Framework Of Possibility Theory Research Articles

Bernoulli Filter for Extended Target Tracking in the Framework of Possibility Theory

Extended objects give rise to a varying number of noisy measurements from its reflection (scattering or feature) points. Due to imperfect detection, only some of the feature points are detected in each scan of input data, while false alarms can also be present. The optimal sequential Bayesian state estimator in the framework of random set theory is the Bernoulli filter for an extended target (BF-X). In this paper we formulate and derive the analog of the BF-X in the framework of possibility theory, where uncertainty is represented using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">possibility</i> functions, rather than <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">probability</i> distributions. Possibility functions have the capacity to model partial (imprecise) probabilistic specifications and thus the main advantage of the proposed possibilistic BF-X, is enhanced robustness in the absence of precise measurements or dynamic models.

Selecting accepted assertions in partially ordered inconsistent DL-Lite knowledge bases

The problem of inconsistency handling in knowledge bases has received considerable research attention in several settings and has met many applications. Inconsistency usually stems from the fact that the information available for reasoning, query answering and decision-making tasks is obtained from different, scattered and potentially unreliable information sources. Performing these tasks is often computationally expensive, especially for knowledge bases in which the data is endowed with preferences and/or uncertainty. In the context of formal ontologies, enriching the assertions (i.e. the data pieces) with conceptual knowledge makes the query answering task harder computationally. Tractable results have been successfully established for the lightweight languages of the DL-Lite family. In this paper, we present two tractable methods for handling inconsistency in partially ordered lightweight ontologies. In essence, both methods are inspired from the well-known Intersection of ABox Repair (IAR) semantics, which proceeds by replacing the initial inconsistent knowledge base with the intersection of all of its maximally consistent sub-bases. The first method assumes a partial preorder over the knowledge base and identifies the accepted assertions, called ‘elected’, in polynomial time in data complexity. The second method captures uncertainty quantitatively in the framework of possibility theory and identifies the so-called ‘π-accepted’ assertions, also in polynomial time in data complexity. However, both characterisations require the conflict set to be readily available. This constitutes a serious limitation in frameworks and applications where the conflict set cannot be exhibited efficiently. In this paper, we propose a new and equivalent characterisation for each method. We show that the elected assertions and also the π-accepted assertions can be identified in polynomial time in the size of the dataset for lightweight ontologies.

Computing a Possibility Theory Repair for Partially Preordered Inconsistent Ontologies

We address the problem of handling inconsistency in uncertain knowledge bases that are specified in the lightweight fragments of description logics DL-Lite. More specifically, we assume that the TBox component is coherent, stable, and fully reliable. However, the ABox component may be inconsistent with respect to the TBox, partially preordered and uncertain. Uncertainty is encoded in the framework of possibility theory. In this context, we propose an extension of standard possibilistic DL-Lite. We represent the ABox as a symbolic weighted base, where the weights attached to the assertions are ordered according to a strict partial order. We define a tractable method for computing a single possibilistic repair for a partially preordered weighted ABox. The idea is to consider the possibilistic compatible bases of such an ABox, which intuitively encode all the possible extensions of a partial order, and compute the possibilistic repair of each compatible base. We then compute the intersection of all these possibilistic repairs to obtain a single repair for the initial ABox. We also provide an equivalent characterization by introducing the notion of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pi$</tex-math></inline-formula> -accepted assertions. This ensures that the computation of the partially preordered possibilistic repair can be achieved in polynomial time in DL-Lite.

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.

Target tracking in the framework of possibility theory: The possibilistic Bernoulli filter

The Bernoulli filter is a Bayes filter for joint detection and tracking of a target in the presence of false and miss detections. This paper presents a mathematical formulation of the Bernoulli filter in the framework of possibility theory, where uncertainty is represented using possibility functions, rather than probability distributions. Possibility functions model the uncertainty in a non-additive manner, and have the capacity to deal with partial (incomplete) problem specification. Thus, the main advantage of the possibilistic Bernoulli filter, derived in this paper, is that it can operate even in the absence of precise measurement and/or dynamic model parameters. This feature of the proposed filter is demonstrated in the context of target tracking using multi-static Doppler shifts as measurements.

