Possibility Theory Research Articles

Improved particle swarm optimization for mean-variance-Yager entropy-social responsibility portfolio with complex reality constraints

PurposeThe weighted evaluation function method with normalized objective functions is used to transform the proposed multi-objective model into a single objective one, which reflects the investors' preference for returns, risks and social responsibility by adjusting the weights. Finally, an example is given to illustrate the solution steps of the model and the effectiveness of the algorithm.Design/methodology/approachBased on the possibility theory, assuming that the future returns of each asset are trapezoidal fuzzy numbers, a mean-variance-Yager entropy-social responsibility model is constructed including piecewise linear transaction costs and risk-free assets. The model proposed in this paper includes six constraints, the investment proportion sum, the non-negativity proportion, the ceiling and floor, the pre-assignment, the cardinality and the round lot constraints. In addition, considering the special round lot constraint, the proposed model is transformed into an integer programming problem.FindingsThe effects of different constraints and transaction costs on the effective frontier of the portfolio are analyzed, which not only assists investors to make decisions close to their expectations by setting appropriate parameters but also provides constructive suggestions through the overall performance of each asset.Originality/valueThere are two improvements in the improved particle swarm optimization algorithm: one is that the complex constraints are specifically satisfied by using a renewable 0–1 random constraint matrix and random scaling factors instead of fixed ones; the other is eliminating the particles with poor fitness and randomly adding some new particles that satisfy all the constraints to achieve the goal of global search as much as possible.

Efficiency Assessment Using Common-Weight FDEA under a Multi-Objective Approach as Well as Possibility and Necessity Theory

Efficiency Assessment Using Common-Weight FDEA under a Multi-Objective Approach as Well as Possibility and Necessity Theory

Powers, possibility, and the essential cosmological argument

AbstractOne classical version of cosmological argument, defended famously by Thomas Aquinas and Duns Scotus, deduces the existence of a First Cause from the existence of a particular sort of causal series: one that is ‘essentially ordered’. This argument has received renewed defence in recent work by Feser (2013), Cohoe (2013), and Kerr (2015). I agree with these philosophers that the argument is sound. I believe, however, that the standard defence given of the ECA in these philosophers can be complemented by a formulation that appeals to the powers theory of possibility. This approach to possibility has been defended in recent years by, for example, Pruss (2002), Jacobs (2010), and Vetter (2015). In this article, I show how this modal theory allows us to defend the ECA in a way that is dialectically advantageous as well as clarifying.

Observer control for bearings-only tracking using possibility functions

Bearings-only tracking using passive sensors is important for covert surveillance of moving targets. This paper adopts a mathematical formulation of bearings-only tracking in the framework of possibility theory, where uncertainties are represented using possibility functions, rather than usual probability distributions. Possibility functions have the capacity to deal robustly with partial (incomplete) specification of mathematical models and have been found particularly useful in model mismatch situations. The paper explores the design of reward functions which provide information gain in the context of observer motion control, in the framework of possibilistic recursive filter for bearings-only tracking. Numerical results demonstrate that in the presence of a model mismatch, the proposed framework performs better than the Bayesian probabilistic framework for stochastic filtering and control.

Uncertainty modelling and computational aspects of data association

A novel solution to the smoothing problem for multi-object dynamical systems is proposed and evaluated. The systems of interest contain an unknown and varying number of dynamical objects that are partially observed under noisy and corrupted observations. In order to account for the lack of information about the different aspects of this type of complex system, an alternative representation of uncertainty based on possibility theory is considered. It is shown how analogues of usual concepts such as Markov chains and hidden Markov models (HMMs) can be introduced in this context. In particular, the considered statistical model for multiple dynamical objects can be formulated as a hierarchical model consisting of conditionally independent HMMs. This structure is leveraged to propose an efficient method in the context of Markov chain Monte Carlo (MCMC) by relying on an approximate solution to the corresponding filtering problem, in a similar fashion to particle MCMC. This approach is shown to outperform existing algorithms in a range of scenarios.

