I.B. Turksen

Interval valued fuzzy sets based on normal forms

Abstract Interval valued fuzzy sets are proposed for the representation of combined concepts based on normal forms where linguistic connectives as well as variables are assumed to be fuzzy. It is shown that the proposed representation (1) exists for certain families of the conjugate pairs of t-norms and t-conorms; and (2) resolves some of the difficulties associated with particular interpretations of conjunction, disjuntion, and implication in fuzzy set theories.

Single-objective and two-objective genetic algorithms for selecting linguistic rules for pattern classification problems

Abstract This paper proposes various methods for constructing a compact fuzzy classification system consisting of a small number of linguistic classification rules. First we formulate a rule selection problem of linguistic classification rules with two objectives: to maximize the number of correctly classified training patterns and to minimize the number of selected rules. Next we propose three methods for finding a set of non-dominated solutions of the rule selection problem. These three methods are based on a single-objective genetic algorithm. We also propose a method based on a multi-objective genetic algorithm for finding a set of non-dominated solutions. We examine the performance of the proposed methods by applying them to the well-known iris data. Finally we propose a hybrid algorithm by combining a learning method of linguistic classification rules with the multi-objective genetic algorithm. High performance of the hybrid algorithm is demonstrated by computer simulations on the iris data.

Measurement of membership functions and their acquisition

Abstract Three basic views of the representation of membership functions are reviewed, together with fundamental measurement of linguistic terms of linguistic variables. The conclusion is that such measurements are either on an ‘ordinal’ or an ‘interval’ scale based on whether the appropriate axioms are validated by the empirical data, with an allowance for stochastic variation. The conjoint measurement is introduced for the case of multi-dimensional linguistic variables whose linguistic terms are compositions of two or finitely many more component linguistic terms of distinct (independent) linguistic variables. It is shown that any composition of distinct (independent) component linguistic terms of component linguistic variables by any t-norm or s-norm or any finite convext linear combination preserves the monotonic weak order property of the components in the composite. Once the scale properties of the measurement values of particular terms are validated, composition procedures may be applied to the experimental data to obtain compound membership functions of fuzzy sets induced by meaningful representations of compositions of linguistic terms of linguistic variables. Finally, four methods of membership acquisition and construction are reviewed from the perspective of fundamental measurement.

An approximate analogical reasoning approach based on similarity measures

An approximate analogical reasoning schema (AARS) which exhibits the advantages of fuzzy set theory and analogical reasoning in expert systems development is described. The AARS avoids going through the conceptually complicated compositional rule of inference. It uses a similarity measure of fuzzy sets as well as a threshold to determine whether a rule should be fired and a modification function inferred from a similarity measure to deduce a consequent. Some numerical examples to illustrate the operation of the schema are presented. Finally, the proposed schema is compared with conventional expert systems and existing fuzzy expert systems. >

A model for the measurement of membership and the consequences of its empirical implementation

Representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set, when the domain of discourse is order-dense. The conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership. Some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions. They suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as ‘not’, ‘very’, and ‘antonym’ are not sufficiently general to account for the range of individual perceptions of vague attributes. A more general mathematical model is proposed. The question of the meaningfulness of statements about membership functions is also discussed, in the light of the assertion that membership can be fundamentally measured at best on an interval scale. Some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested, involving the use of a function which is derived from membership but is on an absolute scale.

An approximate analogical reasoning schema based on similarity measures and interval-valued fuzzy sets

An Approximate Analogical Reasoning Schema (AARS) is proposed which exhibits the advantages of Fuzzy Sets Theory and Analogical Reasoning in Expert Systems development. The AARS avoids going through the conceptually complicated Compositional Rule of Inference. It uses a Similarity Measure of Fuzzy Sets and a threshold τ0 to determine whether a rule should be fired and a Modification Function inferred from a Similarity Measure to deduce a consequent. Linguistic Variables and Terms, rather than Quantitative Variables are used to represent decision rules and/or decision variables. The schema is extended to Interval-Valued Fuzzy Sets where additional uncertainty can be handled in a formal model. Finally, the proposed AARS is implemented using Texas Instruments PC+ Expert System Shell. The Expert System AARS is applied to an Aggregate Production Planning Prototype study (HMMS Paint Factory Model), where intuitive decision-making seems to be common practice. The systems performance is compared with other approaches and the result is encouraging.

A review and comparison of six reasoning methods

Abstract Five fuzzy reasoning methods are reviewed and their performance is compared with respect to a fuzzy control system model developed by an objective method based on three sets of real-life data. It is found from the investigation that: the reasoning precision, the calculation time and the number of possible input states to which a given reasoning method responds differ according to each reasoning method. Generally, the point-valued reasoning methods which are based on the assumption that connectives are crisp give better precision and shorter calculation time. When the connectives are assumed to be linguistic, reasoning with interval-valued fuzzy sets are more appropriate to represent linguistic uncertainty. For this reason, type II fuzziness generated with interval-valued fuzzy sets are better handled with interval-valued reasoning methods.

Interval-valued fuzzy sets and “compensatory AND”

Abstract It is shown that the ‘Compensatory And ’, based on Zimmermanns experimental results, is included within the boundaries of the ‘Interval-Valued Fuzzy Set And ’ and the ‘Interval-Valued Fuzzy Set Or ’, proposed by Turksen, based on normal forms. The containment relation is true in general for a class of t-norms T and t-conorms, S , and for a range of values of the parameter γ ϵ[0, 1] of the Compensatory And . The relation is seen to clarify the expressive power of Compensatory And . The two well known t-normed classes of operators max-min and algebraic sum-product, are investigated as particular cases of ‘Exponential-Compensatory And ’ and ‘Convex-Linear-Compensatory And ’.

A type-2 fuzzy rule-based expert system model for stock price analysis

In this paper, a type-2 fuzzy rule based expert system is developed for stock price analysis. Interval type-2 fuzzy logic system permits us to model rule uncertainties and every membership value of an element is interval itself. The proposed type-2 fuzzy model applies the technical and fundamental indexes as the input variables. This model is tested on stock price prediction of an automotive manufactory in Asia. Through the intensive experimental tests, the model has successfully forecasted the price variation for stocks from different sectors. The results are very encouraging and can be implemented in a real-time trading system for stock price prediction during the trading period.

A novel feature selection approach: Combining feature wrappers and filters

Abstract Feature selection is one of the most important issues in the research fields such as system modelling, data mining and pattern recognition. In this study, a new feature selection algorithm that combines feature wrapper and feature filter approaches is proposed in order to identify the significant input variables in systems with continuous domains. The proposed method utilizes functional dependency concept, correlation coefficients and K -nearest neighbourhood (KNN) method to implement the feature filter and feature wrappers. Four feature selection methods independently select the significant input variables and the input variable combination, which yields best result with respect to their corresponding evaluation function, is selected as the winner. This is similar to the basic information fusion notion of integrating the information collected from different sources. All of the four feature selection methods are performed in two stages: (i) pre-selection, (ii) selection. Two of the four feature selection methods utilize KNN method for evaluating the candidates. These two methods use sequential forward and sequential backward search mechanism, respectively, in pre-selection stage. Whereas, the third feature selection method uses correlation coefficients in the pre-selection stage. It is common to have outliers and noise in real-life data. In order to make the proposed feature selection algorithm noise and outlier resistant, approximate functional dependencies are used by utilizing membership values that inherently cope with uncertainty in the data. Thus, the fourth feature selection method makes use of approximate functional dependencies to evaluate candidates in pre-selection stage. All of these four methods apply KNN method with exhaustive search strategy in order to find the most suitable input variable combination with respect to a performance measure.