Choquet Capacities Research Articles

An axiomatic approach to default risk and model uncertainty in rating systems

In this paper, we deal with an axiomatic approach to default risk. We introduce the notion of a default risk measure, which generalizes the classical probability of default (PD), and allows to incorporate model risk in various forms. We discuss different properties and representations of default risk measures via monetary risk measures, families of related tail risk measures, and Choquet capacities. In a second step, we turn our focus on default risk measures, which are given as worst-case PDs and distorted PDs. The latter are frequently used in order to take into account model risk for the computation of capital requirements through risk-weighted assets (RWAs), as demanded by the Capital Requirement Regulation (CRR). In this context, we discuss the impact of different default risk measures and margins of conservatism on the amount of risk-weighted assets.

Associated Statistical Parameters’ Aggregations in Interactive MADM

From recent studies, the concept of “monotone expectation” (ME) of Interactive Multi-Attribute Decision Making (MADM) is well known, which was developed for the case of different fuzzy sets. This article develops the concept of “monotone expectation” for such statistical parameters as variance, k-order moment and covariance. We investigate the problem of the definition of some statistical parameters, when the uncertainty is represented by a monotone measure—a fuzzy measure—instead of an additive measure. The study presents the concept of the definition of monotone statistical parameters based on the Choquet finite integral for the definition of monotone expectation, monotone variance, monotone k-order moment and monotone covariance. Associated statistical parameters are also presented—expectation, variance, k-order moment and covariance—which are defined in relation to associated probabilities of a fuzzy measure. It is shown that the monotone statistical parameters defined in the study are defined by one particular relevant associated statistical parameter out of the total number n! of such parameters. It is also shown that the aggregations with monotone statistical parameters used in interactive MADM models take into account interactions of the focal elements of only one consonant structure from the n! consonant structures of attributes. In order to take into account the interactions of the focal elements of all n! consonant structures of attributes, the monotone statistical parameters were expanded into the F-associated statistical parameters. Expansion correctness implies that if dual second-order Choquet capacities are taken as the fuzzy measures of aggregation of the F-associated statistical parameters, then the F-associated statistical parameters coincide with the corresponding monotone statistical parameters. A scheme for embedding new aggregation operators, monotone statistical parameters and F-associated statistical parameters into the interactive MADM model has been developed. Specific numerical examples are presented to illustrate the obtained results.

SMAA-Choquet-FlowSort: A novel user-preference-driven Choquet classifier applied to supplier evaluation

The Choquet integral has been used as an aggregation operator to deal with interacting criteria in different types of problems. For ordinal classification problems, the majority of Choquet integral-based models proposed in the literature are built upon a supervised machine learning perspective, where a training data set is considered. Despite the effectiveness of these classification algorithms, their training-data-dependency may be considered a drawback in some decision-making problems. Our goal is then to propose a new multiple criteria Choquet classifier to conduct sorting based on user preference information. The classifier is built inside the FlowSort framework and, as such, aggregates intensity of preferences with respect to pairs of criteria instead of directly aggregating the criteria evaluations. This characteristic allows criteria and limiting profiles to be assessed by heterogeneous scales. In addition, we apply the Stochastic Multi-criteria Acceptability Analysis to the proposed classifier in order to elicit the Choquet capacities, model uncertain input data and analyze robustness of the results. The proposed classifier is applied to a real decision problem regarding the evaluation of pharmaceutical suppliers.

Agile Multicriteria Decision Analysis considering Interaction

An agile system of multiple criteria decision analysis is proposed here. Its main component avoids the collection of field direct information about the importance of the criteria and of the interaction between criteria. This is replaced by a derivation of Choquet capacities to the sets of criteria from the preferences they assign to the alternatives evaluated. This indirect derivation is made possible by a rule to assess importance and interaction based on a principle of preference concentration. The agile applicability of the system is enhanced by the admission of the sampling of initial assessments by pairwise trichotomic comparisons. This possibility reduces the need to precisely measure distances and the care about transitivity. Finally, an ordering of the decision rules based on the facility of explanation of their results provides agility for the final step of the decision. An example of application of the system is given by the study of a case of supplier evaluation.

