A hybrid fuzzy regression model and its application in hydrology engineering

The goal of this paper is to handle the large variation issues in fuzzy data by constructing a variable spread multivariate adaptive regression splines (MARS) fuzzy regression model with crisp parameters estimation and fuzzy error terms. It deals with imprecise measurement of response variable and crisp measurement of explanatory variables. The proposed method is a two-phase procedure which applies the MARS technique at phase one and an optimization problem at phase two to estimate the center and fuzziness of the response variable. The proposed method, therefore, handles two problems simultaneously: the problem of large variation issue and the problem of variation spreads in fuzzy observations. A realistic application of the proposed method is also presented, by which the suspended load is modeled using discharge in a hydrology engineering problem. Empirical results demonstrate that the proposed approach is more efficient and more realistic than some well-known least-squares fuzzy regression models.     

Synthesis of optimal fuzzy estimation algorithms

Conclusion The article shows that the well-known ellipsoidal estimation formalism can be constructively extended to the case of estimating sets of a more general form. Mathematical models of these sets are defined not only by positive definite quadratic forms (as in the case with ellipsoids), but also by Lyapunov, Bellman, and other functions. We have considered the parametric properties of estimates represented by fuzzy estimating sets. A formalism approximating the set-theoretical intersection operation has been developed for these fuzzy estimates. An important feature of the proposed approach is that the intersection procedure is optimized not in each step or cycle, but over the entire sequence of steps in accordance with an additive criterion which is fairly natural for the relevant class of problems. We have thus established a relationship between optimal fuzzy set-theoretical estimation procedures and standard optimal algorithms. In the concluding part of the article we have considered state estimation of a static object from observations distorted by additive noise. The state estimation problem has been solved in the class of fuzzy ellipsoidal estimates with a so-called tolerant part. The linear “size“ or “radius“ of the tolerant part provides an efficient measure of the “degree of fuzziness“ of the estimate. On the whole, the estimation (filtering) algorithm proposed in this article, like other fuzzy estimation algorithms, is robust to prior errors. The study has been supported by US CRDF grant VE 2300. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 167–183, July–August, 1998.     

Random fuzzy alternating renewal processes

Random fuzzy theory offers an appropriate mechanism to model random fuzzy phenomena, with a random fuzzy variable defined as a function from a credibility space to a collection of random variables. Based on this theory, this paper presents the results of an investigation into the representation of properties of alternating renewal processes that are described by sequences of positive random fuzzy vectors. It provides a theorem on the limit value of the average chance of a given random fuzzy event in terms of “system being on at time t”. The resultant model coincides with that attainable by stochastic analysis when the random fuzzy vectors degenerate to random vectors.     

Non-commutative fuzzy Galois connections

 Fuzzy Galois connections were introduced by Bělohlávek in [4]. The structure considered there for the set of truth values is a complete residuated lattice, which places the discussion in a “commutative fuzzy world”. What we are doing in this paper is dropping down the commutativity, getting the corresponding notion of Galois connection and generalizing some results obtained by Bělohlávek in [4] and [7]. The lack of the commutative law in the structure of truth values makes it appropriate for dealing with a sentences conjunction where the order between the terms of the conjunction counts, gaining thus a temporal dimension for the statements. In this “non-commutative world”, we have not one, but two implications ([15]). As a consequence, a Galois connection will not be a pair, but a quadruple of functions, which is in fact two pairs of functions, each function being in a symmetric situation to his pair. Stating that these two pairs are compatible in some sense, we get the notion of strong L-Galois connection, a more operative and prolific notion, repairing the “damage” done by non-commutativity. Dedicated to Prof. Ján Jakubík on the occasion of his 80th birthday.     

Fuzzy linear regression model based on fuzzy scalar product

The new concept and method of imposing imprecise (fuzzy) input and output data upon the conventional linear regression model is proposed in this paper. We introduce the fuzzy scalar (inner) product to formulate the fuzzy linear regression model. In order to invoke the conventional approach of linear regression analysis for real-valued data, we transact the α-level linear regression models of the fuzzy linear regression model. We construct the membership functions of fuzzy least squares estimators via the form of “Resolution Identity” which is a well-known formula in fuzzy sets theory. In order to obtain the membership value of any given least squares estimate taken from the fuzzy least squares estimator, we transform the original problem into the optimization problems. We also provide two computational procedures to solve the optimization problems.     

Fuzzy knowledge representation and reasoning using a generalized fuzzy petri net and a similarity measure

In the study of weighted fuzzy production rules (WFPRs) reasoning, we often need to consider those rules whose consequences are represented by two or more propositions connected by “AND” or “OR”. To enhance the representation capability of those rules, this paper proposes two types of knowledge representation parameters, namely, the input weight and the output weight, for a rule. A Generalized Fuzzy Petri Net (GFPN) is also presented for WFPR reasoning. Furthermore, this paper gives a similarity measure to improve the evaluation method of WFPRs and the multilevel fuzzy reasoning in which the consequences and their certainty factors are deduced synchronously by using a GFPN.     

