Abstract—In our opinion, and in accordance with current literature, the precise contribution of genetic fuzzy systems to the corpus of the machine learning theory has not been clearly stated yet. In particular, we question the existence of a set of problems for which the use of fuzzy rules, in combination with genetic algorithms, produces more robust models, or classifiers that are inherently better than those arising from the Bayesian point of view. We will show that this set of problems actually exists, and comprises interval and fuzzy valued datasets, but it is not being exploited. Current genetic fuzzy classifiers deal with crisp classification problems, where the role of fuzzy sets is reduced to give a parametric definition of a set of discriminant functions, with a convenient linguistic interpretation. Provided that the customary use of fuzzy sets in statistics is vague data, we propose to test genetic fuzzy classifiers over imprecisely measured data and design experiments well suited to these problems. The same can be said about genetic fuzzy models: the use of a scalar fitness function assumes crisp data, where fuzzy models, a priori, do not have advantages over statistical regression. Index Terms—Fuzzy fitness function, fuzzy rule-based classifiers, fuzzy rule-based models, genetic fuzzy systems, vague data. I.

SUMMARY 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 R 3 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.

This paper is written with two purposes in mind. First, it brings together some recent results in the area of mean variance theory model validation for fuzzy systems in the existence of subjective measures suggested by experts. The central idea of the methods presented here is to map random uncertainty given a portfolio selection model into fuzzy random uncertainty description which is useful from an application and analysis point of view. Secondly, this paper also presents a brief self-contained glimpse of empirical representations to practitioners unfamiliar with the field of fuzzy modeling. It is hoped that the expositions such as this one will open new collaborations between other branches of fuzzy mathematics (in particular, operations research which deals with large scale static uncertainty modeling) and asset pricing theories.

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The variance of fuzzy random variables often appears in fuzzy random programming problems. Based on the definition of the variance of a fuzzy random variable, this paper attempts to deduce several formulas for the variances of trapezoidal fuzzy random variables, in which the randomness is characterized by uniform distribution. Firstly, we give the moment formulas for trapezoidal fuzzy variables. Then, according to the obtained results, we deduce the variance formulas for trapezoidal fuzzy random variables. The obtained formulas are useful in studying the properties of fuzzy random programming problems. At last, we also provide some applications of the obtained formulas through three numerical examples.

Abstract Over the last four decades, several estimation issues of the beta have been discussed extensively in a large literature. An emerging consensus is that the betas are time-varying and their estimates are impacted upon the return interval and the length of the estimation period. These findings lead to the prominence of the practical implementation of the Capital Asset Pricing Model. Our goal in this paper is two-fold: After studying the impact of the return interval on the beta estimates, we analyze the sample size effects on the preceding estimation. Working in the framework of fuzzy set theory, we first associate the returns based on closing prices with the intraperiod volatility for the representation by the means of a fuzzy random variable in order to incorporate the effect of the interval period over which the returns are measured in the analysis. Next, we use these fuzzy returns to estimate the beta via fuzzy least square method in order to deal efficiently with outliers in returns, often caused by structural breaks and regime switches in the asset prices. A bootstrap test and an asymptotic test are carried out to determine whether there is a linear relationship between the market portfolio fuzzy return and the given asset fuzzy return. Finally, the empirical results on French stocks reveal that our beta estimates seem to be more stable than the ordinary least square (OLS) estimates when the return intervals and the sample size change.

Fuzzy random variables have been introduced as an imprecise concept of numeric values for characterizing the imprecise knowledge. The descriptive parameters can be used to describe the primary features of a set of fuzzy random observations. In fuzzy environments, the expected values are usually represented as fuzzy-valued, interval-valued or numeric-valued descriptive parameters using various metrics. Instead of the concept of area metric that is usually adopted in the relevant studies, the numeric expected value is proposed by the concept of distance metric in this study based on two characters (fuzziness and randomness) of FRVs. Comparing with the existing measures, although the results show that the proposed numeric expected value is same with those using the different metric, if only triangular membership functions are used. However, the proposed approach has the advantages of intuitiveness and computational efficiency, when the membership functions are not triangular types. An example with three datasets is provided for verifying the proposed approach.

Abstract — When questionnaires are designed, each factor under study can be assigned a set of different items. The answers to these questions must be merged in order to obtain the level of that input. Therefore, it is typical for data acquired from questionnaires that each of the inputs and outputs are not numbers, but sets of values. In this paper, we represent the information contained in such a set of values by means of a fuzzy number. A fuzzy statistics-based interpretation of the semantic of a fuzzy set will be used for this purpose, as we will consider that this fuzzy number is a nested family of confidence intervals for the value of the variable. The accuracy of the model will be expressed by means of an intervalvalued function, derived from a recent definition of the variance of a fuzzy random variable. A multicriteria genetic learning algorithm, able to optimize this interval-valued function, is proposed. As an example of the application of this algorithm, a practical problem of modeling in marketing is solved. I.

Abstract — Multicriteria genetic algorithms can produce fuzzy models with a good balance between their precision and their complexity. The accuracy of a model is usually measured by the mean squared error of its residual. When vague training data is used, the residual becomes a fuzzy number, and it is needed to optimize a combination of crisp and fuzzy objetives in order to learn balanced models. In this paper, we will extend the NSGA-II algorithm to this last case, and test it over a practical problem of causal modeling in marketing. Different setups of this algorithm are compared, and it is shown that the algorithm proposed here is able to improve the generalization properties of those models obtained from the defuzzified training data. I.

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