Interactive Fuzzy Random Two-level Linear Programming through Fractile Criterion Optimization

This paper considers two-level linear programming problems involving fuzzy random variables. Having introduced level sets of fuzzy random variables and fuzzy goals of decision makers, following fractile criterion optimization, fuzzy random two-level programming problems are transformed into deterministic ones. Interactive fuzzy programming is presented for deriving a satisfactory solution efficiently with considerations of overall satisfactory balance

A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation

Let $X$ be a fuzzy set--valued random variable (\frv{}), and \huku{X} the family of all fuzzy sets $B$ for which the Hukuhara difference X\HukuDiff B exists $\mathbb{P}$--almost surely. In this paper, we prove that $X$ can be decomposed as X(\omega)=C\Mink Y(\omega) for $\mathbb{P}$--almost every $\omega\in\Omega$, $C$ is the unique deterministic fuzzy set that minimizes $\mathbb{E}[d_2(X,B)^2]$ as $B$ is varying in \huku{X}, and $Y$ is a centered \frv{} (i.e. its generalized Steiner point is the origin). This decomposition allows us to characterize all \frv{} translation (i.e. X(\omega) = M \Mink \indicator{\xi(\omega)} for some deterministic fuzzy convex set $M$ and some random element in \Banach). In particular, $X$ is an \frv{} translation if and only if the Aumann expectation $\mathbb{E}X$ is equal to $C$ up to a translation. Examples, such as the Gaussian case, are provided.Comment: 12 pages, 1 figure. v2: minor revision. v3: minor revision; references, affiliation and acknowledgments added. Submitted versio

A new integral for capacities

A new integral for capacities, different from the Choquet integral, is introduced and characterized. The main feature of the new integral is concavity, which might be interpreted as uncertainty aversion. The integral is then extended to fuzzy capacities, which assign subjective expected values to random variables (e.g., portfolios) and may assign subjective probability only to a partial set of events. An equivalence between minimum over sets of additive capacities (not necessarily probability distributions) and the integral w.r.t. fuzzy capacities is demonstrated. The extension to fuzzy capacities enables one to calculate the integral also when there is information only about a few events and not about all of them.new integral, capacity, choquet integral, fuzzy capacity, concavity

A linear regression model for imprecise response

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R3 with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies. © 2010 Elsevier Inc. All rights reserved

Properties of Statistical Depth with Respect to Compact Convex Random Sets: The Tukey Depth

We study a statistical data depth with respect to compact convex random sets, which is consistent with the multivariate Tukey depth and the Tukey depth for fuzzy sets. In addition, it provides a different perspective to the existing halfspace depth with respect to compact convex random sets. In studying this depth function, we provide a series of properties for the statistical data depth with respect to compact convex random sets. These properties are an adaptation of properties that constitute the axiomatic notions of multivariate, functional, and fuzzy depth-functions and other well-known properties of depth.For L.G.-D.L.F. and A.N.-R., this research was supported by grant MTM2017-86061-C2-2-P funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”. P.T. was supported by the Ministerio de Economía y Competitividad grant MTM2015-63971-P, the Ministerio de Ciencia, Innovación y Universidades grant PID2019-104486GB-I00, and the Consejería de Empleo, Industria y Turismo del Principado de Asturias grant GRUPIN-IDI2018-000132

Fuzzification of crisp domains

summary:The present paper is devoted to the transition from crisp domains of probability to fuzzy domains of probability. First, we start with a simple transportation problem and present its solution. The solution has a probabilistic interpretation and it illustrates the transition from classical random variables to fuzzy random variables in the sense of Gudder and Bugajski. Second, we analyse the process of fuzzification of classical crisp domains of probability within the category $ID$ of $D$-posets of fuzzy sets and put into perspective our earlier results concerning categorical aspects of fuzzification. For example, we show that (within $ID$) all nontrivial probability measures have genuine fuzzy quality and we extend the corresponding fuzzification functor to an epireflector. Third, we extend the results to simplex-valued probability domains. In particular, we describe the transition from crisp simplex-valued domains to fuzzy simplex-valued domains via a “simplex” modification of the fuzzification functor. Both, the fuzzy probability and the simplex-valued fuzzy probability is in a sense minimal extension of the corresponding crisp probability theory which covers some quantum phenomenon

Selecting Informative Features with Fuzzy-Rough Sets and its Application for Complex Systems Monitoring

One of the main obstacles facing current intelligent pattern recognition appli-cations is that of dataset dimensionality. To enable these systems to be effective, a redundancy-removing step is usually carried out beforehand. Rough Set Theory (RST) has been used as such a dataset pre-processor with much success, however it is reliant upon a crisp dataset; important information may be lost as a result of quantization of the underlying numerical features. This paper proposes a feature selection technique that employs a hybrid variant of rough sets, fuzzy-rough sets, to avoid this information loss. The current work retains dataset semantics, allowing for the creation of clear, readable fuzzy models. Experimental results, of applying the present work to complex systems monitoring, show that fuzzy-rough selection is more powerful than conventional entropy-based, PCA-based and random-based methods. Key words: feature selection; feature dependency; fuzzy-rough sets; reduct search; rule induction; systems monitoring.