The concept of the differential species dates back more than a century to the work of Josias Braun-Blanquet and forms the foundation of the tabulation technique. Differential species are those with restricted distributions among the field samples, or relevés, under comparison: they are present in one or more emerging groups of relevés but absent from at least one group. In manual tabulation, researchers manipulate matrices of relevés, seeking subsets that can be distinguished floristically by these species patterns. The process, sometimes called tablework or table sorting, is potentially highly subjective and particularly complex even for moderately sized tables, because the number of groups is expected to emerge naturally during the manipulation.

The new approach hinges on a distinction that earlier numerical methods have overlooked: the difference between two kinds of species absences. Absences from some relevés within a group, where the species is otherwise present, are generally attributed to stochastic processes, such as the random failure of a species to establish in a plot despite being in the local species pool. Absences from all relevés representing a group, by contrast, are differentiating absences, attributed to a complete absence from the species pool of that group, for example due to biogeographical causes or strong environmental filtering. By leveraging the distribution of these differentiating absences among groups, the method quantifies exclusiveness degrees and distinguishes differential from non-differential species.

Technically, the Differential Value, or DiffVal, of a species given a partition of relevés is calculated by averaging partial differential values across the groups in which the species occurs. Each partial value multiplies the relative frequency of the species’ presences within a group, its constancy, by the relative frequency of its differentiating absences outside that group. The ratio of presences reaches its maximum of one only when the species occurs in every relevé of the group, while the absence ratio reaches one only when the species is completely absent from all other groups. A factor accounting for the number of groups in which the species occurs keeps DiffVal between zero and one. The Total Differential Value is then simply the mean of the DiffVal outcomes for all species in the table, providing a global measure of how well the groups can be distinguished by differential species.

Because only differential species contribute to TDV, the index reflects the strength of the differential species patterns in a classified table. Species that occur in all vegetation groups yield a DiffVal of zero and are effectively ignored, which means the optimization focuses solely on the subset of the matrix where differential species may occur. This property is central to the method’s ability to detect reticulate patterns, misaligned subsets of species that produce multiple sound groupings of relevés. Since multiple runs of the optimization starting from random partitions have a good chance of finding such patterns among local optima, the method can reveal different valid classifications within the same data set.

Optimizing TDV is a formidable combinatorial challenge. The number of possible partitions of a set with n elements corresponds to the nth Bell number, which grows astonishingly fast: just 10 relevés can be partitioned in 115,975 different ways, and 64 relevés into three groups in roughly 7.6 times ten to the eighteenth different solutions. Complete enumeration is therefore impracticable, and the author employs local search-based metaheuristics, including variations of the hill climbing algorithm, to maximize the index. An R package called diffval is now available, implementing DiffVal and TDV calculation as well as TDV-optimization through functions for hill climbing, simulated annealing, and, for the two-group case, mixed-integer linear programming. The author is also working on a mathematical proof showing that finding a partition that maximizes TDV is NP-hard for two or more groups.

The study demonstrates the method on both an artificial data set and a well-known real-world example. The artificial data set contained 107 species and 64 relevés with two distinct reticulated patterns of differential species distributed across three groups each. When compared against ten clustering and partitioning methods using seven statistics, TDV-optimization was the only method capable of converging to local optima that identified both artificial patterns. However, the highest-TDV partition isolated two single-relevé groups, and enforcing a minimum group size of four relevés placed the partitions capturing the artificial patterns in the first and second positions of the returned solutions, highlighting how the parameter can yield more balanced group sizes.

The real-world test used the classical Arrhenatheretum data set, which Ellenberg used in 1956 to meticulously illustrate the tabulation technique and which has since become a standard benchmark. Ellenberg highlighted 26 differential species in the data set, and TDV-optimization found a partition differing from Ellenberg’s three-group classification by only one discrepant relevé. This is a relevant result, as similar attempts found in the literature have consistently shown poorer performances, with earlier methods producing between two and eight discrepant relevés. Notably, the result was achieved using presence-absence data alone, suggesting that the long-held idea that such data were insufficient to classify this data set appears flawed, since Braun-Blanquet vegetation units are based on the presence of diagnostically relevant species while cover is used for physiognomic characterization.

