Uncertainty Modeling

Credal predictors are epistemic-uncertainty-aware models that produce a convex set of probabilistic predictions. They provide a principled framework for quantifying predictive epistemic uncertainty (EU) and have been shown to improve model robustness across a range of settings. However, most state-of-the-art (SOTA) methods primarily define EU as disagreement induced by random training initializations, which mainly reflects sensitivity to optimization randomness rather than uncertainty from more substantive sources. In response, we formulate EU as disagreement between models trained under different degrees of relaxation of the i.i.d. assumption between the training and test distributions. Building on this idea, we propose CreDRO, which learns an ensemble of plausible models via distributionally robust optimization. As a result, CreDRO captures EU arising not only from training randomness but also from informative disagreement due to potential train-test distribution shifts. Empirically, CreDRO consistently outperforms SOTA credal approaches on downstream tasks, including outof-distribution detection on extensive benchmarks and selective classification in medical settings.

This paper presents an extended application of the Quantum Indeterminate Set (QIS) framework to the problem of commercial passenger aircraft selection for airline fleet planning. Unlike conventional multi-criteria decision-making (MCDM) methods that rely on real-valued or additive uncertainty models, the QIS approach encodes each criterion's evaluation as a complexvalued membership function µ(x) = a + bi with phase angle ϕ(x) = arg(µ(x)), capturing both the strength of support and the semantic direction of uncertainty. We provide a comprehensive theoretical exposition of QIS, including its algebraic structure, normalization properties, phase-sensitivity bounds, and comparison with classical fuzzy, intuitionistic, and neutrosophic sets. The decision model incorporates five key criteria: fuel efficiency, range capability, passenger capacity, acquisition cost, and noise emission (EPNdB). Phase angles are assigned using a two-tier linguistic prior and data-driven dispersion adjustment, reflecting constructive or destructive interference among conflicting objectives. Using normalized complex aggregation, we compute the collapsed probability PQ(x) = |µ(x)| 2 for each aircraft candidate. A detailed comparison with fuzzy TOPSIS is provided, including the complete mathematical formulation of TOPSIS. Sensitivity analysis demonstrates ranking stability under ±5% perturbations in weights and phases. The results show that the QIS framework provides richer semantic discrimination than conventional fuzzy MCDM, making it suitable for high-stakes transportation planning under deep uncertainty.Beyond mere ranking, the QIS framework acts as a diagnostic tool; the computed phase angles explicitly identify semantic conflicts between high-performance benefits and operational penalties. This allows decision-makers to distinguish between 'naturally aligned' candidates and those whose suitability is compromised by destructive interference.

This study presents a novel Type-3 Fuzzy Logic (T3FL) control framework for Unmanned Aerial Vehicles (UAVs) operating under turbulent and stochastic environmental conditions. Unlike conventional Type-1 and Interval Type-2 fuzzy systems that represent uncertainty as static intervals, the proposed T3FL architecture employs Bayesian membership modeling to actively learn and adapt to environmental uncertainties in real-time. The framework integrates four core components: (1) Bayesian sequential updating of membership functions using Beta distributions; (2) Monte Carlo-based stochastic inference for rule firing under uncertainty; (3) risk-averse defuzzification using quantile-based decision making; and (4) a deep-uncertainty guidance layer applicable to stealth aerial platforms operating in sensor-denied environments. The methodology is extended to Multi-Criteria Decision Making (MCDM) for swarm leader selection, where each agent's trustworthiness is modeled probabilistically. Simulation results demonstrate that the proposed system achieves an 83% improvement in flight safety metrics, reduces Mean Absolute Error (MAE) by 57% compared to Type-1 baseline controllers, and maintains high calibration accuracy with Expected Calibration Error (ECE) below 0.05. The framework's L 2 stability and variance convergence are mathematically proven, and the hardware architecture for real-time implementation is provided. Extension to dynamic guidance of uncrewed stealth aircraft under deep uncertainty confirms the framework's operational versatility across diverse autonomous aerial platforms.

This paper presents a dynamic approach to Transmission Reliability Margin (TRM) estimation, addressing challenges from load variability and uncertainties in modern power systems. Traditional methods use static confidence factors, often leading to inaccuracies. Using 24-hour real-world load profiles from the New Zealand electricity grid, a time-series rolling window analysis with hourly updates captures real-time load fluctuations. The framework dynamically optimizes the confidence factor based on recent load behavior, ensuring adaptive TRM values. Integrating these values into Available Transfer Capability (ATC) calculations enhances precision, improves transmission capacity utilization, and maintains reliability. Test scenarios show that this method mitigates over-and underestimation issues, supporting more dependable grid operations. This real-time TRM estimation framework offers a practical solution for modern power systems with high operational variability.