Multicriteria decision support for the evaluation of electricity supply resilience: Exploration of interacting criteria

The resilience of an energy system and especially the national electricity supply is a complex and multidimensional concept, which is receiving growing attention at a European level. In particular, the increasing risks of extended electricity supply disruptions and/or severe electricity price fluctuations are stressing the need for an assessment of the European countries’ electricity supply resilience. This paper proposes an elaborative multicriteria decision support methodological framework, based on three major resilience dimensions, namely: “resist”, “restabilise” and “recover”. In total, 35 European countries are evaluated and ranked according to their performance on 17 interacting evaluation criteria. The overall evaluation of the countries is established based on a synergy of two complementary multicriteria decision aid (MCDA) methods. Specifically, the Choquet Integral is approached and assessed as an importance indicator to properly accommodate and incorporate the interacting criteria into the value system, negating therefore their arbitrary effects on the final benchmarking. Simultaneously, a procedure based on the Simos method supports the robust elicitation of the Choquet capacities. All preferential information required for the implementation of the MCDA framework has been elicited from an expert, highly knowledgeable of the European energy system. This research work aims to support energy policymakers in Europe and provide guidelines and areas for improvement at a country level.

Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss and Simpson’s Paradox

Imprecise probabilities alleviate the need for high-resolution and unwarranted assumptions in statistical modeling. They present an alternative strategy to reduce irreplicable findings. However, updating imprecise models requires the user to choose among alternative updating rules. Competing rules can result in incompatible inferences, and exhibit dilation, contraction and sure loss, unsettling phenomena that cannot occur with precise probabilities and the regular Bayes rule. We revisit some famous statistical paradoxes and show that the logical fallacy stems from a set of marginally plausible yet jointly incommensurable model assumptions akin to the trio of phenomena above. Discrepancies between the generalized Bayes (B) rule, Dempster’s (D) rule and the Geometric (G) rule as competing updating rules for Choquet capacities of order 2 are discussed. We note that (1) B-rule cannot contract nor induce sure loss, but is the most prone to dilation due to “overfitting” in a certain sense; (2) in absence of prior information, both B- and G-rules are incapable to learn from data however informative they may be; (3) D- and G-rules can mathematically contradict each other by contracting while the other dilating. These findings highlight the invaluable role of judicious judgment in handling low-resolution information, and the care that needs to be take when applying updating rules to imprecise probability models.

Comment: On Focusing, Soft and Strong Revision of Choquet Capacities and Their Role in Statistics

We congratulate Ruobin Gong and Xiao-Li Meng on their thought-provoking paper demonstrating the power of imprecise probabilities in statistics. In particular, Gong and Meng clarify important statistical paradoxes by discussing them in the framework of generalized uncertainty quantification and different conditioning rules used for updating. In this note, we characterize all three conditioning rules as envelopes of certain sets of conditional probabilities. This view also suggests some generalizations that can be seen as compromise rules. Similar to Gong and Meng, our derivations mainly focus on Choquet capacities of order 2, and so we also briefly discuss in general their role as statistical models. We conclude with some general remarks on the potential of imprecise probabilities to cope with the multidimensional nature of uncertainty.

A Nonlinear Extension of Korovkin’s Theorem

In this paper, we extend the classical Korovkin theorems to the framework of comonotone additive, sublinear, and monotone operators. Based on the theory of Choquet capacities, several concrete examples illustrating our results are also discussed.

A PRINCIPLE OF PREFERENCE CONCENTRATION APPLIED TO THE UNSUPERVISED EVALUATION OF THE IMPORTANCE OF MULTIPLE CRITERIA

ABSTRACT The influence of a principle of preference concentration in the process of decision making in different contexts is discussed here. A procedure exploring the concentration of preferences to derive probabilities and another to derive, with higher generality, Choquet capacities are presented. An important feature of this last procedure is that it derives capacities in a space of criteria only from matrices of probabilities of preference according to each criterion, precluding the effort of directly trying to measure the interactions between them.