The author argues that TDV-optimization is conceptually distinct from previously published clustering approaches on four counts. It does not incorporate a hierarchical structure, it does not rely on geometric features such as distances to centroids or medoids, it does not treat all species as equally informative, and it ensures the exclusiveness of informative species to one or more emerging groups. Among biclustering methods, the closest known approach is that of Češka and Roemer, whose procedure could ensure species exclusivity with a particular parameter setting but was not explored by the authors and does not return a partition of relevés. The traditional tabulation technique seeks exclusive-row and -column biclusters, and the new method aligns with a relaxed exclusive-rows, non-exhaustive-in-the-rows structure that accommodates dominant and companion species spanning all groups.

The author cautions that TDV-optimization is not an instantaneous method yielding a single answer but a tool for exploring vegetation data in search of meaningful classifications. The highest-TDV partitions may isolate outlier or extreme relevés, which could arise from stochastic processes, data errors, or genuinely distinct under-sampled vegetation types, and decisions about such cases must always be guided by expert knowledge not embedded in the data table. The method is expected to be particularly useful in floristically rich regions with high geographical turnover or strong environmental filtering, such as the Mediterranean region, while its utility may be limited where floristic variation is reflected mainly in differences in species cover-abundance. Applications could also extend to other scientific fields focused on partitioning binary data, such as defining subpopulations through exclusive gene blocks.

Subject of Research: A novel numerical index and optimization method for phytosociological vegetation tabulation based on differential species

Article Title:  TDV -optimization: A novel numerical method for phytosociological tabulation

Article References: Monteiro-Henriques, T. (2025). TDV-optimization: A novel numerical method for phytosociological tabulation. Vegetation Classification and Survey, 6, 99-127. https://doi.org/10.3897/vcs.140466

Image Credits: AI Generated

DOI: 10.3897/VCS.140466

Keywords: TDV-optimization, differential species, phytosociological tabulation, vegetation classification, Total Differential Value, biclustering, relevé partitioning, Braun-Blanquet, R package diffval, Arrhenatheretum data set, reticulate patterns, vegetation science

The analysis covers the EU-27 member states plus Norway and Iceland from the European Economic Area, along with NUTS-compatible statistical regions in Switzerland, Turkey, the United Kingdom, Montenegro, North Macedonia, and Serbia. The researchers worked at the NUTS 2 level, the standardized territorial units of roughly 800,000 to 3 million inhabitants that form the backbone of EU cohesion policy. Their raw material was the I4.0+ indicator system, a dataset of 101 indicators compiled from European statistical sources between 2008 and 2019, spanning education levels, youth employment, Erasmus student mobility, graduate specialization, scientific and technological capacity, and even the tone of media coverage of industry drawn from the GDELT database.

Because the underlying data were incomplete, the team faced a substantial data-cleaning challenge. Nearly 32 percent of the values in the initial 2018 table were missing, so the researchers applied a temporally ordered substitution rule, filling gaps with the nearest available year’s observation. After screening out regions and indicators with excessive missingness, the final analysis matrix contained 71 indicators for 320 regions, with only 3.26 percent of cells still unavailable. The authors are transparent about the limits of these choices, noting that zero-encoding of residual gaps could affect marginal league memberships, particularly in the low-performance group, and they have released a full audit identifying every affected cell.

The methodological core of the study is a two-stage sequence. First, seriation reorders the region-by-indicator matrix so that similar cells sit close together, and biclustering algorithms then carve out homogeneous submatrices, each pairing a set of regions with a set of indicators on which those regions perform similarly. These submatrices are the industrial leagues. League A captures comparatively high-performing region-indicator combinations, League C captures low-performing ones, and League B, identified by a variance-minimizing algorithm called BicARE, captures an intermediate group of emerging regions. Crucially, the biclustering is deliberately blind to geography. Only in the second stage does the team test whether league membership is spatially clustered, using Moran’s I, a statistic that measures whether nearby regions have more similar values than expected by chance.