A modified Bayes’ theorem for reliable conformity assessment in industrial metrology

Numerous papers in the literature recommend the use of Bayes’ theorem to merge the available a priori knowledge about a measurand with the measurement result associated to the measurand itself. The obtained a posteriori value should be used in conformity assessment, rather than the measured one, to reduce the risk of erroneous decisions. The assumption behind Bayes’ theorem is that the distributions of both the a priori knowledge and the measurements are correct and truly reliable. This means that no deviations or systematic errors in either the manufacturing process or the measuring instrument are considered.The main question is hence whether these assumptions are satisfied or not in the practical industrial cases. And, if they are not satisfied, as it often occurs in the industrial practice how can deviations and systematic errors be considered when applying Bayes’ theorem? This paper proposes a solution based on a modified Bayes’ theorem using Random-Fuzzy Variables (RFVs) and possibility theory. Different simulations and an experimental case are reported showing the effectiveness of the modified Bayes’ theorem in the context of industrial metrology.

Physical Possibility and Determinate Number Theory

ABSTRACT It is currently fashionable to take Putnamian model-theoretic worries seriously for mathematics, but not for discussions of ordinary physical objects and the sciences. However, I will argue that (under certain mild assumptions) merely securing determinate reference to physical possibility suffices to rule out the kind of nonstandard interpretations of our number talk Putnam invokes. So, anyone who accepts determinate reference to physical possibility should not reject determinate reference to the natural numbers on Putnamian model-theoretic grounds.

Beyond probabilities: A possibilistic framework to interpret ensemble predictions and fuse imperfect sources of information

AbstractEnsemble forecasting is widely used in medium‐range weather predictions to account for the uncertainty that is inherent in the numerical prediction of high‐dimensional, nonlinear systems with high sensitivity to initial conditions. Ensemble forecasting allows one to sample possible future scenarios in a Monte‐Carlo‐like approximation through small strategical perturbations of the initial conditions, and in some cases stochastic parametrization schemes of the atmosphere–ocean dynamical equations. Results are generally interpreted in a probabilistic manner by turning the ensemble into a predictive probability distribution. Yet, due to model bias and dispersion errors, this interpretation is often not reliable and statistical postprocessing is needed to reach probabilistic calibration. This is all the more true for extreme events which, for dynamical reasons, cannot generally be associated with a significant density of ensemble members. In this work we propose a novel approach: a possibilistic interpretation of ensemble predictions, taking inspiration from possibility theory. This framework allows us to integrate in a consistent manner other imperfect sources of information, such as the insight about the system dynamics provided by the analogue method. We thereby show that probability distributions may not be the best way to extract the valuable information contained in ensemble prediction systems, especially for large lead times. Indeed, shifting to possibility theory provides more meaningful results without the need to resort to additional calibration, while maintaining or improving skills. Our approach is tested on an imperfect version of the Lorenz '96 model, and results for extreme event prediction are compared against those given by a standard probabilistic ensemble dressing.

Uncertainty-based approaches for the lateral vibration serviceability assessment of slender footbridges

Slender footbridges with long spans and light materials may lead to serviceability problems in lateral vibrations due to walking pedestrians. The phenomenon, known as the “lock-in effect” occurs when the number of pedestrians on the footbridge reaches the “critical number”. Considering the mode of vibration in which the instability is more likely to develop, the footbridge response depends on the structural natural frequency and modal damping ratio. These parameters may not have a fixed value during the footbridge overall life cycle due to environmental and operational changes. Therefore, lateral vibration serviceability procedures should include the uncertainty in the footbridge modal parameters in the most simple and applicable way, as the dynamic response of slender footbridges is very sensitive to these uncertainties.Hence, this paper proposes a simple approach, useful for practical engineering applications, to assess the lateral vibration serviceability of slender footbridges, considering the uncertainties in natural frequency and modal damping ratio. The uncertainty is considered by two approaches: probability theory and possibility theory (fuzzy sets). As a result, the study allows checking the vibration serviceability criteria over a range of levels of uncertainty, suggesting that the external factors that affect the modal parameters are more significant when the crowd density exceeds the critical number. The study also shows the simplicity and effectiveness of the possibility theory to deal with the uncertainty in modal parameters, although for similar level of uncertainty, the dynamic response is slightly higher than using the probability theory. Finally, the proposal is applied in two real footbridges, showing a good agreement with the experimental data.

