Universal integrals based on copulas

For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures, including as special cases Choquet and Sugeno integrals. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the greatest universal integrals are given. Finally, another construction method for obtaining universal integrals is introduced.

Integrals on finite spaces (e.g., sets of criteria in multicriteria decision support) based on capacities are discussed, axiomatized and examplified. We introduce first the universal integrals, covering the Choquet, Shilkret and Sugeno integrals. Based on optimization approach, we discuss decomposition and superdecomposition integrals. We introduce integrals which are universal and decomposition (superdecomposition) ones and integrals constructed by means of copulas. Several distinguished integrals are represented as particular functionals. Finally, we recall also OWA operators and some their generalizations.KeywordsCapacityChoquet integralCopulaCopula–based integralDecomposition integralSugeno integralUniversal integral

The Sugeno integral. A point of view

This paper provides a formal generalization of Nash equilibrium for games under Knightian uncertainty. The paper is devoted to counterparts of the results of Glycopantis and Muir (Econ Theory 13:743–751, l999, Econ Theory 16:239–244, 2000) for capacities. We prove that the expected payoff defined as the integral of a payoff function with respect to the tensor product of capacities on compact Hausdorff spaces of pure strategies is continuous if so is the payoff function. We prove also an approximation theorem for Nash equilibria when the expected utility payoff functions are defined on the space of capacities.

With the advent of Web 3.0 era, the number and complexity of Web pages in Bayesian networks have shown an explosive growth trend. Accompanying this is the geometric growth of information contained in Web pages. Web text data in Bayesian networks usually hide rich knowledge and rules of user value. However, due to the semi-structured, real-time and discrete characteristics of Web text data, it is difficult for users to obtain the knowledge they need directly from such complex data sets. The emergence of fuzzy mathematics provides a good research idea and method for solving such problems. It can use the idea of fuzzy mathematics to analyze the practical problems in text data. Therefore, how to effectively mine the Web text data information and knowledge that users really care about from Bayesian network, and present it in a way that users can understand, it is a very popular research topic at present. In this paper, we select the text of Bayesian network: microblog data for experiments. User data model of microblog is established by using relevant knowledge of fuzzy theory. The concept of fuzzy measure is introduced to calculate the non-additive measure value under the interaction relationship between the detection indicators. Determine the membership function relationship between the detection user and the text data, calculate the integral values of Choquet integral, Sugeno integral and Wang integral of the membership function under the non-additive measure, the final valuable web text data is judged by integral value. On the basis of the above research contents, the research results of Web text mining technology and fuzzy arithmetic mathematics are combined, design and implement information acquisition and analysis for Bayesian network community. The recall rate obtained by the experimental method in this paper is as low as 4%, and tends to be more stable.

In this note, we introduce the concepts of support disjointness super-\(\oplus \)-additivity and positively super-\(\otimes \)-homogeneity of a functional (with respect to pan-addition \(\oplus \) and pan-multiplication \(\otimes \), respectively). By means of these two properties of functionals, we discuss the characteristics of pan-integrals and present an equivalent definition of the pan-integral. As special cases, we obtain the equivalent definitions of the Shilkret integral, the \(+,\cdot \)-based pan-integral, and the Sugeno integral.

In this paper, based on the foundation of the discussion of the Sugeno integral for real functions which was presented by Wu Congxin and Mamadou in 2003, we introduce and consider the concept of the absolute additivity of the Sugeno integral for real functions. We also discuss the fuzzy additivity of the Sugeno integral for real functions. For these two kinds of additivity, we obtain two necessary and sufficient conditions, a sufficient condition, a necessary condition and several counter examples.KeywordsGenetic AlgorithmReal FunctionNonnegative FunctionFuzzy MeasureSolute AdditivityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A hierarchical family of copula-based integrals is introduced and discussed. When considering the product copula, a family of decomposition integrals independently introduced by Even and Lehrer, and by Mesiar and Stupnanova, is recovered. Boundary members are distinguished universal integrals introduced by Klement at al. in 2010.

