In a paper published in Nonlinear Processes in Geophysics, V. Mithlesh Kumar, Anil Yildiz, and Julia Kowalski of RWTH Aachen University introduce a Bayesian data selection workflow that quantifies, in a rigorous information-theoretic sense, how much each candidate dataset actually contributes to calibrating a landslide runout model. Their central finding is striking: more data does not automatically mean better calibration. What matters is whether an observation captures the specific dynamics that a given parameter governs. An observation that seems superficially relevant — say, the total runout distance, when the ultimate goal is predicting impact area — may be nearly useless for constraining certain parameters, while a less obvious choice, such as a time series of velocity, can be dramatically more informative.

The technical backbone of the approach is Bayesian calibration. Rather than producing a single best-fit parameter value, Bayesian methods represent uncertainty about a parameter as a probability distribution, called the prior, and then update it with observational data to yield a narrower posterior distribution. For landslide models, whose parameters are conceptual rather than physically measurable, this framework is especially attractive because it explicitly handles both measurement noise and model inadequacy — the two dominant sources of discrepancy between what a model predicts and what is actually observed. The catch is computational: Bayesian calibration requires enormous numbers of forward model evaluations, which is why the field has increasingly turned to fast surrogate models, such as Gaussian process emulators, to stand in for the expensive simulation.

Kumar and colleagues go a step further by asking a question that earlier studies had only addressed qualitatively: given several possible observational datasets, which one is most informative? Their answer draws on the Kullback-Leibler divergence, a classic measure from information theory that quantifies how much one probability distribution differs from another. In the Bayesian setting, the KL divergence between the posterior and the prior has a direct interpretation as information gain — it measures how much the data reduced uncertainty about the parameters. A dataset that produces a large KL divergence has done more work; one that leaves the posterior barely changed has contributed little, no matter how abundant or precise it appears.

Computing this quantity is far from trivial, because it involves intractable integrals over posterior distributions. The team sidestepped this with a k-nearest-neighbor-based divergence estimator, and they made the whole pipeline feasible by embedding it in a three-phase workflow built on the Python package PSimPy: first a Gaussian process surrogate is trained on model outputs and validated by cross-validation, then Markov Chain Monte Carlo sampling produces posterior distributions for each candidate dataset, and finally the KL divergences are computed and compared. The result is a matrix that, for every combination of parameter and observation type, quantifies the informational value of that pairing — a systematic tool for deciding where scarce measurement resources should be spent.

To demonstrate the method, the researchers used an idealized lumped mass model governed by the Voellmy rheology, which combines a dry Coulomb friction coefficient that resists basal motion with a velocity-dependent turbulent friction term. They generated synthetic observations with known ground-truth parameters, deliberately isolating the effect of data selection from model error. The experiments compared aggregated observations — maximum velocity and runout distance — against time series of velocity and position, and probed how the length and temporal resolution of the time series influenced the outcome.

The results revealed a clean division of labor. Calibrating with maximum velocity strongly constrained the turbulent friction coefficient but left the Coulomb coefficient nearly untouched, while runout distance did the opposite: it pinned down the Coulomb coefficient while leaving the turbulent coefficient badly biased, with its most probable estimate falling far from the true value. The explanation lies in the physics — runout distance varies almost entirely with the Coulomb friction coefficient, whereas peak velocity is governed by the turbulent term. Time series data, whether of velocity or position, outperformed both aggregated datasets, delivering substantial information about both parameters simultaneously, because they capture the evolving acceleration and deceleration phases that the friction parameters actually control.

Perhaps the most practically valuable insight concerns diminishing returns. As the researchers lengthened the velocity time series used for calibration, information gain rose — but the rate of increase fell steeply and plateaued. Remarkably, the optimal observation window corresponded to the time the sliding mass needed to reach its maximum velocity, the point at which the system’s dynamics have evolved and stabilized. Data collected beyond that threshold added progressively less. This echoes findings from hydrology, where longer calibration records and finer temporal resolution improve parameter inference only up to a point, and it suggests that monitoring campaigns could be designed around the dynamics that matter rather than around maximal data volume.