Why do we need q‐rung orthopair fuzzy sets? Some evidence established via mass assignment

Intuitionistic fuzzy sets (IFSs) have advantage over fuzzy sets and made it possible to describe imprecise information considering its positive and negative aspects simultaneously. In an information system mass assignment and possibility theory are very useful to assign membership grades to elements in a fuzzy set. Unfortunately the situation differs for IFSs in assigning membership function (MF) and nonmembership function (NMF). In this paper, it is shown that the above-mentioned theories fail to produce the MF and NMF for IFSs. Aim of this paper is to present an alternate algorithm to generate these grading functions based on q-rung orthopair fuzzy set. Consequently, it will be extremely convenient to model imprecise and vague information using this approach.

Uncertainty Expression and Propagation in the Risk Assessment of Uncertain Random System

Expressions and propagations of uncertainties have been the core topics in the development of risk science in last decades, whose purpose is to provide decision-makers with clear information about the uncertainty of system's risk. Uncertainty theory is an emerging theory that has great advantages in the expression of epistemic uncertainty, compared to the theories such as subjective probability, evidence theory, and possibility theory. A new framework of uncertainty expression is proposed in this article to properly express different uncertainty sources that exist in the system, where the uncertainty theory is used to express epistemic uncertainties and frequentist probability is used to express aleatory uncertainties. A general propagation method is further developed to address the joint calculations of proposed uncertainty expressions, which can be degenerated into pure probabilistic approach and pure uncertainty-based approach. A case study is conducted to show the effectiveness of proposed method, whose results indicate that the application of proposed method will lead to a result which is easier to be understood than results of theories such as evidence theory and possibility theory, and is more robust than results of Bayesian approach.

A research into the development of models of random variables as part of the structural reliability analysis performed in the absence of some statistical information

Introduction. The scientific review article addresses the approaches to the modeling of random variables performed as part of the structural reliability analysis of elements provided that some statistical information missing (limited). The objectives of the research include the statement of the problem of the probabilistic structural reliability analysis subject to incomplete statistical data, the study of the development of approaches to the generation of models of random variables within the framework of this problem, as well as the assessment of the current state of affairs in this field and some development prospects for the coming years.
 Materials and methods. The principal model of a random variable, considered in the article, represents a p-box (pro­bability box) model. A p-box is an area of possible functions of distributed probabilities of a random variable generated by the two boundary functions of the probability distribution. The article addresses p-boxes generated using the fuzzy set theory, the probability theory, Kolmogorov–Smirnov boundaries, etc.
 Results. The approaches, considered in the article, are illustrated by the numerical examples of p-boxes that use the same statistical data. P-boxes, based on the probability theory, allow to accurately simulate a random variable; however, a priori information about the type of the distribution function is needed. P-boxes, based on the possibility theory, can be used even if an extremely small amount of statistical data is available, and it is also necessary to carefully address the issue of assigning the cutoff (risk) level. P-boxes based on the Chebyshev inequality and the Kolmogorov–Smirnov statistics allow to effectively simulate random variables regardless of the type of the probability distribution. However, these approaches may generate an assessment that is too uninformative for decisions to be made in a number of tasks.
 Conclusions. The choice of a probabilistic model of a random variable for the further reliability analysis of structural elements will depend on the amount and type of statistical data obtained about the random variable. In particular cases, if the statistical information represents a subset of intervals, special approaches based on the Dempster–Shafer theory can be used. A promising and relevant method that underlies both the development of probabilistic models of random variables and the analysis of structural reliability in case of missing statistical information encompasses the employment of numerical modeling methods that employ surrogate models (kriging, Bayesian networks, interval predictors, etc.) and neural network algorithms.

