4)
ssIndval (α = 0.05)	10.55	5.67	10.16	6.04	9.68	11.61	10.81	6.97	10.16	8.59	5.69	5.12	8.07	8.64
nsIndic (α = 0.05)	20.82	8.00	19.00	9.00	17.83	21.83	20.34	9.00	19.00	13.43	8.04	7.11	13.00	15.00
avISAMIC	0.6006	0.5997	0.6047	0.5832	0.5940	0.6150	0.5939	0.6429	0.6047	0.6090	0.5997	0.7071	0.5808	0.5824
nEll-not-diff-but-excl	9	8	12	9	6	9	5	11	12	8	8	15	6	5
	(41%)	(36%)	(55%)	(41%)	(27%)	(41%)	(23%)	(50%)	(55%)	(36%)	(36%)	(68%)	(27%)	(23%)
nEll-diff-found	23	25	23	24	24	24	26	22	23	24	25	26	26	26
	(88%)	(96%)	(88%)	(92%)	(92%)	(92%)	(100%)	(85%)	(88%)	(92%)	(96%)	(100%)	(100%)	(100%)
NDR’EllPar’	8	3	7	4	8	5	8	9	7	7	3	5	1	0
TDV	0.1821	0.2409	0.1909	0.2471	0.1838	0.2051	0.1854	0.2380	0.1909	0.2330	0.2409	0.2511	0.2285	0.2127

¹ For OptimClass 1, I provide in parentheses the number of significant species that are exclusive to one or two groups.

Figure 4. 

Visualization of the exclusive species for each group (or combination of two groups) in partition k, as generated by the “tabulation” function. Relevé and species names have been added, with Ellenberg’s differential species highlighted in orange. See Ellenberg (1956) or Mueller-Dombois and Ellenberg (1974) for species names.

TDV-optimization provides a means of identifying partitions that are likely to be relevant for vegetation classification. However, the researcher must still examine the solutions, possibly assessing the distinctiveness/separability of groups in the geographical space or environmental hyperspace (Orlóci 1975; Schmidtlein et al. 2010). Indices such as those proposed by Cerdeira et al. (2012) or Feoli and Ganis (2019) may assist in this evaluation.

By design, TDV-optimization should be robust to spatial autocorrelation, as the species pools associated with a given biogeographical or environmental unit are not expected to change with the inclusion of geographically close relevés. However, a substantial number of geographically proximal relevés sharing a significant number of exclusive species may ultimately form a group. This, however, is a desirable outcome, as it reflects a genuine floristic pattern that should be detected and analysed.

It is also worth noting that, when applied to two relevés, the DiffVal reduces to the Jaccard distance, i.e., the complement of the Jaccard index (Jaccard 1901; see Murphy 1996, for an earlier reference). For an elegant generalization of the Jaccard index, see Koch (1957).

Since the inception of vegetation science, different schools of thought have emerged across regions, closely tied to their respective regional ecological contexts. By identifying patterns of differential species in vegetation data, TDV-optimization is expected to be particularly useful in floristically rich regions with high geographical turnover and/or strong environmental filtering, such as the Mediterranean region. In areas where floristic variation is primarily reflected in differences in species cover-abundance or fluctuations in species relative frequency, TDV-optimization may have limited utility, unless broad environmental gradients are analysed.

TDV-optimization is not an instantaneous method yielding a single answer; rather, it is a tool for exploring vegetation data in search of meaningful and useful classifications, potentially revealing different patterns within the same data (see Dale 1988; Schmidtlein et al. 2010), that may be harder to spot with other methodologies. These patterns can tell specific stories about the same data set.

The partitioning method presented in this article, along with all its specific details, emerged from within a vegetation science conundrum, generally referred to as vegetation classification (see Lötter et al. 2013). While it is evident that this method could have applications in other scientific fields focused on partitioning binary data (e.g., defining subpopulations through exclusive genes blocks), I strongly believe that it was essential to present the method within the framework of vegetation science, where it has been sought for so long. This ensures that readers are familiar with the foundational research and key researchers whose work has progressively built an impressive body of knowledge, upon which this study is based.

