Some of the **indicators** described above have already been generalized for **discrete** **Choquet** **integrals**, the case for example of the degree of orness and the dispersion indicator (Shannon entropy). The purpose of this article is to propose **indicators** that have not yet been deﬁned for the **Choquet** integral. However, we describe the existing **indicators** here in order to provide a complete compilation of **indicators** for characterizing **Choquet** **integrals**. Hereinafter, the **indicators** are considered from two perspectives -the global and the local. Broadly speaking, a global indicator does not depend on the input values to be aggregated while a local one does. This terminology is taken from Dujmovi´ c [5], who proposes a classiﬁcation of orness **indicators** by means of a three-letter code 5 X/Y/Z, 5 This terminology is also adopted by Koles´ arov´ a and Mesiar [15] who provide an elegant explanation

Show more
43 Read more

given in terms of continuity and comonotonic additivity, showing that pos- itive homogeneity can be replaced for continuity (Theorem 3.2). The main result of this paper, Theorem 3.3, presents a **characterization** of families of signed **Choquet** **integrals** in terms of necessary and sufficient conditions which:

The paper is organized as follows. In Section 2 , we recall basic notions and terminology related to the concept of signed **Choquet** **integrals** and present some preliminary **characterization** results. In Section 3 , we survey several results that culminate in a description of comonotonic modularity and establish connections to other well studied properties of aggregation functions. These results are then used in Section 4 to provide characterizations of the various classes of functions considered in the previous sections, as well as of classes of functions extending Sugeno **integrals**.

Show more
14 Read more

Necessity (resp. possibility) measures are very simple min-decomposable (resp. max-decomposable) representations of epistemic uncertainty due to incomplete knowledge. They can be used in both quantitative and qualitative settings. In the present work, we revisit **Choquet** and Sugeno **integrals** as criteria for decision under uncertainty and propose new axioms when uncertainty is representable in possibility theory. First, a **characterization** of **Choquet** integral with respect to a possibility or a necessity measure is proposed. We respectively add an optimism or a pessimism axiom to the axioms of the **Choquet** integral with respect to a general capacity. This new axiom enforces the maxitivity or the minitivity of the capacity without requiring the same property for the functional. It essentially assumes that the decision-maker preferences only reflect the plausibility ordering between states of nature. The obtained pessimistic (resp. optimistic) criterion is an average maximin (resp. maximax) criterion of Wald across cuts of a possibility distribution on the state space. The additional axiom can be also used in the axiomatic approach to Sugeno integral and generalized forms thereof to justify possibility and necessity measures. The axiomatization of these criteria for decision under uncertainty in the setting of preference relations among acts is also discussed. We show that the new axiom justifying possibilistic **Choquet** **integrals** can be expressed in this setting. In the case of Sugeno integral, we correct a **characterization** proof for an existing set of axioms on acts, and study an alternative set of axioms based on the idea of non-compensation.

Show more
18 Read more

Abstract. This paper deals with non-monotonic **Choquet** integral, a generalization of the regular **Choquet** integral. The **discrete** non-monotonic **Choquet** integral is considered under the viewpoint of aggregation. In particular we give an axiomatic **characterization** of the class of non-monotonic **Choquet** integrals.We show how the Shapley index, in contrast with the monotonic case, can assume positive values if the criterion is in average a benefit, depending on its effect in all the possible coalition coalitions, and negative values in the opposite case of a cost criterion.

Show more
12 Read more

This paper is organized as follows. In Section 2, we recall definitions and give examples of entwining struc- tures and entwined modules. In Section 3, we introduce the **integrals** of entwining structure and analyse its pro- perties generalizing the results of [6]. Finally, in Sec- tion 4 we derive the dual form of the **integrals** of entwin- ing structure and its properties.

We have shown that a general form of pessimistic **Choquet** preferences under risk with linear utility leads to optimal portfolio allocation problems that can be solved by quantile regression methods. These portfolios may also be interpreted as maximiz- ing expected return subject to a “coherent” measure of risk in the sense of Artzner, Delbaen, Eber, and Heath (1999). By providing a general method for computing pessimistic portfolios based on **Choquet** expected returns we hope to counter the impression occassionally seen in the literature that alternatives to the dominant ex- pected utility view are either too vague or too esoteric to be applicable in problems of practical significance. There is a growing body of scholarship suggesting that pes- simistic attitudes toward risk offer a valuable complement to conventional views of risk aversion based on expected utility maximization; portfolio allocation would seem to provide a valuable testing ground for further evaluation of these issues.

Show more
18 Read more

and the preferred choice between the two equivalent forms will depend on context, in particular whether it is more convenient for expressions to contain stochastic or classical **integrals**. We now survey the major rules and briefly illustrate their use in simplification. More detailed information and unit tests can be found in ItoIntegralTests.nb.

12 Read more

To cite this version : Dubois, Didier and Prade, Henri and Rico, Agnés and Teheux, Bruno Generalized Sugeno **Integrals**. (2016) In: 16th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2016), 20 June 2016 - 24 June 2016 (Eindhoven, Netherlands).

13 Read more

Adem Kılıçman: Department of Mathematics, University of Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia Current address: Institute of Advanced Technology, University of Putra Mala[r]

To establish the theorem 1, expressing the multivariable Gimel-function in the Mellin-Barnes multiple **integrals** contour with the help of (1.1) and interchanging the order of integrations which is justified under the conditions mentioned above, evaluating the inner -integral with the help of the lemma 2 and evaluating the inner -integral with the help of the lemma 4. and interpreting the resulting multiple **integrals** contour with the help of (1.1) about the gimel-function of r-variables, we obtain the desired theorem 4.