Reconstructing Marx’s Theory of Credit and Payment Crises under Simple Circulation

ABSTRACT There is general agreement amongst scholars of Marx that his monetary theory is incomplete, especially in his most detailed writings on credit in the third volume of Capital. Moreover, in these unfinished notes Marx takes sides with the banking school approach, notable for its opacity compared to the clear axioms of its currency school counterpart. A reconstruction is proposed based on Marx’s step-by-step method, commencing with a critique of Say’s Law under simple commodity circulation, these foundations formalised here using the model of pure labour developed by Pasinetti (1993). Piecing together the fragments, and filling in some of the gaps in Marx’s writings on money, the analysis builds from commodity money and private debt contracts, to the modelling of pure credit and pure banking systems. Adapting the Pasinetti model of a real economy, its endogenous money requirements provide an alternative to the exogenous money approach of the currency school: a streamlined analytical core to the banking school approach, as interpreted by Marx. In addition, the structure of payment crises — as an extension of Marx’s possibility theory of crises — is examined with money as a means of payment required to settle debts between producers and the banking system.

A Redundancy Metric Set within Possibility Theory for Multi-Sensor Systems.

In intelligent technical multi-sensor systems, information is often at least partly redundant—either by design or inherently due to the dynamic processes of the observed system. If sensors are known to be redundant, (i) information processing can be engineered to be more robust against sensor failures, (ii) failures themselves can be detected more easily, and (iii) computational costs can be reduced. This contribution proposes a metric which quantifies the degree of redundancy between sensors. It is set within the possibility theory. Information coming from sensors in technical and cyber–physical systems are often imprecise, incomplete, biased, or affected by noise. Relations between information of sensors are often only spurious. In short, sensors are not fully reliable. The proposed metric adopts the ability of possibility theory to model incompleteness and imprecision exceptionally well. The focus is on avoiding the detection of spurious redundancy. This article defines redundancy in the context of possibilistic information, specifies requirements towards a redundancy metric, details the information processing, and evaluates the metric qualitatively on information coming from three technical datasets.

A universal approach to imprecise probabilities in possibility theory

Possibility theory is a computationally efficient framework for reasoning with imprecise probabilities. Before performing any possibilistic analysis, however, the (imprecise) probabilistic information about the experiment needs to be expressed in the form of a possibility distribution. In this paper, we propose a novel Imprecise Probability-to-Possibility Transformation. This method unifies many results in quantitative possibility theory concerning information modeling, data analysis, and the construction of joint distributions. Furthermore, we show how it enables new results about possibilistic information aggregation and how it may refine frequentist inference in (im-)precise statistical models. The approach is characterized by a clear distinction between the elementary events' well-known objective possibilities from quantitative possibility theory and the elementary events' subjective plausibilities, a reimagination of qualitative possibility measures, which helps overcome several non-uniqueness issues.

A Possibility Theory-Based Approach to the Ranking of Generalized Fuzzy Numbers

The ranking of fuzzy numbers plays a remarkable role in some application systems such as approximate reasoning, decision analysis, optimization and forecasting under fuzzy environments. In this paper, we propose a novel possibility degree formula of ranking generalized fuzzy numbers based on the possibility theory. The combined effects of the possibilistic mean and the variance/standard deviation on the ranking of generalized fuzzy numbers are considered. The axiomatic properties of the proposed ranking method are further verified. It is found that the possibilistic mean exhibits the dominant role as compared to the possibilistic variance or standard deviation. Some comparisons with the existing approaches are reported by carrying out lots of numerical examples. The observations reveal that the shortcomings in an existing method can be overcome. Generalized fuzzy numbers can be distinguished using a possibility degree. The developed ordering procedures of fuzzy numbers are consistent with human intuition, where the inherent uncertainty of fuzzy quantities is revealed.