Guaranteed cost control for multi-inventory systems with uncertain demand

In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann–Liouville integral of a continuous function f(x) of order v(v > 0) which is written as D −v f(x) has been proved to still be continuous and bounded. Furthermore, upper box dimension of D −vf(x) is no more than 2 and lower box dimension of D −v f(x) is no less than 1. If f(x) is a Lipshciz function, D −v f(x) also is a Lipshciz function. While f(x) is differentiable on [0, 1], D −v f(x) is differentiable on [0, 1] too. With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f(x) satisfying Holder condition, upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x). Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying Holder condition of order v(v > 0) is strictly less than 2 − v. Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.

A monotonicity of Interval Sugeno Integrals proposed in “X. Pu, R. Mesiar, R. R. Yager, L. Jin, Interval Sugeno integral with preference,” IEEE Trans. Fuzzy Syst. vol. 28, pp. 597-601, 2020 has been found incorrect with a counter example. This letter presents the counter example and analyzes the reason that causes such nonmonotonicity. In addition, two new related forms of Interval Sugeno Integrals are also proposed, which feature the synchronizations about values and subsets, respectively. The similar monotonicities for the new forms are proved to be correct.

Hypopharyngeal carcinoma is a very aggressive cancer that is generally diagnosed at advanced stages and consequently has a poor prognosis and low survival rate. The treatment of these tumors is designed to cure the patient with the cancer and to preserve or restore physiologic function of the laryngo-pharyngeal complex as well. However, the need for extensive ablative surgery often coupled with radiation therapy renders many patients incapable of performing the basic human functions of swallowing and speech. Loss of such integral functions has a dramatically negative influence on the patients' quality of life, which has already been threatened by the aggressive nature of this disease. Therefore, it is imperative to use a reliable surgical strategy with low morbidity that will allow expedient restoration of speech and swallowing. Consequently, conservation surgery for hypopharynx carcinoma offers a wider resection with promising functional results in hypopharyngeal cancer patients. Recently several procedures were introduced for this purpose and we review the introduction and indications of those conservation methods in hypopharyngeal cancer surgery.

Coordinate interleaved orthogonal designs (CIOD) (Khan and Rajan in IEEE Trans Inform Theory 52(5):2062---2091, 2006) can offer desirable properties, such as full rate, full diversity and single-symbol maximum likelihood decoding for two, three and four transmit antennas under quasi-static fading channels. When fading is time-selective, zero-forcing decoder can applied to achieve good performance while still maintain low decoding complexity. In this paper, theoretical analysis of symbol error rate performance for CIOD codes over time-selective fading channels with a zero-forcing linear receiver is derived. Firstly, a closed-form expression (i.e., not in integral form) is derived for the average symbol pair-wise error probability (SPEP) in time-selective frequency-nonselective independent identically distributed (i.i.d.) Rayleigh fading channels. Then, the SPEP is used to derive a tight upper bound (UB) for the symbol-error rate (SER) of CIOD codes. Simulation results indicate that our theoretical UB often coincides (within 0.05 dB) with the true SER obtained via Monte-Carlo simulation. The UB can thus be used to accurately predict and optimize the performance of CIOD codes over time-selective fading channels.

In this correspondence, we correct an erroneous result on the achievability part of the Renyi common information with order $1+s\in (1,2]$ in (L. Yu and V. Y. F. Tan, “Wyner’s common information under Renyi divergence measures,” IEEE Trans. Inf. Theory, vol. 64, no. 5, pp. 3616–3632, May 2018). The new achievability result (upper bound) of the Renyi common information no longer coincides with Wyner’s common information. We also provide a new converse result (lower bound) in this correspondence for the Renyi common information with order $1+s\in (1,\infty]$ . Numerical results show that for doubly symmetric binary sources, the new upper and lower bounds coincide for the order $1+s\in (1,2]$ and they are both strictly larger than Wyner’s common information for this case.

We determine the asymptotic capacity of a Gaussian multiple-relay channel as the number of relays tends to infinity. The upper bound is an application of the cut-set theorem, and the lower bound follows from an argument involving uncoded transmission. Hence, this paper gives one more example where the cut-set bound is achievable, and one more example where uncoded transmission achieves optimal performance. In the latter sense, the result is an extension of the work of Gastpar, Rimoldi and Vetterli (see IEEE Trans. Info. Theory, May 2001). The arguments of this paper are also relevant to wireless networks, yielding an asymptotic capacity result.