Sébastien Destercke

Unifying practical uncertainty representations -- I: Generalized p-boxes

There exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Fersons p-boxes and Neumaiers clouds. Both for theoretical and practical considerations, it is very useful to know whether one representation is equivalent to or can be approximated by other ones. In this paper, we define a generalized form of usual p-boxes. These generalized p-boxes have interesting connections with other previously known representations. In particular, we show that they are equivalent to pairs of possibility distributions, and that they are special kinds of random sets. They are also the missing link between p-boxes and clouds, which are the topic of the second part of this study.

Toward an Axiomatic Definition of Conflict Between Belief Functions

Recently, the problem of measuring the conflict between two bodies of evidence represented by belief functions has known a regain of interest. In most works related to this issue, Dempsters rule plays a central role. In this paper, we propose to study the notion of conflict from a different perspective. We start by examining consistency and conflict on sets and extract from this settings basic properties that measures of consistency and conflict should have. We then extend this basic scheme to belief functions in different ways. In particular, we do not make any a priori assumption about sources (in)dependence and only consider such assumptions as possible additional information.

Unifying practical uncertainty representations. II: Clouds

There exist many simple tools for jointly capturing variability and incomplete information by means of uncertainty representations. Among them are random sets, possibility distributions, probability intervals, and the more recent Fersons p-boxes and Neumaiers clouds, both defined by pairs of possibility distributions. In the companion paper, we have extensively studied a generalized form of p-box and situated it with respect to other models. This paper focuses on the links between clouds and other representations. Generalized p-boxes are shown to be clouds with comonotonic distributions. In general, clouds cannot always be represented by random sets, in fact not even by two-monotone (convex) capacities.

Probability boxes on totally preordered spaces for multivariate modelling

A pair of lower and upper cumulative distribution functions, also called probability box or p-box, is among the most popular models used in imprecise probability theory. They arise naturally in expert elicitation, for instance in cases where bounds are specified on the quantiles of a random variable, or when quantiles are specified only at a finite number of points. Many practical and formal results concerning p-boxes already exist in the literature. In this paper, we provide new efficient tools to construct multivariate p-boxes and develop algorithms to draw inferences from them. For this purpose, we formalise and extend the theory of p-boxes using Walleys behavioural theory of imprecise probabilities, and heavily rely on its notion of natural extension and existing results about independence modeling. In particular, we allow p-boxes to be defined on arbitrary totally preordered spaces, hence thereby also admitting multivariate p-boxes via probability bounds over any collection of nested sets. We focus on the cases of independence (using the factorization property), and of unknown dependence (using the Frechet bounds), and we show that our approach extends the probabilistic arithmetic of Williamson and Downs. Two design problems-a damped oscillator, and a river dike-demonstrate the practical feasibility of our results.

An extension of Universal Generating Function in Multi-State Systems Considering Epistemic Uncertainties

Many practical methods and different approaches have been proposed to assess Multi-State Systems (MSS) reliability measures. The universal generating function (UGF) method, introduced in 1986, is known to be a very efficient way of evaluating the availability of different types of MSSs. In this paper, we propose an extension of the UGF method considering epistemic uncertainties. This extended method allows one to model ill-known probabilities and transition rates, or to model both aleatory and epistemic uncertainty in a single model. It is based on the use of belief functions which are general models of uncertainty. We also compare this extension with UGF methods based on interval arithmetic operations performed on probabilistic bounds.

A flexible bipolar querying approach with imprecise data and guaranteed results

In this paper, we propose an approach to query a database when the user preferences are bipolar (i.e., express both constraints and wishes about the desired result) and the data stored in the database are imprecise. Results are then completely ordered with respect to these bipolar preferences, giving priority to constraints over wishes. Additionally, we propose a treatment that allows us to guarantee that any query will return a result, even if no element satisfies all constraints specified by the user. Such a treatment may be useful when users constraints are unrealistic (i.e., cannot be all satisfied simultaneously) and when the user desires a guaranteed result. The approach is illustrated on a real-world problem concerning the selection of optimal packaging for fresh fruits and vegetables.