Possibly extreme, probably not: Is possibility theory the route for risk‐averse decision‐making?

AbstractEnsemble forecasting has become popular in weather prediction to reflect the uncertainty about high‐dimensional, nonlinear systems with extreme sensitivity to initial conditions. By means of small strategical perturbations of the initial conditions, sometimes accompanied with stochastic parameterisation schemes of the atmosphere–ocean dynamical equations, ensemble forecasting aims at sampling possible future scenario and ideally at interpreting them in a Monte‐Carlo‐like approximation. Traditional probabilistic interpretations of ensemble forecasts do not take epistemic uncertainty into account nor the fact that ensemble predictions cannot always be interpreted in a density‐based manner due to the strongly nonlinear dynamics of the atmospheric system. As a result, probabilistic predictions are not always reliable, especially in the case of extreme events. In this work, we investigate whether relying on possibility theory, an uncertainty theory derived from fuzzy set theory and connected to imprecise probabilities, can circumvent these limitations. We show how it can be used to compute confidence intervals with guaranteed reliability, when a classical probabilistic postprocessing technique fails to do so in the case of extreme events. We illustrate our approach with an imperfect version of the Lorenz 96 model and demonstrate that it is promising for risk‐averse decision‐making.

Multisource basic probability assignment fusion based on information quality

Information quality has received extensive attention recently. Yager and Petry proposed an information quality suitable for the framework of probability theory, and proposed a method of fusing multisource information, which can improve the information quality required for decision-making. Then, Bouhamed et al. extended information quality to the theory of possibility. However, the basic probability assignment (BPA) in evidence theory can deal with uncertainty more effectively. Therefore, this work provides a companion paper that makes the method applicable to evidence theory. This method uses vector notation to represent B P A. A fusion method is designed to select the best quality subset based on two factors: information quality and source credibility function, and using the score function to verify the quality of each subset. Finally, a numerical example details the eight steps of the method, and uses the Iris data set and banknote authentication data set to illustrate the application of the method in pattern recognition.

Did Bach Compose Musical Works? Thinking with Adorno through Paradigms of Possibility

Abstract This essay presents Theodor W. Adorno’s case pro and contra the regulation of the work-concept in a tradition that long sustained a great 1800 divide. The case is made through the provocation offered in Lydia Goehr’s Imaginary Museum of Musical Works that “Bach did not intend to compose musical works.” The essay investigates the many references to Johann Sebastian Bach in Adorno’s aesthetic, social, and philosophical writings to show that his Bach case was not merely illustrative but paradigmatic of every case he made for a critical theory of possibility. Adorno’s case was an urgent matter of rescue and justice, made by linking Bach’s compositions to Arnold Schoenberg’s paradigmata of a possible music through the mediation of the work-concept for which Ludwig van Beethoven was made in the tradition paradigmatically to stand.

Disjunctive attribute dependencies in formal concept analysis under the epistemic view of formal contexts

This paper considers an epistemic interpretation of formal contexts, interpreting blank entries in the context matrix as absence of information, which is in agreement with the usual focus on the extraction of implications between attributes. After recalling non-classical connections induced by rough sets and possibility theory in formal concept analysis (FCA), and the standard theory of attribute implications in FCA, this paper presents the notion of disjunctive attribute implications, which reflect additional information that can be extracted from an epistemic context. We show that they can be computed like standard attribute implications from the complementary context. The paper also recalls the logic of classical attribute implications, relying on works pertaining to functional dependencies in database theory, and proposes a dual logic for disjunctive attribute implications. A method for extracting the latter kind of rules from a formal context is proposed, using a counterpart of pseudo-intents. Lastly, the paper outlines a generalization of both conjunctive and disjunctive attribute implications under the form of rules, with a conjunction of conditions in the body and a disjunction of conditions in the head, that hold in a formal context under the epistemic view.