I dedicate this work to three professors who were instrumental in shaping this study. First, Dieter Mueller-Dombois (posthumously), whom I was fortunate to meet in 2005 during the annual meeting of the International Association for Vegetation Science. Dieter left a lasting impression on me, and his seminal work, alongside that of Heinz Ellenberg, inspired this research many years ago. Second, José Carlos Costa, whose profound knowledge of flora and chorology guided me in the practice of the tabulation technique and the intricate description of vegetation communities. He patiently answered the myriad questions I raised throughout my PhD, enabling me to gain a deep understanding of the underlying principles of the method. Finally, Jorge Orestes Cerdeira, a brilliant and sharp mathematical mind with an exceptional commitment to interdisciplinary dialogue. His great patience and deeply insightful teachings on discrete mathematics – particularly graph theory, combinatorics, logic, optimization, and computer science – equipped me with the tools needed to develop DiffVal and TDV, addressing the complexity of the tabulation method in what I hope is a useful and mathematically elegant manner.

I acknowledge Professor David W. Roberts for his thorough review of an earlier version of this article, his clarifications on the index notation, and his cordial exchange of long letters, which guided me toward a clearer, better-grounded manuscript. I also thank three anonymous reviewers, whose critiques, comments, and generous suggestions have enhanced the manuscript considerably.

T.M.H. was partially funded by the European Social Fund (POCH and NORTE 2020) and by National Funds (MCTES), through a FCT – Fundação para a Ciência e a Tecnologia postdoctoral fellowship (SFRH/BPD/115057/2016), as well as by National Funds, through the same foundation, under the project UIDB/04033/2020.

Initial remarks

Most of the issues discussed in this appendix stem from correspondence with Professor David W. Roberts, who kindly exchanged ideas with me after the submission of the first version of this article. Anticipating that these issues would likely be raised by other researchers, Professor Roberts recommended their discussion. I summarize some of the viewpoints that I have defended, highlighting subtleties that I believe merit consideration and that underpin the development of this method.

DiffVal, TDV, and TDV-optimization have the potential to reignite long-standing debates on the nature of vegetation communities. The use of the word ‘differential’ in their names may also resurrect old terminological discussions. Precisely because such debates have persisted for over a century, without reaching a clear consensus among vegetation scientists, I intentionally avoided delving into these epistemological or terminological matters in the body of the article. The sole objective of the article is to present a method for vegetation classification, using a block-based approach. As with any other method, the suitability of the approach for the intended analysis must be assessed by the user. Furthermore, the article makes no claim to resolve long-standing issues about the nature of vegetation communities.

Going back to the seminal concepts of vegetation science

Characteristic species, fidelity, and differential species: The notion of exclusiveness

At the beginning of the 20^th century, characteristic species were those that indicated a particular environment or site. To refine this concept, Braun and Furrer (1913, p. 21) proposed that characteristic species should be those located exclusively, or almost exclusively, in a given plant community. Shortly thereafter, the notion of species fidelity to a plant community emerged, and around 1918, Braun-Blanquet proposed five degrees of fidelity (see Braun-Blanquet 1921, p. 316). Braun-Blanquet emphasized the role of faithful species in distinguishing and demarcating plant communities, using the first three degrees of fidelity to define three types of characteristic species: exclusive (exclusif), elective (électif), and preferring (préférant) (see Braun-Blanquet 1921, p. 316). These were later renamed: exclusive (treue), selective (feste), and preferential (holde) (Braun-Blanquet 1932, p. 59).

The concept of fidelity relies on how the species are distributed across different plant communities. Braun-Blanquet refers to this as the “sociological distribution of species” (Braun-Blanquet 1932, p. 58), which is determined by examining (i) the distribution of the species presences among plant communities, or (ii) the distribution of species abundance or vitality across those communities.