The integration of renewable energy sources (RESs), such as wind and solar, introduces significant uncertainties into power system operations, complicating Available Transfer Capability (ATC) assessment. A key factor in ATC determination, the Transmission Reliability Margin (TRM), accounts for uncertainties like load variations, generation fluctuations, and network dynamics. The traditional deterministic TRM methods often fail to capture the stochastic nature of modern grids, leading to inaccurate estimations. This paper reviews the TRM assessment methodologies, emphasizing probabilistic approaches that enhance accuracy in high-RES environments. It explores adaptive statistical techniques, such as rolling window analysis, for dynamic TRM computation. Key challenges, emerging trends, and potential solutions are discussed to support the development of robust ATC modeling frameworks for secure and efficient renewable energy integration.

The rapid integration of renewable energy sources (RES), particularly wind, together with fluctuating demand, has introduced significant uncertainty into power system operation, challenging traditional approaches for estimating Transmission Reliability Margin (TRM) and Available Transfer Capability (ATC). This paper proposes a fully adaptive TRM estimation framework that leverages rolling-window statistical analysis of net-load forecast errors to capture real-time uncertainty fluctuations. By continuously updating both the confidence factor and window length based on evolving forecast-error statistics, the method adapts to changing grid conditions. The framework is validated on the IEEE 30-bus system with 80 MW wind (42.3% penetration) and assessed for scalability on the IEEE 118-bus system (40.1% wind penetration). Comparative analysis against static TRM, fixedconfidence rolling-window, and Monte Carlo Simulation (MCS)-based methods shows that the proposed approach achieves 88.0% reliability coverage (vs. 81.8% for static TRM) while providing enhanced transfer capability for 31.5% of the operational day (7.5 h). Relative to MCS, it yields a 20.1% lower mean TRM and a 2.5% higher mean ATC, with an adaptation ratio of 18.8:1. Scalability assessment confirms preserved adaptation (12.4:1) with sub-linear computational scaling (1.82 ms to 3.61 ms for a 3.93× network size increase), enabling 1 min updates interval.

In this work, the GESCONDA system is presented. Initially it was conceived as a system for knowledge discovery and Data Mining, but currently, the system supports two new functionalities. A case-based reasoning engine and a rule-based reasoning shell are provided. These new skills of GESCONDA makes it a suitable prototype tool for the deployment of Intelligent Decision Support Systems, including all main steps like data preparation and filtering, data mining, model validation, reasoning abilities to generate solutions, and predictive models to support final users. The purpose of the paper is to present its architecture as well as its functionalities.

Three-dimensional reconstruction of objects, particularly buildings, within an aerial scene is still a challenging computer vision task and an importance component of Geospatial Information Systems. In this paper we present a new homographybased approach for 3D urban reconstruction based on virtual planes. A hybrid sensor consisting of three sensor elements including camera, inertial (orientation) sensor (IS) and GPS (Global Positioning System) location device mounted on an airborne platform can be used for wide area scene reconstruction. The heterogeneous data coming from each of these three sensors are fused using projective transformations or homographies. Due to inaccuracies in the sensor observations, the estimated homography transforms between inertial and virtual 3D planes have measurement uncertainties. The modeling of such uncertainties for the virtual plane reconstruction method is described in this paper. A preliminary set of results using simulation data is used to demonstrate the feasibility of the proposed approach.

Most Galaxy-sized systems (M host ≃ 10 12 M ⊙ ) in the ΛCDM cosmology are expected to have interacted with at least one satellite with a total mass M sat ≃ 10 11 M ⊙ ≃ 3M disk in the past 8 Gyr. Analytic and numerical investigations suggest that this is the most precarious type of accretion for the survival of thin galactic disks because more massive accretion events are relatively rare and less massive ones preserve thin disk components. We use high-resolution, dissipationless N -body simulations to study the response of an initially-thin, fully-formed Milky-Way type stellar disk to these cosmologically common satellite accretion events and show that the thin disk does not survive. Regardless of orbital configuration, the impacts transform the disks into structures that are roughly three times as thick and more than twice as kinematically hot as the observed dominant thin disk component of the Milky Way. We conclude that if the Galactic thin disk is a representative case, then the presence of a stabilizing gas component is the only recourse for explaining the preponderance of disk galaxies in a ΛCDM universe; otherwise, the disk of the Milky Way must be uncommonly cold and thin for its luminosity, perhaps as a consequence of an unusually quiescent accretion history.