Abstract This paper is devoted to studying the singular integrals and Marcinkiewicz integrals with mixed homogeneity along surfaces, which contain many classical surfaces as model exampl[r]

23 Read more

In this paper, we adopt another approach to model uncertainty, in order to avoid some limits inherent to the maxmin criterion. As usual, the decision maker expresses preferences relative to the uncertain payoffs generated by a real option project at various dates. But this time they are taken into account in a different way: we refer to capacities (instead of additive probabilities) to weight likelihood of events and rely on discounted **Choquet** **integrals** to compute payoffs value 19 . Let’s first clarify these key notions before showing how the dynamics of the real option cash

Show more
37 Read more

By completeness of propositional ω -logic [15, 20] and the validity of the propositions in all models, i.e. measures or **integrals**, of the theory we see that, classically, there should be a proof in the theory that these are indeed interpretations. We will provide such a constructive proof in Theorem 3. This treatment is different from the classical one; see e.g. [19]. We take the topological/computational aspects into account by distinguishing between lower reals and Dedekind reals, moreover we do not use the extension of a valuation to a measure on the Borel sets. Our result is more general: not only is it constructive, and hence valid in any topos, but it also abstracts from a lattice of sets to a general lattice.

Show more
22 Read more

To use the **Choquet** integral preliminary we have to identify the fuzzy measure on the basis of expert knowledge. This identification is complicated by exponential increasing complexity in the sense that it is necessary to set a value of fuzzy measure for each subset of criteria. Setting the values of all coefficients of the fuzzy measure is very difficult or even impossible for the expert. Note that even in case of three criteria for determining the fuzzy measure it is necessary to obtain 2 3 8 coefficients. Despite this complexity **Choquet** integral still can be applied in practice. For this Grabisch proposed the concept of - order fuzzy measure or - additive fuzzy measure [12]. This order can be less than the number of aggregated criteria, . Essence of the - additivity concept consists in simplification of the task of fuzzy measures determining by excluding from consideration the dependencies between more than criteria. According to the - additivity concept in most practical cases it is possible to use the **Choquet** integral with respect to 2-order fuzzy measure or, equivalently, the 2-order **Choquet** integral because it allows to model the interaction between the criteria while remaining relatively simple [12]. The paper [13] is entirely devoted to the question under what conditions such a simplification (using of the 2-order **Choquet** integral) is correct. This paper presents necessary conditions that should satisfy the expert preferences in order that they can be formalized using the 2-order **Choquet** integral.

Show more
In the previous chapter, we have studied to find the area bounded by the curve y = f (x), the ordinates x = a, x = b and x-axis, while calculating definite integral as the limit of a sum. Here, in this chapter, we shall study a specific application of **integrals** to find the area under simple curves, area between lines and arcs of circles, parabolas and ellipses (standard forms only). We shall also deal with finding the area bounded by the above said curves.

20 Read more

This part four begins with a quick recall of basic probability, including con- ditional expectation, random processes, constructions of probability measures and ending with short comments on martingale in **discrete** time, in a way, this is an enlarged review of part three. Chapter 2 deals with stochastic processes in continuous times, martingales, L´ evy processes, and ending with integer random measures. In Chapters 3 and 4, we introduce the stochastic calculus, in two iterations, beginning with stochastic integration and passing through stochas- tic differentials and ending with stochastic flows. Chapters 5 is more like an appendix, where Makrov process are discussed in a more ’analysis’ viewpoint, which ends with a number of useful examples of transition functions.

Show more
457 Read more

The block diagram of the proposed model is shown in Fig. 1. The proposed fuzzy rule is similar to that of T-S model as far as premise part is concerned but is different in the consequent part. The formation of h x k ( ) i which is the input to the **Choquet** Integral is entirely unique thus giving better output. This input information is obtained by fuzzifying the input data and then sorting the fuzzified data to form the source of information to the **Choquet** Integral system.

22 Read more

In admission process to any stream it is difficult to rank the student because the seats are limited. Normally the admissions are given on the basis of performance of student in previous examination or on the basis of entrance test. But it could not give proper justice to student because intelligence quotient, subject linking, responsibility etc. varies student to student and subject to subject. Fuzzy measures and **integrals** are appropriate tools to collect the information.

Thus far the algorithms proposed in this thesis, consider monotonicity, i.e., the al- gorithms enforce monotonicity by auxiliary constraints. Since there are m2 m−1 constraints for assuring monotonicity (given m attributes), for large number of at- tributes solving the optimization problem indeed is quite difficult and time con- suming. Chapter 1 clearly states that this thesis provides several algorithms and approaches to deal with monotone data. Since monotonicity in our case is a prereq- uisite, the datasets which are desirable are monotone. This means that, they have monotone structures. The ultimate goal of learning is to bias the observations. In this regard, the expectation is that the model can capture the properties of data. Since monotonicity is a kind of data-property, the model can capture such proper- ties. Note that there is no guarantee, that the model can capture whole monotonicity property, but at least there is a chance to capture it partially. To this end, this section serves the idea of learning the monotone models underlying the **Choquet** integral without any monotonicity constraints. This problem in the literature is addressed under “relaxation”. Basically the core idea of relaxation is to learn the optimal pa- rameters for a fuzzy measure without enforcing any monotonicity constraints and finally making corrections for the optimal learned parameters. It is quite obvious, that there is no guarantee, that learned parameters satisfy monotonicity. Now the non trivial question is, how is possible to monotonize the learned parameters in the end? Let us consider more in the details the structure of fuzzy measure. Basically, the structure of a fuzzy measure can be seen as a DAG (directed acyclic graph) structure (S), where V = {V 1 , . . . , V p }, the vertices, correspond to the elements

Show more
170 Read more