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How To Combine Probabilistic And Fuzzy Uncertainty: Theoretical Explanation Of Clustering-Related Empirical Result, Lázló Szilágyi, Olga Kosheleva, Vladik Kreinovich
How To Combine Probabilistic And Fuzzy Uncertainty: Theoretical Explanation Of Clustering-Related Empirical Result, Lázló Szilágyi, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In contrast to crisp clustering techniques that assign each object to a class, fuzzy clustering algorithms assign, to each object and to each class, a degree to which this object belongs to this class. In the most widely used fuzzy clustering algorithm -- fuzzy c-means -- for each object, degrees corresponding to different classes add up to 1. From this viewpoint, these degrees act as probabilities. There exist alternative fuzzy-based clustering techniques in which, in line with the general idea of the fuzzy set, the largest of the degrees is equal to 1. In some practical situations, the probability-type fuzzy …
Why Cauchy Membership Functions: Efficiency, Javier Viana, Stephan Ralescu, Kelly Cohen, Anca Ralescu, Vladik Kreinovich
Why Cauchy Membership Functions: Efficiency, Javier Viana, Stephan Ralescu, Kelly Cohen, Anca Ralescu, Vladik Kreinovich
Departmental Technical Reports (CS)
Fuzzy techniques depend heavily on eliciting meaningful membership functions for the fuzzy sets used. Often such functions are obtained from data. Just as often they are obtained from experts knowledgable of the domain and the problem being addressed. However, there are cases when neither is possible, for example because of insufficient data, or unavailable experts. What functions should one choose and what should guide such choice? This paper argues in favor of using Cauchy membership functions, thus named because their expression is similar to that of the Cauchy distributions. The paper provides a theoretical explanation for this choice.
For Quantum And Reversible Computing, Intervals Are More Appropriate Than General Sets, And Fuzzy Numbers Than General Fuzzy Sets, Oscar Galindo, Vladik Kreinovich
For Quantum And Reversible Computing, Intervals Are More Appropriate Than General Sets, And Fuzzy Numbers Than General Fuzzy Sets, Oscar Galindo, Vladik Kreinovich
Departmental Technical Reports (CS)
Need for faster and faster computing necessitates going down to quantum level -- which means involving quantum computing. One of the important features of quantum computing is that it is reversible. Reversibility is also important as a way to decrease processor heating and thus, enable us to place more computing units in the same volume. In this paper, we argue that from this viewpoint, interval uncertainty is more appropriate than the more general set uncertainty -- and, similarly, that fuzzy numbers (for which all alpha-cuts are intervals) are more appropriate than more general fuzzy sets. We also explain why intervals …
How To Fully Represent Expert Information About Imprecise Properties In A Computer System -- Random Sets, Fuzzy Sets, And Beyond: An Overview, Hung T. Nguyen, Vladik Kreinovich
How To Fully Represent Expert Information About Imprecise Properties In A Computer System -- Random Sets, Fuzzy Sets, And Beyond: An Overview, Hung T. Nguyen, Vladik Kreinovich
Departmental Technical Reports (CS)
To help computers make better decisions, it is desirable to describe all our knowledge in computer-understandable terms. This is easy for knowledge described in terms on numerical values: we simply store the corresponding numbers in the computer. This is also easy for knowledge about precise (well-defined) properties which are either true or false for each object: we simply store the corresponding "true" and "false" values in the computer. The challenge is how to store information about imprecise properties. In this paper, we overview different ways to fully store the expert information about imprecise properties. We show that in the simplest …
Ellipsoids And Ellipsoid-Shaped Fuzzy Sets As Natural Multi-Variate Generalization Of Intervals And Fuzzy Numbers: How To Elicit Them From Users, And How To Use Them In Data Processing, Vladik Kreinovich, Jan Beck, Hung T. Nguyen
Ellipsoids And Ellipsoid-Shaped Fuzzy Sets As Natural Multi-Variate Generalization Of Intervals And Fuzzy Numbers: How To Elicit Them From Users, And How To Use Them In Data Processing, Vladik Kreinovich, Jan Beck, Hung T. Nguyen
Departmental Technical Reports (CS)
In this paper, we show that ellipsoids are natural multi-variate generalization of intervals and ellipsoid-shaped fuzzy sets are a natural generalization of fuzzy numbers. We explain how to elicit them from users, and how to use them in data processing.
On Approximation Of Fuzzy Sets By Crisp Sets: From Continuous Control-Oriented Defuzzification To Discrete Decision Making, Hung T. Nguyen, Witold Pedrycz, Vladik Kreinovich
On Approximation Of Fuzzy Sets By Crisp Sets: From Continuous Control-Oriented Defuzzification To Discrete Decision Making, Hung T. Nguyen, Witold Pedrycz, Vladik Kreinovich
Departmental Technical Reports (CS)
In this paper, we show that the necessity to make crisp decisions in uncertain (fuzzy) situations leads to the necessity to "approximate" fuzzy sets by crisp sets. We show that seemingly natural approximation ideas - such as using alpha-cut for a given alpha - often do not work, and we describe new approximations which not only work, but which are optimal in some reasonable sense
Multi-Criteria Optimization - An Important Foundation Of Fuzzy System Design, Hung T. Nguyen, Vladik Kreinovich
Multi-Criteria Optimization - An Important Foundation Of Fuzzy System Design, Hung T. Nguyen, Vladik Kreinovich
Departmental Technical Reports (CS)
In many real-life design situations, there are several different criteria that we want to optimize, and these criteria are often in conflict with each other. Traditionally, such multi-criteria optimization situations are handled in an ad hoc manner, when different conflicting criteria are artificially combined into a single combination objective that is then optimized. The use of unnatural ad hoc tools is clearly not the best way of describing a very natural aspect of human reasoning. Fuzzy logic describes a much more natural way of handling multi-criterion optimization problems: when we cannot maximize each of the original conflicting criteria 100%, we …