The first three degrees of fidelity imply that (i) all, or almost all, of the presences are confined to a single plant community; (ii) a higher frequency is confined to a single plant community; or (iii) the higher abundances or vitalities are confined to a single plant community. In short, when any of these three conditions hold, the species is said to be centred in that community (see Whittaker 1962, p. 132).

Since the early stages of proposition of the phytosociological method and before the notions of fidelity and differential species emerged, Braun-Blanquet and Furrer were well aware that characteristic species were not enough to differentiate all plant communities (Braun and Furrer 1913, p. 26). In fact, a few years later, in 1925, the idea that some other species could be used for the sharp differentiation of plant communities, without requiring such strict confinement within a single community, matured into the concept of differential species. According to Braun-Blanquet (1925, p. 134), the term was proposed together with Walo Koch.

Differential species are those appearing “only in one of two or more related societies” (Braun-Blanquet 1925, p. 134, 1932, p. 59), i.e., a concept clearly grounded in the notion of exclusiveness. Not by chance the most faithful species are called exclusive species by Braun-Blanquet. Also not by chance, the word exclusiveness is given as a synonym of fidelity by Whittaker (1962, p. 132) and by Mueller-Dombois and Ellenberg (1974, p. 203). All these concepts (characteristic species, fidelity, and differential species) are grounded in the exclusiveness notion, irrespective of their different purposes and utilities.

The loss of the characteristic species at the association level

Braun-Blanquet presents fidelity as a pivotal concept in the study of plant communities at the association level. For him, fidelity is a synthetic quality of plant communities, determined after the associations are established by tabulating each association side by side (Braun-Blanquet 1932, pp. 52, 58).

The initial community-unit hypothesis put forward by Clements (1916), contemporary with Braun-Blanquet’s early texts on characteristic species, proposed a strong overlap of species within a given community and almost perfect non-overlap with species from neighbouring communities (see, e.g., Collins et al. 1993 for a graphical representation). At that time, it was believed that at least one exclusive species could be found for most associations, in a univocal manner. If such a clear relationship between an association and a single exclusive (exclusif/treue) species truly existed, delimiting associations would also entail identifying their exclusive species, i.e., associations would be delimited simultaneously with the identification of first-degree fidelity. Naturally, this would require comprehensive knowledge of flora distribution. The second and third degrees of fidelity could be determined subsequently.

From the 1940s to the 1960s, the narrowing of the association concept, as described in Mueller-Dombois and Ellenberg (1974, p. 206), implied that such a univocal relationship could not exist in practice at the association level. Eventually, after considerable discussions during the 20^th century (see e.g. Whittaker 1962; Westhoff and van der Maarel 1978), the idea of globally (or even regionally) valid characteristic species at the association level was considered unfeasible. Likewise, the notion that the higher frequencies (or higher abundances or vitalities) could be confined to a single plant community was also considered unfeasible, as a species may perform well in more than one association. Species are simply not centred within the contemporary conception of plant communities.

At this stage, it no longer made sense to refer to characteristic species at the association level. In fact, Poore (1956) and Moravec (1993) highlighted that, in 1941, Gams asserted there is no fundamental difference between characteristic and differential species, thus triggering “the decline of the character species doctrine” (Moravec 1993). Furthermore, Moravec notes that characteristic species were frequently reassessed as differential species, which led to the introduction of the neutral term ‘diagnostic species’ by Whittaker in 1962.

Differential species, as initially defined, became crucial in the delimitation of associations. Eventually, the floristic distinction between associations was based entirely on differential species. In practice, phytosociologists find combinations of species, with these species potentially serving as differential species in other associations (Ellenberg 1956). Without the univocal relationship, the complexity increases considerably, and first-degree fidelity can no longer be determined simultaneously with the delimitation of the association.

Even though the narrowing of the association concept compromised the identification of characteristic species at the association level, there was considerable reluctance to abandon the doctrine in the following decades (Moravec 1993). After the 1960s, several authors proposed adjustments to the original concepts, possibly in an attempt to salvage the notion of characteristic species at the association level. Moreover, as differential species came to be seen as essential for delimiting associations – yet clearly dependent on the context of comparison – some authors proposed refinements to the concept. Even if subtle, changes in the concepts of fidelity and differential species over the years have had significant consequences when applied in numerical approaches. Therefore, I believe a full exposition is necessary for clarity.