Intuitionistic fuzzy recognizers and intuitionistic fuzzy finite automata

Intuitionistic fuzzy recognizers and intuitionistic fuzzy finite automata are discussed. The notions of intuitionistic fuzzy recognizer, complete accessible intuitionistic fuzzy recognizer, intuitionistic fuzzy finite automata, deterministic intuitionistic fuzzy finite automata, and intuitionistic fuzzy language are introduced. It is shown that the languages recognized by intuitionistic fuzzy recognizer are regular, and the intuitionistic fuzzy languages recognized by the intuitionistic fuzzy finite automaton and the intuitionistic fuzzy languages recognized by deterministic intuitionistic fuzzy finite automaton are equivalent. This work is supported by National Science Foundation of China (Grant No.10571112), “TRAPOYT” of China and National 973 Foundation Research Program(Grant No.2002CB312200).     

Lattice structure on some fuzzy algebraic systems

In this paper, we study the lattice structure of some fuzzy algebraic systems such as (G-)fuzzy groups, some fuzzy ordered algebras and fuzzy hyperstructures. We prove that under suitable conditions, these structures form a distributive or modular lattice. This research partially is supported by the “ Fuzzy Systems and its Applications Center of Excelence, Shahid Bahonar University of Kerman, Iran”.     

The sufficient and necessary condition for chance distribution of bifuzzy variable

Fuzzy sets and fuzzy variables have undergone several different extensions overtime. One of them involved including a “bifuzzy variable” as a fuzzy element for describing the more complete systems. The properties of bifuzzy variable were obtained by introducing the concept of “chance distribution”. In this paper, we will present a sufficient and necessary condition for chance distribution of bifuzzy variable. Here we present a constructive proof base on credibility theory for the sufficient part.     

A new approach to formulate fuzzy regression models

A fuzzy regression model is developed to construct the relationship between the response and explanatory variables in fuzzy environments. To enhance explanatory power and take into account the uncertainty of the formulated model and parameters, a new operator, called the fuzzy product core (FPC), is proposed for the formulation processes to establish fuzzy regression models with fuzzy parameters using fuzzy observations that include fuzzy response and explanatory variables. In addition, the sign of parameters can be determined in the model-building processes. Compared to existing approaches, the proposed approach reduces the amount of unnecessary or unimportant information arising from fuzzy observations and determines the sign of parameters in the models to increase model performance. This improves the weakness of the relevant approaches in which the parameters in the models are fuzzy and must be predetermined in the formulation processes. The proposed approach outperforms existing models in terms of distance, mean similarity, and credibility measures, even when crisp explanatory variables are used.     

Gradual elements in a fuzzy set

The notion of a fuzzy set stems from considering sets where, in the words of Zadeh, the “transition from non-membership to membership is gradual rather than abrupt”. This paper introduces a new concept in fuzzy set theory, that of a gradual element. It embodies the idea of fuzziness only, thus contributing to the distinction between fuzziness and imprecision. A gradual element is to an element of a set what a fuzzy set is to a set. A gradual element is as precise as an element, but the former is flexible while the latter is fixed. The gradual nature of an element may express the idea that the choice of this element depends on a parameter expressing some relevance or describing some concept. Applications of this notion to fuzzy cardinality, fuzzy interval analysis, fuzzy optimization, and defuzzification principles are outlined.     

A fuzzy logic system based on Schweizer-Sklar t-norm

In recent years, the basic research of fuzzy logic and fuzzy reasoning is growing ac- tively day by day, such as the basic logic system BL proposed by Hajek[1]; fuzzy logic system MTL proposed by Esteva and Godo[2]; fuzzy reasoning, implication operators …     

Evolution of fuzzy logic: from intelligent systems and computation to human mind

Inspired by human’s remarkable capability to perform a wide variety of physical and mental tasks without any measurements and computations and dissatisfied with classical logic as a tool for modeling human reasoning in an imprecise environment, Lotfi A. Zadeh developed the theory and foundation of fuzzy logic with his 1965 paper “Fuzzy sets” (Zadeh in Inf Control 8:378–53, 1965) and extended his work with his 2005 paper “Toward a generalized theory of uncertainty (GTU)—an outline” (Zadeh in Inf Control, 2005). Fuzzy logic has at least two main sources over the past century. The first of these sources was initiated by Peirce in the form what he called a logic of vagueness in 1900s, and the second source is Lotfi’s A. Zadeh work, fuzzy sets and fuzzy Logic in the 1960s and 1970s.     

A fuzzy extension of the XPath query language

Today the current state of the art in querying XML data is represented by XPath and XQuery, both of which rely on Boolean conditions for node selection. Boolean selection is too restrictive when users do not use or even know the data structure precisely, e.g. when queries are written based on a summary rather than on a schema. In this paper we describe a XML querying framework, called FuzzyXPath, based on Fuzzy Set Theory, which relies on fuzzy conditions for the definition of flexible constraints on stored data. A function called “deep-similar” is introduced to replace XPath’s typical “deep-equal” function. The main goal is to provide a degree of similarity between two XML trees, assessing whether they are similar both structure-wise and content-wise. Several query examples are discussed in the field of XML based metadata for e-learning.     