Weak convergence of probability measures to Choquet capacity functionals

In the definition of weak convergence of probability measures it is assumed that the limit is a probability measure as well. We get rid of this assumption and require that the limit merely needs to be a Choquet-capacity functional. In terms of random variables this means that the distributional limit no longer is a random point, but a random closed set, namely one uniquely determined by the Choquet capacity. For our extended notion of weak convergence there is a counterpart of the portmanteau theorem. Moreover, we demonstrate basic relations to the theory of random closed sets with emphasis on weak convergence in hyperspace topologies including two correspondence theorems. Finally, the approach carries over to sequences of Choquet capacities.

A novel multicriteria decision aiding method based on unsupervised aggregation via the Choquet integral

In multicriteria decision aiding (MCDA), the Choquet integral has been used as an aggregation operator to deal with the case of interacting decision criteria. In this context, a practical problem that arises is related to the identification of the parameters associated with the Choquet integral, which are known as the Choquet capacities. In this paper, we address the problem of capacity identification by means of unsupervised learning, which, in MCDA, refers to the situations in which only the decision matrix is available. Our contribution is twofold. First, we discuss the extension of some previous works on the subject as well as some of their limitations. Then, we introduce a novel method, which is able to associate the parameters of the Choquet integral with the decision table correlation structure. As attested by numerical experiments, the proposed approach is conceptually simple to be implemented and can detect interactions between criteria in a data-driven fashion.

Non-additivity in Accounting Valuation: Theory and Applications

This paper has three objectives. First, to introduce a theoretical solution to the issue of non-additivity between assets in place, relying on an accounting-based valuation approach. Second, to explain how such an approach can be implemented empirically by measuring synergies between assets. Third, to present the properties of this non-additive valuation technique. We use Choquet capacities, that is, non-additive aggregation operators, to measure the interactions between assets and apply our methodology to a sample of US firms from the capital goods industry. To operationalize our approach we examine the relationships between synergies—captured by Choquet capacities—and the market-to-book ratio (proxying for growth options), and show how interactions between assets are consistently linked to a firm’s market-to-book ratio. We also measure firm-specific productive efficiency relative to the industry and firm size. For large firms, efficiency, as defined by our approach, is positively associated with higher future operating cash flows. For small firms, efficiency is positively associated with higher future sales growth. We document that the non-additive approach appears to be better able to identify expected relationships between efficiency and future performance than a simpler approach based on the market-to-book ratio.

On the Semantics and Complexity of Probabilistic Logic Programs

We examine the meaning and the complexity of probabilistic logic programs that consist of a set of rules and a set of independent probabilistic facts (that is, programs based on Sato's distribution semantics). We focus on two semantics, respectively based on stable and on well-founded models. We show that the semantics based on stable models (referred to as the "credal semantics") produces sets of probability measures that dominate infinitely monotone Choquet capacities; we describe several useful consequences of this result. We then examine the complexity of inference with probabilistic logic programs. We distinguish between the complexity of inference when a probabilistic program and a query are given (the inferential complexity), and the complexity of inference when the probabilistic program is fixed and the query is given (the query complexity, akin to data complexity as used in database theory). We obtain results on the inferential and query complexity for acyclic, stratified, and normal propositional and relational programs; complexity reaches various levels of the counting hierarchy and even exponential levels.

Choquet expected utility with affine capacities

This paper studies decisions under ambiguity when attention is paid to extreme outcomes. In a purely subjective framework, we propose an axiomatic characterization of affine capacities, which are Choquet capacities consisting in an affine transformation of a subjective probability. Our main axiom restricts the well-known Savage’s Sure-Thing Principle to a change in a common intermediate outcome. The representation result is then an affine combination of the expected utility of the valued act and its maximal and minimal utilities.

Choquet integral capacities-based data fusion for system health monitoring

Data fusion is a cornerstone step for monitoring equipment health state within a prognostics and health management (PHM) vision. It enables gathering complementary information from several data sources in order to build a global view for the situation evolving over time. Herein, an advanced information fusion method for health monitoring is proposed. It is based on the unsupervised approach from Choquet capacities elicitation and aims at evaluating the global health/performance of the monitored system.