SPOCC: Scalable POssibilistic Classifier Combination - toward robust aggregation of classifiers

We investigate a problem in which each member of a group of learners is trained separately to solve the same classification task. Each learner has access to a training dataset (possibly with overlap across learners) but each trained classifier can be evaluated on a validation dataset.We propose a new approach to aggregate the learner predictions in the possibility theory framework. For each classifier prediction, we build a possibility distribution assessing how likely the classifier prediction is correct using frequentist probabilities estimated on the validation set. The possibility distributions are aggregated using an adaptive t-norm that can accommodate dependency and poor accuracy of the classifier predictions. We prove that the proposed approach possesses a number of desirable classifier combination robustness properties. Moreover, the method is agnostic on the base learners, scales well in the number of aggregated classifiers and is incremental as a new classifier can be appended to the ensemble by building upon previously computed parameters and structures. A python implementation can be downloaded at this link https://github.com/john-klein/SPOCC.

A Machine-Learning-Based Epistemic Modeling Framework for Textile Antenna Design

A novel machine-learning-based framework to evaluate the effect of design parameters affected by epistemic uncertainty on the performance of textile antennas is presented in this letter. In particular, epistemic variations are characterized in the framework of possibility theory, which is combined with Bayesian optimization to accurately and efficiently perform uncertainty quantification. A suitable application example validates the proposed method.

Possibilistic randomisation in strategic-form games

Since the seminal work of John Nash, convex combinations of actions are known to guarantee the existence of equilibria in strategic-form games. This paper introduces an alternative notion of randomisation among actions – possibilistic randomisation – and investigates the mathematical consequences of doing so. The framework of possibility theory gives rise to two distinct notions of equilibria both of which are characterised in our main results: a qualitative one based on the Sugeno integral and a quantitative one based on the Choquet integral. Then the two notions of equilibrium are compared against a coordination game with payoff-distinguishable equilibria known as the Weak-link game.

Formalization of Fuzzy Control in Possibility Theory via Rule Extraction

Possibility system has been recently recognized as a potential foundational theory for fuzzy set theory though the concept of possibility originated from the membership function of fuzzy sets. As an application of fuzzy set theory, fuzzy control has been widely used in engineering practices, where control laws are described by fuzzy if-then rules. This paper intends to formulate fuzzy control in the framework of possibility theory by regarding the fuzzification procedure of fuzzy control as extraction process of the fuzzy variable from crisp input, and the fuzzy if-then rules as extracted rules between fuzzy input/output. Typical procedures of fuzzy control, such as fuzzification, implication, aggregation, and defuzzification, are eventually formulated as a series of conditional possibilities operated by “max-product” operators. The reformulated fuzzy controller has a more general form, which encompasses the Mamdani fuzzy controller as a special case. Some fundamental concepts of possibility theory, such as randomness/fuzziness, possibility, and conditional possibility, are also discussed, which may be helpful for the correct understanding of possibility theory. Efforts have been made to bridge the two systems of possibility theory and fuzzy sets by the derivation of composition rule of fuzzy relations from conditional possibility. All results are derived for both normalized possibility and non-normalized possibility. This paper strengthened the role of possibility theory as a foundation for fuzzy sets, and as a complementary method to probability theory for handling information with fuzzy uncertainty.

Uncertain lightweight ontologies in a product-based possibility theory framework

This paper investigates an extension of lightweight ontologies, encoded here in DL-Lite languages, to the product-based possibility theory framework. We first introduce the language (and its associated semantics) used for representing uncertainty in lightweight ontologies. We show that, contrarily to a min-based possibilistic DL-Lite, query answering in a product-based possibility theory is a hard task. We provide equivalent transformations between the problem of computing an inconsistency degree (the key notion in reasoning from a possibilistic DL-Lite knowledge base) and the weighted maximum 2-Horn SAT problem. The last part of the paper provides an encoding of the problem of computing inconsistency degree in product-based possibility DL-Lite as a weighted set cover problem and the use of a greedy algorithm to compute an approximate value of the inconsistency degree. This encoding allows us to provide an approximate algorithm for answering instance checking queries in product-based possibilistic DL-Lite. Experimental studies show the quality of the approximate algorithms for both inconsistency degree computation and instance checking queries.

Inconsistency Management from the Standpoint of Possibilistic Logic

Uncertainty and inconsistency pervade human knowledge. Possibilistic logic, where propositional logic formulas are associated with lower bounds of a necessity measure, handles uncertainty in the setting of possibility theory. Moreover, central in standard possibilistic logic is the notion of inconsistency level of a possibilistic logic base, closely related to the notion of consistency degree of two fuzzy sets introduced by L. A. Zadeh. Formulas whose weight is strictly above this inconsistency level constitute a sub-base free of any inconsistency. However, several extensions, allowing for a paraconsistent form of reasoning, or associating possibilistic logic formulas with information sources or subsets of agents, or extensions involving other possibility theory measures, provide other forms of inconsistency, while enlarging the representation capabilities of possibilistic logic. The paper offers a structured overview of the various forms of inconsistency that can be accommodated in possibilistic logic. This overview echoes the rich representation power of the possibility theory framework.