This separation is what gives the framework its policy punch. Spatially constrained clustering methods, which force only adjacent regions into the same group, cannot distinguish a genuinely concentrated pattern from a dispersed one. By first identifying leagues without spatial constraints and then testing their spatial dependence, the researchers can tell a policymaker whether a problem calls for a place-based intervention, such as a regional innovation hub, or a sector-wide program, such as a pan-European education initiative. The authors formalize this as a two-part decision rule: league membership identifies which regions and indicators need attention, and the corrected Moran evidence determines the appropriate spatial scale of action.

The results reveal a striking asymmetry. League A, the high-performance group, shows strong and statistically significant spatial clustering across most threshold settings, with Moran statistics between roughly 0.40 and 0.57 after false-discovery-rate correction. Employment and population-education indicators dominate this league and carry the highest global autocorrelation values in the entire dataset, with education indicators P01 and P02 topping the list at 0.7716. This is the familiar core-periphery geography of European development: advantages concentrate in places like Inner London, Bratislava, and Prague, which emerged as the highest-performing capital regions, and spill over to their neighbors.

League C tells a very different story. Its regional membership pattern is spatially diffuse, failing significance tests under every nonconstant threshold setting, even though many of its component indicators, particularly in science and education, still show significant individual spatial structure. The low-performing league draws disproportionately from Turkey, Greece, North Macedonia, southern Italy, and northern England, and its characteristic weaknesses include the number of industry-related research studies, R&D personnel in the business sector, research institutes, and their reflection in media coverage. In other words, the continent’s innovation deficit is not a regional problem in the traditional sense; it is a shared weakness distributed across distant and dissimilar places, which argues for broad framework programs rather than geographically targeted ones, supplemented where indicator-level evidence warrants local action.

Perhaps the most counterintuitive finding concerns overlap. At the baseline threshold, 87.5 percent of regions belong to both the high-performing and low-performing leagues simultaneously, meaning most European regions excel on some indicators while lagging on others. Only four capital regions sit exclusively in the top league. Indicators, by contrast, are never assigned to both League A and League C, a mutual exclusivity built into the method’s construction. This region-indicator distinction dissolves the simplistic picture of rich and poor regions and replaces it with a more granular one in which nearly every territory has a specific, identifiable pathway of indicators to improve in order to advance to a higher league.

The timing of such a framework is significant. The EU’s Recovery and Resilience Facility is channeling 723.8 billion euros into member states with explicit mandates for digital and green transitions, and the effectiveness of that spending depends on knowing which regions can absorb and deploy digital-transformation investments. The authors emphasize that their 2018 snapshot should be read as a historical baseline rather than a current ranking, since structural gaps in digital infrastructure, R&D capacity, and workforce skills tend to persist over multiyear periods. The principal contribution is methodological and reusable: the biclustering-plus-spatial-testing pipeline can be reapplied as newer I4.0+ data become available.

The study is candid about its limitations, including the unrecorded random seed of the original stochastic runs, the unequal country representation of between 1 and 41 regions, and the fact that the 320 regions, not the 22,720 region-indicator cells, are the true statistical units. The spatial weight matrix required deterministic fallback connections for five remote regions, including an unusually long edge linking the French overseas territory of Réunion to Cyprus. Yet the released reproduction archive, with archived memberships, seed manifests, and full spatial diagnostics, sets a high bar for transparency. For regional development agencies, the message is practical: benchmark against comparable regions in your league, identify the specific indicators that gate advancement, and let the geography of the evidence, not administrative habit, decide whether the remedy should be local, continental, or both.

Subject of Research: A combined biclustering and spatial autocorrelation framework for assessing Industry 4.0 readiness across European NUTS 2 regions

Article Title: Combining spatial and aspatial methods to identify European industrial leagues: Development perspectives

Article References: Kosztyán, Z. T., Banász, Z., Kurbucz, M. T., Katona, A. I., Pribojszki-Németh, A., & Jakobi, Á. (2026). Combining spatial and aspatial methods to identify European industrial leagues: Development perspectives. Heliyon, 12(15), Article e45501. https://doi.org/10.1016/j.heliyon.2026.e45501

Image Credits: AI Generated

DOI: 10.1016/j.heliyon.2026.e45501

Keywords: Industry 4.0, regional development, biclustering, spatial autocorrelation, Moran's I, NUTS 2 regions, European Union, cohesion policy, I4.0+ indicators, industrial leagues, digital transformation, core-periphery