Changes in postulates over the years

Nuances still under the exclusiveness notion

Ellenberg (1956) illustrates the practice of phytosociology, particularly the tabulation technique, detailing how it relies on identifying co-occurrence patterns of differential species. This practical exemplification is available in English in Mueller-Dombois and Ellenberg (1974).

Ellenberg introduces a slightly modified concept of differential species. While Braun-Blanquet and Walo Koch define differential species as species appearing “only in one of two or more related societies”, Ellenberg’s approach allows a differential species to occur in more than one vegetation unit, provided it is absent from at least one of the units under comparison. In other words, the species remains exclusive to one or some of the emerging groups. This adjustment is particularly relevant to the present work, as it is incorporated into the DiffVal formulation.

A shift from the exclusiveness notion to the concentration notion

Whittaker (1962, p. 132), while discussing the application of the phytosociological method, makes no significant changes to Braun-Blanquet’s concept of fidelity. However, on pp. 130–131, Whittaker associates the identification of characteristic species with the specific set of relevés under tabulation – a much more limited context compared to Braun-Blanquet’s broader approach. When addressing differential species (p. 131), Whittaker introduces some modifications:

“If there are no character-species for a grouping; but certain species are present in samples of this grouping and absent or clearly less important in those of another, closely related grouping, these are differential-species for a unit usually of lower rank than the association.” [my own bold emphasis]

Compared to Braun-Blanquet’s initial definition, there are two modifications: (i) Whittaker’s retains the possibility that a species may be entirely absent from one grouping but relaxes this criterion by adding: “or clearly less important”. In contrast, Braun-Blanquet strictly required complete absence from one (or more) of the compared groups. (ii) Differential species are now considered useful for distinguishing units below the association level, whereas Braun-Blanquet also applied the concept at the association level (note that, at Braun-Blanquet’s time, the association concept was even broader).

Westhoff and van der Maarel (1978, p. 324) later defined fidelity as “the degree to which a species is concentrated in one” syntaxon. They further described differential species as those that “distinguish two closely related syntaxa by presence in most samples of the one and absence in most samples of the other” (p. 372). This definition fully relaxes the requirement that differential species be completely absent from one or more compared plant communities. Moreover, they restrict the use of differential species to ranks below the association level.

The concentration notion was also employed by Feoli and Orlóci (1979) and Podani and Feoli (1991), in their analysis of concentration. Neither of these works refers to fidelity or differential species, although the latter briefly mentions tabular sorting.

Barkman (1989, p. 105) defines fidelity as “the degree of preference of a species for a given association”. By introducing the concept of preference, Barkman’s definition moves away from restricting fidelity to a single plant community, giving it a continuous, statistical flavour. In p. 106, Barkman refers to a “wide sense” definition of differential species, which is commonly used in practice (the one I follow in the article body). He also provides a “more strict and formal” definition: “taxa that have an optimum i[n] only one syntaxon within the next higher syntaxon”. The introduction of the concept of optimum aligns with the concentration notion.

Barkman extends the concept of fidelity to that of differential species, asserting that this extension is implicit in a work by Koch in 1926, and stating that “there is no reason not to use the fidelity classes for differential taxa as well”. At that time, and still today, differential species form the foundation of vegetation classification, as evidenced by the works of Ellenberg (1956) and Mueller-Dombois and Ellenberg (1974). Therefore, extending the fidelity classes to differential taxa/species would be epistemologically catastrophic. Barkman’s formal definition of differential species, if accepted, would cripple their use in the delimitation of plant communities. For instance, at the association level, Barkman’s formal definition requires knowing the alliance circumscription before establishing the associations. However, an alliance can only be established after the associations have been determined and compared, generating a clear paradox.