Methods for Constructing Choquet-Capacity Preserving and Ergodic Systems: Examples

The Choquet capacity of a random closed set on a metric space is regarded as or related to a non-additive measure, an upper probability, a belief function, and a counterpart of the distribution functions of ordinary random vectors. Unlike the ordinary measures which are additive, Choquet capacities are generally non-additive. The purpose of this paper is to generalize the traditional measure preserving and measure ergodic systems to non-additive settings: Choquet-capacity preserving and Choquet-capacity ergodic systems, by constructing solid and standard capacity systems.

Elliptic operators and Choquet capacities

Choquet capacities generated by solutions of certain elliptic partial differential equations are discussed. Bibliography: 11 titles.

On the upper semicontinuity of Choquet capacities

The Choquet capacity T of a random closed set X on a metric space E is regarded as or related to a non-additive measure, an upper probability, a belief function, and in particular a counterpart of the distribution functions of ordinary random vectors. While the upper semicontinuity of T on the space of all closed subsets of E (hit-or-miss topology) is highly desired, T is not necessarily u.s.c. if E is not compact, e.g. E = R n . For any locally compact separable metric space E, this controversial situation can be resolved in the probabilistic context by stereographically projecting X into the Alexandroff compactification E ∞ of E with the “north pole” added to the projection. This leads to a random compact set X ¯ that is defined on the same probability space, takes values in a space homeomorphic to the space of X, and possesses an equivalent probability law. Particularly, the Choquet capacity T ¯ of X ¯ is u.s.c. on the space of all closed subsets of E ∞ . Further, consequences of the upper semicontinuity of T ¯ are explored, and a proof of the equivalence between the upper semicontinuity of T and continuity from above on F ( E ) is provided.

Foundations of neo-Bayesian statistics

We study an axiomatic model of preferences, which contains as special cases Subjective Expected Utility, Choquet Expected Utility, Maxmin and Maxmax Expected Utility and many other models. First, we give a complete characterization of the class of functionals representing these preferences. Then, we show that any such functional can be represented as a Choquet integral I ( f ) = ∫ κ ( f ) d ν where κ : B ( Σ ) → A ( C ) is the canonical mapping from the space of bounded Σ-measurable functions into the space of weak ∗ -continuous affine functions on a weak*-compact, convex set C of probability measures on Σ. Conversely, any preference relation defined by means of such functionals satisfies the axioms of the model we study. Different properties of the capacity give rise to different models. Our result shows that the idea of Choquet integration is general enough to embrace all the models mentioned above. In doing so, it widens the range of applicability of well-known procedures in robust statistics theory such as the Neyman–Pearson lemma for capacities [P.J. Huber, V. Strassen, Minimax tests and the Neyman–Pearson lemma for capacities, Ann. Statist. 1 (1973) 251–263], Bayes' theorem for capacities [J.B. Kadane, L. Wasserman, Bayes' theorem for Choquet capacities, Ann. Statist. 18 (1990) 1328–1339] or of results like the Law of Large numbers for capacities [F. Maccheroni, M. Marinacci, A strong law of large numbers for capacities, Ann. Probab. 33 (2005) 1171–1178].

Valuing future cash flows with non separable discount factors and non additive subjective measures: conditional Choquet capacities on time and on uncertainty

We consider future cash flows that are contingent both on dates in time and on uncertain states. The decision maker (DM) values the cash flows according to its 2 decision criterion: Here the payoffs' expectation with respect to a capacity measure. The subjective measure grasps the DM's behaviour in front of the future, in the spirit of de Finetti's (1930) and of Yaari's (1987) Dual Theory in the case of risk. Decomposition of the criterion into two criteria that represent the DM's preferences on uncertain payoffs and time contingent payoffs are derived from Ghirardato's (1997) results. Conditional Choquet integrals are defined by dynamic consistency requirements and conditional capacities are derived, under some conditions on information. In contrast with other models referring to dynamic consistency, ours doesn't collapse into a linear one because it violates a weak version of consequentialism.