The workflow also proved capable of inferring the noise characteristics of the observations themselves. When the discrepancy parameters of the statistical noise model were calibrated alongside the friction coefficients, the estimates closely matched the known noise levels used to generate the synthetic data — evidence that rich, time-resolved observations can help quantify the very uncertainty they carry, offsetting the effects of imperfect data quality.

The authors are candid about the limitations of their demonstration. The study relies on synthetic data and idealized topography, and it assumes negligible model discrepancy — an assumption that may not hold for real field observations, where structural model errors can cause posteriors to contract toward incorrect parameter regions, inflating KL divergence in misleading ways. They recommend complementing information-theoretic metrics with posterior predictive checks in such cases. Still, the framework is model-agnostic and transferable to complex models and real terrain, and it offers geohazard practitioners something they have lacked: a way to assess, before an expensive field campaign, which measurements will actually sharpen their predictions. For a hazard community working with sparse, costly data, knowing which observations carry the most information may prove as important as the models themselves.

Subject of Research: Bayesian data selection and information-theoretic quantification of observational data value for calibrating landslide runout model parameters

Article Title: Bayesian data selection to quantify the value of data for landslide runout calibration

Article References: Kumar, V. M., Yildiz, A., & Kowalski, J. (2026). Bayesian data selection to quantify the value of data for landslide runout calibration. Nonlinear Processes in Geophysics, 33(3), 425-453. https://doi.org/10.5194/npg-33-425-2026

Image Credits: AI Generated

DOI: 10.5194/npg-33-425-2026

Keywords: landslide runout, Bayesian calibration, Kullback-Leibler divergence, data selection, information gain, Gaussian process emulator, MCMC, Voellmy rheology, friction parameters, natural hazards, uncertainty quantification, surrogate modeling

T-spherical fuzzy sets are a generalization that has been gaining traction because they let modelers record three independent degrees of membership: a positive degree expressing support for an alternative, a neutral or abstinence degree expressing indecision, and a negative degree expressing opposition. Where an ordinary fuzzy set records a single number between zero and one, and intuitionistic or picture fuzzy sets record two or three tightly constrained degrees, the T-spherical framework adds a tunable parameter t that controls how many graded levels each membership degree can take. This makes the framework expressive enough to represent panel opinions in which several experts disagree in structured ways, but it also raises a subtle mathematical question: how do you measure how different two such rich, multi-dimensional judgments are?

That question is answered by divergence measures, the information-theoretic cousins of distance functions. A divergence quantifies the discrepancy between two uncertain descriptions, and it underpins everything from clustering algorithms to decision rules. The most famous example, the Kullback-Leibler divergence, compares probability distributions by averaging the logarithm of their likelihood ratios. Researchers had already extended the Kullback-Leibler idea to T-spherical fuzzy sets, but the new paper documents a serious practical flaw in that extension: whenever any of the membership, non-membership, or abstinence components approaches zero, the logarithm drives the divergence toward an undefined value. Since zero evaluations are common in real-world rating data, the existing measure can simply fail to produce an answer precisely when practitioners need one.

A second existing measure, developed earlier by Huang and colleagues, suffers from the opposite problem. Its formulation is built on exponential terms, which means it produces unbounded positive values. Even small differences between two T-spherical fuzzy values can yield enormous divergence numbers, which makes interpretation difficult and can distort downstream rankings. In other words, the two available tools for measuring disagreement in this framework were either undefined at boundary cases or numerically explosive, leaving a genuine gap in the toolbox for anyone working with T-spherical fuzzy data.

The authors close that gap by constructing new divergence measures and then rigorously verifying their axiomatic properties. In the fuzzy-set literature, a divergence measure is not just any formula that outputs a number; it must satisfy axioms such as being zero only when the two sets are identical, being symmetric in its arguments, and behaving monotonically as the sets become more different. The paper carries out a thorough mathematical investigation to show that the proposed measures meet these requirements, ensuring theoretical soundness rather than merely plausible behavior. This axiomatic grounding matters because downstream algorithms inherit the reliability, or the flaws, of the divergence they are built upon.