GNSS (Global Navigation Satellite System) positioning is not available underwater due to the very short range of electromagnetic waves in the sea water medium. In this article a LBL (Long Base Line) acoustic repeater system of the GNSS positioning is presented. The system is hyperbolic, i.e., based on time differences and it does not need very accurate atomic clocks to synchronize repeaters. The system architecture and system calculations that demonstrate the feasibility of the solution are presented. The system uses four buoys that sequentially transmit their position and the time of the instant of transmission, for which they are equipped with GNSS receivers and acoustic modems. The buoys can be fixed or even drifting, but they are inexpensive devices, which pose no hazard to navigation and can be easily and quickly deployed for a specific underwater mission. The multilateration algorithm used in the receiver is presented. To simplify the algorithm, the depth of the receiver, meas...

We propose to employ evidential reasoning (ER) rule to construct a clinical decision support system (CDSS) to aid physicians to predict the probability of intensive care unit (ICU) admission and in-hospital death for trauma patients once they arrive at a hospital. A generalized Bayesian rule is used to mine evidence from historical data. Evidence is profiled using a format of belief distribution, where the belief degrees of different trauma outcomes are assigned with derived probabilities linked to the corresponding outcomes. Inputs to the CDSS are clinical data of a patient, and output from the system is predicted belief degree of severe trauma, including ICU admission and in-hospital death. The inner logic of the CDSS is that pieces of evidence that match the clinical data of a patient are identified from the evidence base first, and then the ER rule-based evidence aggregation mechanism is utilized to combine the matched evidences to arrive at a prediction. The reliability, weight, and interdependence of clinical evidence are taken into account. Moreover, an evidence weight training module is constructed. The ER rulebased prediction model has superior performance compared with logistic regression and artificial neural network models. An innovative and pragmatic ER rule-based CDSS for trauma outcome prediction is contributed by this article. In the era of big data, this CDSS helps predict patient outcomes based on historical data and helps physicians in emergency departments make proper trauma management decisions.

This Executive Report provides a synopsis of the book Migration and Settlement: A Multiregional Comparative Study, the capstone of nearly 10 years of research on multistate demography at the International Institute for Applied Systems Analysis (IIASA). This work, one of IIASA's most successful projects, was led by Professor Andrei Rogers during his tenure at IIASA. Professor Rogers organized teams of researchers in all of the countries in which IIASA had National Member Organizations, East and West, who left a permanent imprint in the form of methods now incorporated in routine national calculations. Since most of what was accomplished in the course of the Migration and Settlement Study was methodological and descriptive in character, it was natural that future IIASA work in this general area would more intimately involve substance. And this has been the case. During the past year much of substance has appeared, especially in the writings of James Vaupel and Anatoli Yashin. Consequences of heterogeneity in populations, a universal problem of demographic analysis that has been little noticed up to now; social and economic effects of reduced mortality; period and cohort effects in the reduction of mortality; the concentration of reproduction in a surprisingly small fraction of women.

Satellite imaging systems deployed in low-Earth orbit and deep-space missions are inherently exposed to severe environmental disturbances, including cosmic radiation, photon shot noise, solar glare, thermal drift, platform jitter, and intermittent data loss during downlink transmission. These effects significantly degrade image quality and compromise the reliability of downstream analysis, particularly in resource-constrained platforms such as CubeSats. While multi-modal sensor fusion has shown promise in enhancing information content, most existing approaches assume clean, fully observed data distributions and exhibit limited robustness under realistic noise and distribution shift.

The Guided Entropy Principle (GEP) is a mathematical framework for regulating uncertainty in complex information systems through entropy-aware control. This document presents the formal derivation of GEP from first principles, including connections to Shannon entropy, Lyapunov stability theory, and classical control formulations.

The analysis focuses on the conditions under which entropy regulation yields stable convergence, bounded behavior, and controlled information flow in non-neural systems such as databases, routing engines, and decision-support pipelines. No neural network or transformer-specific assumptions are made in this work.

These mathematical foundations serve as the theoretical basis for subsequent experimental extensions, including exploratory applications to attention mechanisms (e.g., WIPER). Such extensions are explicitly treated as hypotheses requiring independent validation and are not assumed by the derivations presented here.

This document is intended as a standalone technical reference for researchers interested in entropy-regulated systems, stability analysis, and control-oriented approaches to information processing.