Development of fuzzy process control charts and fuzzy unnatural pattern analyses

Many problems in scientific investigation generate nonprecise data incorporating nonstatistical uncertainty. A nonprecise observation of a quantitative variable can be described by a special type of membership function defined on the set of all real numbers called a fuzzy number or a fuzzy interval. A methodology for constructing control charts is proposed when the quality characteristics are vague, uncertain, incomplete or linguistically defined. Fuzzy set theory is an inevitable tool for fuzzy control charts as well as other applications subjected to uncertainty in any form. The vagueness can be handled by transforming incomplete or nonprecise quantities to their representative scalar values such as fuzzy mode, fuzzy midrange, fuzzy median, or fuzzy average. Then crisp methods may be applied to those representative values for control chart decisions as “in control” or “out of control”. Transforming the vague data by using one of the transformation methods may result in biased decisions since the information given by the vague data is lost by the transformation. Such data needs to be investigated as fuzzy sets without transformation, and the decisions based on the vague data should not be concluded with an exact decision. A “direct fuzzy approach (DFA)” to fuzzy control charts for attributes under vague data is proposed without using any transformation method. Then, the unnatural patterns for the proposed fuzzy control charts are defined using the probabilities of fuzzy events.     

Dynamic fuzzy OWA model for group multiple criteria decision making

Obtaining relative weights in MCDM problems is a very important issue. The Ordered Weighted Averaging (OWA) aggregation operators have been extensively adopted to assign the relative weights of numerous criteria. However, previous aggregation operators (including OWA) are independent of aggregation situations. To solve the problem, this study proposes a new aggregation model – dynamic fuzzy OWA based on situation model, which can modify the associated dynamic weight based on the aggregation situation and can work like a “magnifying lens” to enlarge the most important attribute dependent on minimal information, or can obtain equal attribute weights based on maximal information. Two examples are adopted in this paper for comparison and showing the effects under different weights.     

On the impact of network latency on distributed systems design

Research in distributed database systems to date has assumed a “variable cost” model of network response time. However, network response time has two components: transmission time (variable with message size) and latency (fixed). This research improves on existing models by incorporating a “fixed plus variable cost” model of the network response time. In this research, we: (1) develop a distributed database design approach that incorporates a “fixed plus variable cost”, network response time function; (2) run a set of experiments to create designs using this model, and (3) evaluate the impact the new model had on the design in various types of networks. This revised version was published online in July 2006 with corrections to the Cover Date.     

/spl alpha/-cut fuzzy arithmetic: simplifying rules and a fuzzy function optimization with a decision variable

The problems of /spl alpha/-cut fuzzy arithmetic have been shown, like in interval arithmetic, that distinct states of fuzzy parameters (or fuzzy variable values) may be chosen and produce an overestimated fuzziness. Meanwhile, local extrema of a function may exist inside the support of fuzzy parameters and cause an underestimation of fuzziness and an illegal fuzzy number's result. Previous approaches to overcoming these problems have appeared in literature. Yet, the computational burden of these approaches became even heavier. Thus, this paper is based on the vertex method in literature and extensively proposes newly devised rules observed greatly useful for simplifying the vertex method. These rules are devised through a function partitioned into subfunctions, distinguishing the types of fuzzy parameter/variable occurrences, and types of subfunctions or functions with the various observations. The improved efficiency has been found able to significantly reduce the combination (vertex) test of the vertex method for the fuzzy parameters' /spl alpha/-cut endpoints possibly to only a few fuzzy parameters' endpoint combinations. Also as related, a procedure for the fuzzy optimization of fuzzy functions with a fuzzy blurred argument (a single variable) is examined with the vertex method as well. A proper and useful preliminary algorithm is proposed. Numerical examples with results are provided.     

Probability representations of fuzzy systems

1 Background of birth of fuzzy systems As is well-known, just considering the great uncertainties of many systems, Zadeh put forward the notion of fuzzy sets and advanced the idea of the fuzzy reasoning by means of fuzzy sets which could describe a system approximately[1]. Those systems that are constructed on the basis of fuzzy reasoning are called fuzzy systems in general[2―4]. The research on fuzzy systems has attracted broad attention[5―7]. For instance, universal ap- proximation proper…     

Induction of descriptive fuzzy classifiers with the Logitboost algorithm

Recently, Adaboost has been compared to greedy backfitting of extended additive models in logistic regression problems, or “Logitboost". The Adaboost algorithm has been applied to learn fuzzy rules in classification problems, and other backfitting algorithms to learn fuzzy rules in modeling problems but, up to our knowledge, there are not previous works that extend the Logitboost algorithm to learn fuzzy rules in classification problems.In this work, Logitboost is applied to learn fuzzy rules in classification problems, and its results are compared with that of Adaboost and other fuzzy rule learning algorithms. Contradicting the expected results, it is shown that the basic extension of the backfitting algorithm to learn classification rules may produce worse results than Adaboost does. We suggest that this is caused by the stricter requirements that Logitboost demands to the weak learners, which are not fulfilled by fuzzy rules. Finally, it is proposed a prefitting based modification of the Logitboost algorithm that avoids this problem