Dufrêne and Legendre (1997) introduced an ecological perspective to fidelity, describing it as “based upon ecological preferences”. Similarly, they defined differential species as “species with clear ecological preferences”. These definitions were inspired by Hill’s (1979) formulation of the TWINSPAN procedure. While Dufrêne and Legendre’s definitions are not far from Braun-Blanquet’s original concepts, their proposal of quantification differs significantly. They suggest that the relative frequency of presences in the vegetation cluster (quantity B) serves as a measure of fidelity. However, on closer examination, only quantity A conceptually aligns with a fidelity measure, as it evaluates the accumulation of abundance in one group relative to other groups (with abundance often replaced by cover in vegetation science). Quantity B, on the other hand, is not relative to other groups, and, therefore, cannot accurately represent a “preference”. This distinction has been highlighted previously (see Tichý and Chytrý 2006; De Cáceres and Legendre 2009; Willner et al. 2009). Nevertheless, quantity A (or the IndVal, A × B, as a whole) still presents a notable difference when compared to Braun-Blanquet’s concept of differential species, a difference I will revisit further below.

Bruelheide (2000) provides a definition of fidelity, again relying on the concentration notion:

“Fidelity is one of the most important concepts of the Braun-Blanquet (Zürich-Montpellier) approach. Generally speaking, fidelity is the degree to which a species is concentrated in a given vegetation unit. The fidelity of a species determines whether it can be considered a differential or character species or just a companion or accidental species. A character species can be interpreted as a special case of a differential species: a differential species shows a distinct accumulation of occurrences in one or more vegetation units; whereas a character species should accumulate in only one single vegetation unit (Barkman 1989: 106).” [my own bold emphasis]

Bruelheide’s definition of fidelity likely draws support from the previously mentioned works, such as the review by Westhoff and van der Maarel (1978), though references were not provided since fidelity was described in general terms. It is worth noting that the accumulation of abundance, as measured by Dufrêne and Legendre’s (1997) quantity A, is analogous to the “accumulation of occurrences” mentioned by Bruelheide. Toward the end of the paragraph, Bruelheide links this fidelity concept with a modified version of the differential species concept, based on Barkman’s (1989) definition. Bruelheide adapted Barkman’s notion of “optimum” into “accumulation of occurrences”.

Considering Bruelheide’s work, I highlight three issues:

1. The fundamental shift from Braun-Blanquet’s original concept: In 1925, differential species were those accumulating absences in one or more vegetation units (i.e., being completely absent from all but one vegetation unit). In contrast, by 2000, differential species were those accumulating presences in one or more vegetation units (and possibly present in all vegetation units).
2. Bruelheide raises concerns about circular reasoning in his own work. However, the primary cause of this circularity stems from a change he made to the definition of fidelity. Since Bruelheide uses the concept to assist in classification, fidelity is defined in relation to any “vegetation unit” – not necessarily ranked units (syntaxa). This represents a significant departure from the long-established phytosociological tradition, where fidelity was studied and determined after the classification process (see Table A1). This decision clearly introduces circularity into the fidelity concept.
3. Bruelheide’s concerns about circularity are not satisfactorily addressed. The author claims that circularity is avoided by optimizing the u measure. However, optimization does not offer an epistemological solution to this circularity, as the definition of fidelity remains circular in relation to the definition of vegetation unit (Bruelheide 2000, p. 167), regardless of the optimization process.

These issues, particularly (i) and (ii), are deeply rooted in the fact that Bruelheide built on Barkman’s idea of extending the fidelity concept to the differential species concept.

As we can see, the exclusiveness notion found in the works of Braun-Blanquet, Whittaker, and Mueller-Dombois and Ellenberg progressively mutated into the idea of concentration, which seems to have become well-established in vegetation science. Chytrý et al. (2002) described fidelity (also in general terms) as “a measure of species concentration in vegetation units” and, building upon Bruelheide’s (2000) u proposal, they illustrate the use of the phi coefficient. As a side note, it should be mentioned that the square value of the phi coefficient was first proposed in the meteorology field by Doolitte in 1885 (see Murphy 1996).