Decision-making theory has developed over many decades at the intersection of economics, mathematics, psychology, and engineering. Its classical foundations include Bernoulli's expected utility theory, von Neumann and Morgenstern's rational choice theory, and the criteria proposed by Savage, Wald, Hurwicz, and others. However, in real-world contexts, decisions are made under uncertainty, incompleteness, and unreliability of information, which classical approaches do not adequately address. To overcome these limitations, modern multi-criteria decision-making methods such as Analytic Hierarchy Process (AHP), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), VlseKriterijumska Optimizacija I Kompromisno Resenje (Compromise solution approach) (VIKOR), and ELimination Et Choix Traduisant la REalité (Elimination and Choice Expressing Reality) (ELECTRE), as well as their fuzzy and Z-number extensions, are widely applied to the modeling and evaluation of complex systems. These Z-number extensions are based on the concept of Z-numbers introduced by Lotfi Zadeh in 2011 to formalize higher-order uncertainty. This study introduces the Z-Regret principle, which extends Savage's regret criterion through the use of Z-numbers. Supported by Rafik Aliev's mathematical justifications concerning arithmetic operations on Z-numbers, the model evaluates regret not only as a loss relative to the best alternative but also by incorporating the degree of confidence and reliability of this evaluation. Calculations for the selection of digital advertising platforms in terms of performance assessment under various scenarios demonstrate that the Z-Regret principle enables more stable and well-founded decision-making under uncertainty.

Uncertainty remains one of the most fundamental challenges in science, philosophy, and artificial intelligence (AI). Classical probability theory provides a means to quantify randomness, while possibility theory offers a way to describe vagueness and incomplete information. This paper explores the theoretical measures of possibility and probability within the fuzzy theoretical framework, focusing on their conceptual distinctions, mathematical relationships, and integrative potential. Drawing on developments in fuzzy set theory, this study recognizes the complementary roles of possibility and probability measures in modelling uncertainty and decision-making systems. It therefore, proposes a hybrid uncertainty representation that unites probabilistic and possibilistic reasoning. Applications in environmental risk analysis and decision systems demonstrate how fuzzy theory that bridges the gap between quantitative and qualitative uncertainty. The results contribute to a unified understanding of uncertainty representation and reasoning under imprecise conditions. The analysis highlights the implications of these measures of uncertainty in reasoning, risk assessment and intelligent system design.

This study aims to use a generically integrated meteorological and hydrodynamic ensemble modelling system to quantitatively assess the effect of uncertainties arising from the numerical weather prediction (NWP) on the tide, surge and wave forecasts. The modelling system was applied to the English and Bristol Channels for the storm events occurred between 25 th and 30 th October 2004. The model results show that the accuracy of the predicted surge and waves can be considerably improved by the ensemble approach, particularly in predicting the storm peaks, when short-term predictions (T+2 day) are used.

We present a formal framework identifying fifteen fundamental topological structures that underlie common-sense reasoning across all domains. These structures-including containment, sequence, adjacency, separation, hierarchy, and ten others-provide a mathematical foundation for knowledge representation that operates in polynomial time and requires minimal training data. Unlike neural approaches that learn implicit representations through extensive examples, our framework makes knowledge structure explicit through formal topological relations. We prove the independence and sufficiency of these fifteen structures, demonstrate their compositional properties, and analyze their computational complexity. This work offers a rigorous foundation for artificial intelligence systems that reason about the world using explicit structural knowledge rather than purely statistical patterns.

We address the problem of controlling a linear system with unknown parameters ranging over a continuum by means of switching among a ÿnite family of candidate controllers. We present a new hysteresis-based switching logic, designed speciÿcally for this purpose, and derive a bound on the number of switches produced by this logic on an arbitrary time interval. The resulting switching control algorithm is shown to provide stability and robustness to arbitrary bounded noise and disturbances and su ciently small unmodeled dynamics.

We address the problem of controlling a linear system with unknown parameters ranging over a continuum by means of switching among a ÿnite family of candidate controllers. We present a new hysteresis-based switching logic, designed speciÿcally for this purpose, and derive a bound on the number of switches produced by this logic on an arbitrary time interval. The resulting switching control algorithm is shown to provide stability and robustness to arbitrary bounded noise and disturbances and su ciently small unmodeled dynamics.