Ranking of fuzzy intervals seen through the imprecise probabilistic lens

Within the fuzzy literature, the issue of ranking fuzzy intervals has been addressed by many authors, who proposed various solutions to the problem. Most of these solutions intend to find a total order on a given collection of fuzzy intervals. However, if one sees fuzzy intervals as descriptions of uncertain quantities, an alternative to rank them is to use ranking rules issued from the imprecise probabilistic literature. In this paper, we investigate ranking rules based on different statistical features (mean, median) and orderings, and relate the obtained (partial) orders to some classical proposals. In particular, we propose a generic expression of stochastic orderings, and then use it to systematically investigate extensions of the most usual stochastic orderings to fuzzy intervals. We also show some relations between those extensions, and explore their relation with existing fuzzy ranking proposals.

Online active learning of decision trees with evidential data

Learning from uncertain data has attracted increasing attention in recent years. In this paper, we propose a decision tree learning method that can not only handle uncertain data, but also reduce epistemic uncertainty by querying the most valuable uncertain instances within the learning procedure. Specifically, we use entropy intervals extracted from the evidential likelihood to query uncertain training instances when needed, with the goal to improve the selection of the splitting attribute. Experimental results under various conditions confirm the interest of the proposed approach. HighlightsActive belief decision trees are learned from uncertain data modelled by belief functions.A query strategy is proposed to query the most valuable uncertain instances while learning decision trees.To deal with evidential data, entropy intervals are extracted from the evidential likelihood.Experiments with UCI data illustrate the robustness of proposed approach to various kinds of uncertain data.

A Decision Support System to design modified atmosphere packaging for fresh produce based on a bipolar flexible querying approach

We define a multi-criteria Decision Support System for designing fresh food packaging.A modified atmosphere packaging simulation module is included in the DSS.A flexible querying module handles imprecise data stored in a packaging database.Using the DSS the user will have only one trial to perform validation step. To design new packaging for fresh food, stakeholders of the food chain express their needs and requirements, according to some goals and objectives. These requirements can be gathered into two groups: (i) fresh food related characteristics and (ii) packaging intrinsic characteristics. Modified Atmosphere Packaging (MAP) is an efficient way to delay senescence and spoilage and thus to extend the very short shelf life of respiring products such as fresh fruits and vegetables. Consequently, packaging O2/CO2 permeabilities must fit the requirements of fresh fruits and vegetable as predicted by virtual MAP simulating tools. Beyond gas permeabilities, the choice of a packaging material for fresh produce includes numerous other factors such as the cost, availability, potential contaminants of raw materials, process ability, and waste management constraints. For instance, the user may have the following multi-criteria query for his/her product asking for a packaging with optimal gas permeabilities that guarantee product quality and optionally a transparent packaging material made from renewable resources with a cost for raw material less than 3?/kg. To help stakeholders taking a rational decision based on the expressed needs, a new multi-criteria Decision Support System (DSS) for designing biodegradable packaging for fresh produce has been built. In this paper we present the functional specification, the software architecture and the implementation of the developed tool. This tool includes (i) a MAP simulation module combining mass transfer models and respiration of the food, (ii) a multi-criteria flexible querying module which handles imprecise, uncertain and missing data stored in the database. We detail its operational functioning through a real life case study to determine the most satisfactory materials for apricots packaging.

A K-nearest neighbours method based on imprecise probabilities

K-nearest neighbours algorithms are among the most popular existing classification methods, due to their simplicity and good performances. Over the years, several extensions of the initial method have been proposed. In this paper, we propose a K-nearest neighbours approach that uses the theory of imprecise probabilities, and more specifically lower previsions. We show that the proposed approach has several assets: it can handle uncertain data in a very generic way, and decision rules developed within this theory allow us to deal with conflicting information between neighbours or with the absence of close neighbour to the instance to classify. We show that results of the basic k-NN and weighted k-NN methods can be retrieved by the proposed approach. We end with some experiments on the classical data sets.