Later, De Cáceres and Legendre (2009) provided the same definition as Chytrý et al. (2002) (also in general terms and without references), while exploring variants of the IndVal and the phi coefficient.

Tables A1, A2 summarize the changes in the concepts over the years, highlighting the subtle differences between the two paradigms: exclusiveness (in light grey) and concentration (in dark grey).

Exclusiveness versus concentration: What are the consequences?

In the previous subchapters, I have examined the emergence of the definitions of fidelity and differential species, as well as their changes over the years, highlighting the existence of two paradigms: one based on exclusiveness and the other on concentration.

Concentration-based fidelity is now used in the examination of syntaxa after a classification has been obtained, rather than during the classification process itself. This is indisputable, as previous attempts to use fidelity directly in classification have led researchers into paradoxes, circular definitions, and begging-the-question misconceptions. Fidelity is derived after the different syntaxa are established (Chytrý et al. 2002), a principle that holds true under both paradigms.

I will continue exploring the fidelity concept here, as some readers might find it interesting to highlight the differences between DiffVal and other indices currently used in the study of concentration-based fidelity. However, I caution the reader that DiffVal, TDV and TDV-optimization are grounded in the exclusiveness-based differential species concept, as implemented in Ellenberg (1956) and in Mueller-Dombois and Ellenberg (1974). They are not based on any of the concentration-based fidelity concepts, even though they are related. TDV-optimization is focused on the floristic delimitation of plant communities (i.e., vegetation classification, particularly at the association level), rather than the study of species fidelity after a classification is completed.

If we closely examine the first three degrees of the original fidelity concept (Braun-Blanquet 1932, p. 59; Mueller-Dombois and Ellenberg 1974, p. 203), we observe a progressive relaxation of the exclusiveness. The first degree is strongly rooted in the exclusiveness of species presences to a single vegetation unit, relaxing this notion only slightly by stating that a species is “completely or almost completely confined to one community” or “exclusively or almost exclusively found only in a single association”. The second degree further relaxes the exclusiveness of species presences, allowing for presence in more communities, but not all. Note that “other” associations/communities are mentioned but not “several”, as in the third degree. Still, a higher frequency is expected in one vegetation unit only. The third degree relaxes exclusiveness the most: the species is present in “several” associations/communities, and the focus shifts to the species’ “better vitality” or being “optimally developed” or “abundant” in one vegetation unit.

The first two degrees of the original fidelity concept can also be described as a concentration or accumulation of presences in one (or a few) groups, with exclusiveness to that group (or those groups) guaranteed. This is where we begin to spot some disparities: IndVal, the phi coefficient, and similar indices do not ensure exclusiveness to the groups and can return intermediate to high values for species with presences in all the groups being compared.

Table A3 shows how the commonly used group-equalized versions of IndVal (IndVal^g_ind), the phi coefficient (r^g_ϕ), and the point-biserial correlation coefficient (r^g_pb) behave based on changes in (i) exclusiveness to groups, (ii) concentration of presences, and (iii) concentration of cover. Calculations for individual groups and for the best group combination are presented, obtained using the “multipatt” function from the R package indicspecies (De Cáceres and Legendre 2009). DiffVal is also included for comparison. From the table, we observe that IndVal, the phi coefficient, and the point-biserial correlation coefficient – while used for the study of concentration-based fidelity – are grounded in a notion that differs essentially from the original definition provided by Braun-Blanquet. The exclusiveness rationale behind the original definition is particularly straightforward. For example, considering Table A3, what would be the diagnostic value of species 5 in the field? Would its presence or absence in a stand allow us to differentiate any of the three groups? Similarly for species 8, 9, and 10, would their presence in the field, say with a cover-abundance value of 2 or 3, allow us to differentiate any of the three groups? The answer is certainly no!