The Monte Carlo Method is a powerful statistical technique for propagating uncertainty in photometric measurements, particularly when models are nonlinear or input variables deviate from Gaussian distributions. This study explores the application of MCM to luminous flux measurements using Type C goniophotometers, with comparisons to the conventional Root Sum of Squares Method. Three distinct Monte Carlo models were implemented. The first classified uncertainties as Type A or Type B, assigning corresponding probability distributions. The second employed Shannon information theory to derive distributions based on available knowledge, while the third exploratory model randomized distribution selection among normal, uniform, and triangular forms. Sensitivity analysis, guided by the Pareto principle, identified the key variables contributing to 80%  of overall variability. Results showed that Root Sum of Squares Method yielded an expanded uncertainty of 3.4%, whereas the first, second, and third Monte Carlo models achieved 3.0%, 2.5%, and 2.3% , respectively. Importantly, the Pareto-based reduction strategy preserved accuracy while lowering computational complexity. These findings demonstrate that Monte Carlo Method provides a more effective framework for uncertainty evaluation in photometry than traditional approaches, with the information theory-based model offering the best balance between accuracy and efficiency. The proposed methodology enhances the reliability of luminaire characterization and supports practical adoption in industrial measurement contexts.

International audienceTo predict the wave characteristics of the periodic media in the presence of fuzzy uncertainties, the wave finite element method in conjunction with fuzzy logic and algebra has been applied. For one-dimensional wave propagation, firstly, the most significant input parameters such as Young's modulus and mass density are identified and then fuzzified using the membership functions. Then, the fuzzy variable is propagated through the numerical model of interval analysis. The dispersion curves for flexural and longitudinal waves with fuzzy parameters have been used to illustrate the generality of the proposed approach also looking at the possibility to have frequency region in which those waves cannot propagate (frequency band gaps). The triangular membership functions have been used in the numerical examples and the obtained results are compared against the classical Monte Carlo simulations (MCS). The approach was presented for very simplified test-cases but it...

The objective of this work is to develop a new tuning strategy for multivariable extended predictive control ͑EPC͒. A natural concern is the problem of ill conditionality in controlling multi-input multi-output ͑MIMO͒ systems. The main advantage of EPC is that it has a simple and effective tuning strategy that results in a well-conditioned system which can achieve tight closed-loop response. Moreover, unlike most existing model predictive control tuning strategies, the proposed strategy establishes a direct relationship between one main tuning parameter for each subprocess of the MIMO system. This tuning method has been derived based on the assumption of an infinite control horizon resulting in powerful stability for the nominal case and in the presence of model uncertainty. This tuning method is applicable to unconstrained multivariable processes, and was proven to have good control on nonsquare systems. The main features of the new tuning strategy are practically illustrated on a MIMO temperature system with improved control performance as compared to move suppressed predictive control.

This paper focuses on searching the best k objects with more attributes according to user preferences in the Web environment. Attributes of an object type are distributed on servers in a disjunctive way, i.e. values of one attribute are stored in only one remote server on the Internet. In our work, every user can express his/her preferences for each attribute by a fuzzy function and mutual relations between the attributes by an aggregation function. We use client/server architecture and communication via Web Services. We deal with the usage of Fagin's NRA algorithm, which can find the best k objects without accessing all the objects. Because of support of sorting objects according to a fuzzy function, an indexing method based on B+-tree is used in each remote server. Moreover, each server is stateless, i.e. independent from any previous request. Our solution is based on cache memory, which loads objects from remote servers in batches and thus reduces the amount of network communication. In this paper we present a system TOPKNET, which can efficiently find the best k objects for various users with data on remote servers.

Summary Although typically large uncertainties are associated with reservoir structure, the reservoir geometry is usually fixed to a single interpretation in history-matching workflows, and focus is on the estimation of geological properties such as facies location, porosity, and permeability fields. Structural uncertainties can have significant effects on the bulk reservoir volume, well planning, and predictions of future production. In this paper, we consider an integrated reservoir-characterization workflow for structural-uncertainty assessment and continuous updating of the structural reservoir model by assimilation of production data. We address some of the challenges linked to structural-surface updating with the ensemble Kalman filter (EnKF). An ensemble of reservoir models, expressing explicitly the uncertainty resulting from seismic interpretation and time-to-depth conversion, is created. The top and bottom reservoir-horizon uncertainties are considered as a parameter for a...

Interim Reports on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. Abstract South Africa faces tremendous demographic challenges, mostly related to HIV/AIDS. Any projection is made even more difficult through significant disagreements over current population size, age structure, fertility and mortality levels. This paper expands methods of expert-based probabilistic population projections to include uncertainty distributions about the current demographic conditions. The result is a probabilistic distribution of future population trends in South Africa combining uncertainty about present starting conditions and future rates. For 2030 the total population of South Africa is projected at 54 million (median) with the 95% uncertainty interval ranging from 36 to 74 million.

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