From the new divergence, the team builds two practical engines. The first is a divergence-based version of TOPSIS, a widely used multi-criteria decision-making technique whose name stands for Technique for Order of Preference by Similarity to Ideal Solution. Classical TOPSIS ranks alternatives by their geometric closeness to an ideal option and their remoteness from an anti-ideal one. The new formulation replaces the usual distance computations with the proposed divergence, so the ranking of alternatives reflects information-theoretic disagreement between fuzzy evaluations rather than simple Euclidean separation. Alongside it, the authors introduce a fresh method for determining the weights of criteria when the underlying data are spherical fuzzy, addressing the perennial problem that some evaluation criteria matter more than others and that those weights are themselves uncertain.

The second application pushes the divergence into machine learning territory: the authors extend a pattern classification algorithm so that it can operate on T-spherical fuzzy descriptions. Classification with fuzzy divergence works by measuring how far an unclassified object’s fuzzy feature profile diverges from the profiles of known classes, assigning the object to the class with the smallest divergence. Because the new measures stay defined across the full range of possible evaluations, the classifier does not stumble on boundary cases that would crash or distort a Kullback-Leibler-based approach.

To demonstrate that the framework survives contact with reality, the researchers validate it through a case study on selecting an all-rounder cricketer for a test-match competition under uncertain and imprecise evaluation conditions. The scenario is a natural fit for the method: cricket all-rounders must be judged simultaneously on batting, bowling, and fielding, and scouts’ opinions about each skill are naturally expressed with degrees of agreement, hesitation, and disagreement. The T-spherical fuzzy representation captures those layered opinions, and the divergence-based TOPSIS method converts them into a defensible ranking of candidates.

Crucially, the authors do not stop at a single worked example. They conduct sensitivity analyses to check whether small perturbations in the fuzzy evaluations or the criterion weights destabilize the ranking, comparative analyses to benchmark the new measures against the existing Kullback-Leibler-style and exponential-style alternatives, and numerical experiments to probe behavior across parameter settings. The reported results confirm that the proposed framework is stable, effective, and superior to prior approaches in the tested scenarios, and that it enhances both the interpretability and the reliability of decisions made under T-spherical fuzzy conditions. Interpretability is the quieter but arguably more important gain: because the divergence values remain bounded and defined, decision makers can actually read the numbers they are given rather than wrestling with infinities or wildly scaled outputs.

The significance of the work extends beyond cricket selection boards. Multi-criteria decision making under uncertainty appears in supplier selection, medical diagnosis, environmental assessment, and recommender systems, and pattern classification with fuzzy features is relevant wherever expert-labeled data carry hesitation and dissent. By supplying divergence measures that are axiomatically sound and numerically robust across the entire evaluation range, the study gives researchers and practitioners a dependable foundation for building such systems on T-spherical fuzzy data. The paper, published open access on 24 September 2026 in the Journal of Big Data, was supported by Rajamangala University of Technology Phra Nakhon, and its authors declare no competing interests. As fuzzy frameworks grow richer to match the messiness of human judgment, the mathematics for comparing those judgments must keep pace, and this contribution marks a substantial step in that direction.

Subject of Research: Divergence measures for T-spherical fuzzy sets applied to multi-criteria decision making and pattern classification

Article Title: Divergence measures for T-spherical fuzzy sets with applications to multi-criteria decision making and pattern classification

Article References: Khan, M. J., Al-Kenani, A. N., Muangchoo, K., & Ekvittayaniphon, S. (2026). Divergence measures for T-spherical fuzzy sets with applications to multi-criteria decision making and pattern classification. Journal of Big Data. https://doi.org/10.1186/s40537-026-01565-8

Image Credits: AI Generated

DOI: 10.1186/s40537-026-01565-8

Keywords: T-spherical fuzzy sets, divergence measures, multi-criteria decision making, TOPSIS, pattern classification, fuzzy set theory, uncertainty modeling, Kullback-Leibler divergence, criteria weights, sports analytics, cricket, Journal of Big Data