Species 5 and 8 demonstrate how the phi-coefficient returns reasonably high values, detecting the concentration of the presences in one of the groups, even though exclusiveness to that group is not ensured. When considering the best group combination, species 4, 7 and 9 show cases where the phi-coefficient behave quite differently from DiffVal. Similarly, the high values of IndVal and the point-biserial correlation coefficient for species 8, 9 and 10 illustrate how these indices detect the concentration of cover in one group, again irrespective of exclusiveness. Note that DiffVal is zero for all three species. When considering the best group combination, species 4, 7, 9 and 10 provide examples where IndVal and the point-biserial correlation coefficient can diverge significantly from DiffVal.

The third degree of the original fidelity concept is the closest to a general notion of concentration, as it is postulated almost independently of exclusiveness (i.e., it assumes the species occurs in “several” associations/communities – though not necessarily in all of them). The third degree is also the closest to the quantity A used in IndVal. However, it is important to note that a species may occur in all the groups under analysis and still show a high IndVal, while its practical diagnostic value may remain questionable (see, e.g., species 8 in Table A3).

While the concentration notion seems to generalize the exclusiveness notion (i.e., a species considered exclusive to one vegetation unit has its presences fully concentrated in it), this is not entirely the case. We can easily think of a species with concentrated presences (or concentrated cover) in one vegetation unit, yet not exclusive to any of the vegetation units under consideration (see, e.g., species 5 and 8 in Table A3). All the previously mentioned indices (Bruelheide’s u, IndVal, phi coefficient, the point-biserial correlation coefficient, and their variants) focus on the concentration without ensuring exclusiveness to any group(s). As shown, these indices can detect concentration of presences (or cover), but are not suitable for detecting differential species as originally defined and sought in phytosociological tabulation.

From Table A3, we can also conclude that IndVal^g_ind, r^g_ϕ, r^g_pb, and DiffVal behave quite similarly for species that are absent from at least one group (species 1 to 4, 6 and 7), particularly when focusing on values for single groups. This suggests that these indices may behave similarly in broader vegetation analyses, spanning vegetation types that are floristically very different from one another. Although characteristic species were (and still are) considered unfeasible at the association level, this is not the case for higher syntaxonomic ranks, where high-fidelity species are found (Westhoff and van der Maarel 1978). As evidenced by the behaviour of these indices in such cases, the detection of this associated high fidelity holds irrespective of the paradigm.

On the significance testing for the association between a species and a group of relevés

Bruelheide (2000), Chytrý et al. (2002), and De Cáceres and Legendre (2009) focus on “statistical inference for species-site group associations”. Nevertheless, while discussing this inference, De Cáceres and Legendre (2009, p. 3572) write:

“Finally, practitioners should keep in mind that if the classification of the sites has been obtained from the species composition itself (for example by K-means partitioning), the site groups would not be completely independent of the species data. In such a case of circularity, we can expect more significantly associated species than expected by chance only.”

We must bear in mind that, in vegetation science, the classification of relevés is expected to emerge from species composition itself (i.e., be floristically based). Therefore, using such inference in this context is not advisable, as it would violate the independence assumption of significance testing (see Milligan 1996, pp. 366–367, and discussion of Example 1). In fact, De Cáceres et al. (2010) completely abandoned the concepts of fidelity and character species in their work and state on p. 1683:

“More indicator species will be found than expected by chance when the classification of sites has been obtained from the species composition itself (De Cáceres and Legendre 2009). In this case, p-values must be taken with caution: they do not result from a genuine test of significance since the classification of sites is not independent from the species data used in the indicator species analysis.”

This is also why p-values are interpreted as a direct measure of fidelity rather than as probabilities in hypothesis testing (see, e.g., Tichý et al. 2010), often with lower thresholds than conventional significance levels. However, this approach significantly affects the method’s ability to detect indicators (see Suppl. material 4).