Quantitative possibility theory: logical- and graphical-based representations

In the framework of quantitative possibility theory, two representation modes were developed: logical-based representation in terms of quantitative possibilistic bases and graphical-based representation in terms of product-based possibilistic networks. This paper deals with logical and graphical representations of uncertain information using a quantitative possibility theory framework. We first provide a deep analysis of the relationships between these two forms of representational frameworks. Then, in the logical setting, we develop syntactic relations between penalty logic and quantitative possibilistic logic. These translations are useful for different applications and are interesting for taking advantage of each format at the inferential level. We also provide an algorithm for reasoning with quantitative possibilistic logic. The algorithm takes inspiration from the syntactic relations between quantitative possibilistic logic bases and penalty logic. Finally, we provide experimental results that compare the uncertainty propagation algorithms developed for the possibilistic product-based networks and the inference algorithms based on WMAXSAT developed in this paper.

Comparing and Weakening Possibilistic Knowledge Bases

Weighted logics are important tools for reasoning under uncertain knowledge. A weighted knowledge base is a set of weighted formulas of the form (φi, αi) where φi is a propositional formula and αi is the certainty degree associated with φi. This paper addresses two issues in the possibility theory framework: i) how to compare two weighted knowledge bases with respect to their information contents? and ii) how to syntactically define the concept of forgetting variables? We show that comparing knowledge bases is the counterpart of the concept of the specificity relation defined between possibility distributions.

The Construction of Joint Possibility Distributions of Random Contributions to Uncertainty

The evaluation and expression of uncertainty in measurement is one of the fundamental issues in measurement science and challenges measurement experts especially when the combined uncertainty has to be evaluated. Recently, a new approach, within the framework of possibility theory, has been proposed to generalize the currently followed probabilistic approach. When possibility distributions are employed to represent random contribution to measurement uncertainty, their combination is still an open problem. This combination is directly related to the construction of the joint possibility distribution, generally performed by means of t-norms. The problem is that joint possibility distributions strongly depend on the considered t-norm, and, therefore, the choice of one particular t-norm over another has to be justified. The first goal of this paper is, hence, to provide support to the choice of a particular t-norm. The second goal is to discuss the construction of the joint possibility distribution when the two random contributions to uncertainty show mutual dependence, with specific reference to a particular class of possibility distributions.

Layers of protection analysis in the framework of possibility theory

An important issue faced by risk analysts is how to deal with uncertainties associated with accident scenarios. In industry, one often uses single values derived from historical data or literature to estimate events probability or their frequency. However, both dynamic environments of systems and the need to consider rare component failures may make unrealistic this kind of data. In this paper, uncertainty encountered in Layers Of Protection Analysis (LOPA) is considered in the framework of possibility theory. Data provided by reliability databases and/or experts judgments are represented by fuzzy quantities (possibilities). The fuzzy outcome frequency is calculated by extended multiplication using α-cuts method. The fuzzy outcome is compared to a scenario risk tolerance criteria and the required reduction is obtained by resolving a possibilistic decision-making problem under necessity constraint. In order to validate the proposed model, a case study concerning the protection layers of an operational heater is carried out

Inference using compiled min-based possibilistic causal networks in the presence of interventions

Qualitative possibilistic causal networks are important tools for handling uncertain information in the possibility theory framework. Contrary to possibilistic networks (Ayachi et al., 2011 [2]), the compilation principle has not been exploited to ensure causal reasoning in the possibility theory framework. This paper proposes mutilated-based inference approaches and augmented-based inference approaches for qualitative possibilistic causal networks using two compilation methods. The first one is a possibilistic adaptation of the probabilistic inference approach (Darwiche, 2002 [13]) and the second is a purely possibilistic approach based on the transformation of the graphical-based representation into a logic-based one (Benferhat et al., 2002 [3]). Each of the proposed methods encodes the network or the possibilistic knowledge base into a propositional base and compiles this output in order to efficiently compute the effect of both observations and interventions. This paper also reports on a set of experimental results showing cases in which augmentation outperforms mutilation under compilation and vice versa.

