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Journal of Computational Science

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M Camarinha-Mtos P. Barcelona, Spain. Institute of Electrical and Electronics Engineers Inc.

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Proc Int Astron Union In: F. Information Science Reference. Porto, Portugal. Plymouth, England. In: 2nd International Conference on Electrical Engineering. Coimbra, Portugal.

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Madeira, Portugal. Vilamoura, Portugal, p In: EVER Vilamoura, Portugal. In: Proceedings of the 15th Portuguese National Meeting. Coimbra, Portugal, p London, United Kingdom. Baltimor, USA.

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The Eemhof, The Netherlands, p Chania, Greece, p In: World Automation Congress. Budapest, Hungary. Paphos, Cyprus. Tomar, Portugal. In: Proceedings of the International conference 9th Fuzzy Days. Dortmund, Germany. Universidade Nova Lisboa.

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Paris, France. Acta Press Inc, Innsbruck, Austria. Costa da Caparica, Portugal. Danesy, S. Poedts, A. It may also be the case that y and z are somewhat similar to each other e. But, in this example, x and z are not similar e. However, the properties of transitive fuzzy relations are often desirable from a mathematical viewpoint and are used here.

The family of normal fuzzy sets produced by a fuzzy partitioning of the universe of discourse can play the role of fuzzy equivalence classes [86]. Extending this to the case of fuzzy equivalence classes is straightforward: objects can be allowed to assume membership values, with respect to any given class, in the interval [0, 1]. A rough-fuzzy set is a generalization of a rough set, derived from the approximation of a fuzzy set in a crisp approximation space.

This corresponds to the case where only the decision attribute values are fuzzy; the conditional values are crisp. Rough-fuzzy sets can be generalized to fuzzy-rough sets [86], where all equivalence classes may be fuzzy. When applied to dataset analysis, this means that both the decision values and the conditional values may be fuzzy or crisp.

Computational Intelligence in Bioinformatics

In [25] the concepts of information theoretic measures are related to rough sets, comparing these to established rough set models of uncertainty. Rough sets may be expressed by a fuzzy membership function to represent the negative, boundary, and positive regions []. All objects in the positive region have a membership of one, and those belonging to the boundary region have a membership of 0. The reason for integrating fuzziness into rough sets is to quantify the levels of roughness in the boundary region by using fuzzy membership values. It is necessary therefore to allow elements in the boundary region to have membership values in the range of 0 to 1, not just the value 0.

However, there is still a need for a method that uses object membership values when dealing with equivalence classes. For a more comprehensive discussion on the hybridization of rough and fuzzy sets, including applications, see Chapter 7. Traditional set-theoretic concepts and operations were initially presented. From these definitions fuzzy sets, rough sets, and fuzzy-rough sets are all constructed. Each of these extensions of set theory attempts to address different uncertainties encountered in real-world data. Learning comes in several general forms: supervised learning, unsupervised learning, and reinforcement learning.

These components are the attributes or features of the database.

This approach is essentially the same as that of the decision systems described in Section 2. Models learned from training data are then evaluated with a different test set in order to determine if they can be generalized to new cases. Supervised learning techniques are the main focus of this chapter. Unsupervised learning concerns problems where there are no explicit target outputs. In addition there may be some explicit or implicit a priori information as to the importance of aspects of the data.


Reinforcement learning is the mechanism used by agents that must learn behavior through trial-and-error interactions with a dynamic environment. Two main strategies exist. The second is to employ dynamic programming and statistical methods to estimate the utility of taking actions in states of the world. One of the main attractions of rule induction is that the rules are much more transparent and easier to interpret than, say, a regression model or a trained neural network. Rules are arguably the most comprehensive concept representation formalism.

Restrictions imposed by the language used to describe the data and the language used to describe the induced ruleset must be taken into account. The data description language imposes a bias on the form of data, and the hypothesis restriction language imposes a bias on the form of the induced rules. First, a complete matching is considered where all attribute-value pairs of a rule must match all values of the corresponding attributes for the example. If this is not possible, a partial matching is done, where some attribute-value pairs of a rule match the values of corresponding attributes.

The best rule is chosen based on probability estimates. Each rule is weighted by the percentage of positive examples in the set of examples covered by it. The weights of rules of the same class are combined to a weight for the entire class and the class with the highest weight will be returned. The algorithm consists of two main procedures: the search procedure and the control procedure. The likelihood ratio statistic is typically used for this purpose.

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This ratio measures the difference between the distribution of class probabilities in the set of objects covered by the rule and the class probability distribution in the set of all training objects. For ordered ruleset induction, the search mechanism looks for the best rule determined by the heuristic and removes all training objects covered by it. A new search is initiated by the search procedure, repeating this process until all objects are covered. For unordered rulesets, rules are induced for each class.

Only objects covered by the rule and belonging to the class are removed, leaving the negative examples in the training data. This prevents CN2 from inducing exactly the same rule. Rules in each ruleset predict the same class. As a result the sets of rulesets produced are not order dependent.

Rules are created while there are objects remaining in the dataset with this class. This rule generation process begins by creating a rule with an empty antecedent part that predicts the class C. This is repeated for all classes.