For those interested in p-values, a time-consuming approach can be used to assess whether a given TDV (and the associated differential species pattern) could arise by chance – i.e., under the null hypothesis that the pattern does not reflect an underlying structure in species distribution. This can be tested with a Monte Carlo permutation test, optimizing TDV multiple times on randomized tables by permuting each species’ presences among the relevés. A p-value is then obtained by computing the permutational probability, following Dufrêne and Legendre (1997), ranking the observed TDV within the distribution of randomly generated TDVs ordered in decreasing order.

Concluding remarks: Different contexts might call for different paradigms

The subtle changes found in Whittaker (1962) and Barkman (1989) bent the original concept of differential species. While these modifications may not be problematic in themselves, the differences – though seemingly minor refinements in the definition – lead to vastly different outcomes. This significantly restricts their application and usefulness for classification. Completely removing the subtlety of exclusiveness has consequences for the handling of vegetation data and has the potential to degrade both the implementation and the understanding of the tabulation technique.

The recognized loss of the exclusiveness notion of characteristic species at the association level should not have been extended to the concept of differential species. Floristics play a key role in the phytosociological approach. As it is currently postulated, concentration-based measures are ineffective in the vegetation classification process at the association level.

The DiffVal includes a measure of how exclusive a species is to one (or more) groups, i.e., a measure of the degree of exclusiveness. It is intended as a single measure for each species in a table, determined solely by the partition of the samples (note that, once a specific partition is given, it is straightforward to identify which species occur exclusively in each group or groups). In the case of IndVal, the phi coefficient, or the point-biserial correlation coefficient, comparisons are always pairwise (target group vs. all other groups). For these indices, it is irrelevant how the species’ presences/absences are distributed among the remaining groups – i.e., outside the target group or combination of groups. This is not true for the DiffVal summands, marking a fundamental difference between DiffVal and the other indices.

I hope I have demonstrated why approaches based on the concentration notion are paradigmatically different from those based on the exclusiveness notion. This is the primary reason why comparisons between them are difficult, or even inconsistent. I believe that the block-based approaches are the only ones truly comparable to TDV-optimization.

Braun-Blanquet (1932, p. 62) presents an interesting reflection on the causes of fidelity. In brief, the author identifies several well-known natural phenomena, which are still considered today as underpinning general biogeographical and ecological patterns. Adapting to the terminology used today, the causes of fidelity are: (i) (paleo)biogeographical and environmental filtering; (ii) extreme-habitat adaptations; (iii) evolution/genetics; (iv) habitat availability; (v) biotic interactions (dependence); (vi) biotic interactions (competition). Centuries of research in botany, biogeography and ecology have shown that groups of species can be recognized as being associated with certain regions or environments throughout Earth’s history. The seminal idea that vegetation communities could be defined floristically now has an inescapable (paleo)biogeographical and (paleo)ecological foundation.

In some regional contexts, particularly those with generally low diversity and low environmental roughness, an approach based on concentration without ensuring exclusiveness might be the only useful way of identifying vegetation types, as there may not be sufficient richness and turnover to find species exhibiting some degree of exclusiveness. In regions with high diversity, rich in endemism, and with significant environmental variation and roughness, an approach that ensures exclusiveness can be more effective for understanding the vegetation phenomenon, helping to reduce noise caused by numerous species with very high frequencies (e.g., dominant and companion species).

It is not my intention to raise any objections to approaches based on concentration. First, when analysing (or tabulating) highly dissimilar vegetation (e.g., higher syntaxonomic ranks), we have observed that the concentration-based measures give similar results to those based on exclusiveness. Second, in regions with low species turnover and low environmental roughness, such approaches have proven useful.

The idea of a universal method to classify vegetation stands is appealing, but not necessarily feasible. Nearly a century ago, Braun-Blanquet (1932, p. 26) recognized this:

“The apparent simplicity of the analysis of the concept of vegetation is directly in contrast with the difficulty of making any universal rules. Sometimes it is entirely impossible to submit different vegetation types to similar methods of treatment.”

The two different paradigms might have to coexist.

• Tiago Monteiro-Henriques (Corresponding author, tmh@isa.ulisboa.pt), ORCID: https://orcid.org/0000-0002-4206-0699