Preferential Semantics for Plausible Subsumption in Possibility Theory

Handling exceptions in a knowledge-based system is an important issue in many application domains, such as medical domain. Recently, there is an increasing interest in nonmonotonic extension of description logics to handle exceptions in ontologies. In this paper, we propose three preferential semantics for plausible subsumption to deal with exceptions in description logic-based knowledge bases. Our preferential semantics are defined in the framework of possibility theory, which is an uncertainty theory devoted to handling incomplete information. We consider the properties of these semantics and their relationships. We also discuss the relationship between two of our preferential semantics and two existing preferential semantics. We extend a description logic-based knowledge base by adding preferential subsumptions. Entailment of plausible subsumptions relative to an extended knowledge base is defined. Properties of the preferential subsumption relations relative to an extended description logic-based knowledge base are discussed. Finally, we show that our semantics for plausible subsumption can be reduced to standard semantics of an expressive description logic. Thus, the problem of plausible subsumption checking under our semantics can be reduced to the problem of subsumption checking under the classical semantics.

Connecting possibilistic prudence and optimal saving

In this paper we study the optimal saving problem in the framework of possibility theory. The notion of possibilistic precautionary saving is introduced as a measure of the way the presence of possibilistic risk (represented by a fuzzy number) influences a consumer in establishing the level of optimal saving. The notion of prudence of an agent in the face of possibilistic risk is defined and the equivalence between the prudence condition and a positive possibilistic precautionary saving is proved. Some relations between possibilistic risk aversion, prudence and possibilistic precautionary saving were established. inequality is proved. In Section 3 the possibilistic two-period model of precautionary saving is studied. The consumer is represented by two utility functions u and v and the risk, present in the second period, is described by a fuzzy number. The expected lifetime utility of the model is defined with the help of the notion of possibilistic expected utility. The main introduced notion is possibilistic precautionary saving. It measures the changes on optimal saving produced by the presence of risk in the second period. If this indicator has a positive value then by adding the risk the consumer will choose a greater level of optimal saving. The main result of the section characterizes the positivity of possibilistic precautionary saving by the condition

Inference in possibilistic network classifiers under uncertain observations

Possibilistic networks, which are compact representations of possibility distributions, are powerful tools for representing and reasoning with uncertain and incomplete information in the framework of possibility theory. They are like Bayesian networks but lie on possibility theory to deal with uncertainty, imprecision and incompleteness. While classification is a very useful task in many real world applications, possibilistic network-based classification issues are not well investigated in general and possibilistic-based classification inference with uncertain observations in particular. In this paper, we address on one hand the theoretical foundations of inference in possibilistic classifiers under uncertain inputs and propose on the other hand a novel efficient algorithm for the inference in possibilistic network-based classification under uncertain observations. We start by studying and analyzing the counterpart of Jeffrey's rule in the framework of possibility theory. After that, we address the validity of Markov-blanket criterion in the context of possibilistic networks used for classification with uncertain inputs purposes. Finally, we propose a novel algorithm suitable for possibilistic classifiers with uncertain observations without assuming any independence relations between observations. This algorithm guarantees the same results as if classification were performed using the possibilistic counterpart of Jeffrey's rule. Classification is achieved in polynomial time if the target variable is binary. The basic idea of our algorithm is to only search for totally plausible class instances through a series of equivalent and polynomial transformations applied on the possibilistic classifier taking into account the uncertain observations.

Modelling parking choice behaviour using Possibility Theory

This article presents a discrete choice model for evaluating parking users’ behaviour. In order to explicitly take into account imprecision and uncertainty underlying a user's choice process, the proposed model has been developed within the framework of Possibility Theory. This approach is an alternative way to represent imperfect knowledge (uncertainty) of users about both parking and transportation system status, as well as the approximate reasoning of the human decision maker (imprecision). The resulting model is a quantitative soft computing tool that could support traffic analysts in planning parking policies and Advanced Traveller Information Systems. In fact, effects of information on user choice can be incorporated into the model itself. Thus, we consider the parking user be a decision maker who assumes a certain choice set (set of perceived parking alternatives); the user has some information about the parking supply system and he/she associates each parking alternative with an approximate perceived cost/utility that is represented by a possibility distribution; and, finally, the user chooses the alternative which minimises/maximises his/her perceived parking cost/utility. The results show how the model is able to represent the effect of various parking policies on users’ behaviour and how the single component of parking policy affects the decision process.