Artificial neural networks are efficient computing models which have shown their strengths in solving hard problems in artificial intelligence. They have also been shown to be universal approximators. Notwithstanding, one of the major criticisms is their being black boxes, since no satisfactory explanation of their behavior has been offered. In this paper, we provide such an interpretation of neural networks so that they will no longer be seen as black boxes. This is stated after establishing the equality between a certain class of neural nets and fuzzy rule-based systems. This interpretation is built with fuzzy rules using a new fuzzy logic operator which is defined after introducing the concept of f-duality. In addition, this interpretation offers an automated knowledge acquisition procedure.

Abstract-In this paper, we consider a fundamental theoretical question, Why does fuzzy control have such good performance for a wide variety of practical problems?. We try to answer this fundamental question by proving that for each fixed fuzzy logic belonging to a wide class of fuzzy logics, and for each fixed type of membership function belonging to a wide class of membership functions, the fuzzy logic control systems using these two and any method of d e f d c a t i o n are capable of approximating any real continuous function on a compact set to arbitrary accuracy. On the other hand, this result can be viewed as an existence theorem of an optimal fuzzy logic control system for a wide variety of problems. It points out that fuzzy control has been effectively used in the context of complex ill-defined processes, specially those which can be controlled by a skilled human operator without the knowledge of their underlying dynamics. In this sense, neural and adaptive fuzzy systems has been compared to classical control methods by B. Kosko in [8]. There, it is remarked that they are model-free estimators, i.e., they estimate a function without requiring a mathematical description of how the output functionally depends on the input; they learn from samples. I. I NTRODUCTI ONHowever, some people criticize fuzzy control because its effectiveness has not been proved. That is, the very fundamental theoretical question "Why does a fuzzy rule-based system have such good performance for a wide variety of practical problems?' remains unanswered. There exist some qualitative explanations, e.g., "fuzzy rules utilize linguistic information", "fuzzy inference simulates human thinking procedure", "fuzzy rule systems capture the approximate and inexact nature of the real world," etc., but mathematical proofs have not been obtained.A first approach to answer this fundamental question in a quantitative way was presented by Wang [ 181. He proved that a particular class of FLC systems are universal approximators,

Abstract-In this paper, we consider a fundamental theoretical question: Is it always possible to design a fuzzy system able of approximating any real continuous function on a compact set with arbitrary accuracy? Moreover, we will research whether the answer to the above question is positive when we restrict to a fixed (but arbitrary) type of fuzzy reasoning and to a subclass of fuzzy relations. This result can be viewed as an existence theorem of an optimal fuzzy system for a wide variety of problems.

This paper presents an extension of the method presented by Benitez et al (1997) for extracting fuzzy rules from an artificial neural network (ANN) that express exactly its behavior. The extraction process provides an interpretation of the ANN in terms of fuzzy rules. The fuzzy rules presented are in accordance with the domain of the input variables. These rules use a new operator in the antecedent. The properties and intuitive meaning of this operator are studied. Next, the role of the biases in the fuzzy rule-based systems is analyzed. Several examples are presented to comment on the obtained fuzzy rule-based systems. Finally, the interpretation of ANNs with two or more hidden layers is also studied.

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Abstract.The use of feature selection can improve accuracy, efficiency, applicability and understandability of a learning process. For this reason, many methods of automatic feature selection have been developed. Some of these methods are based on the search of the features that allows the data set to be considered consistent. In a search problem we usually evaluate the search states, in the case of feature selection we measure the possible feature sets. This paper reviews the state of the art of consistency based feature selection methods, identifying the measures used for feature sets. An in-deep study of these measures is conducted, including the definition of a new measure necessary for completeness. After that we perform an empirical evaluation of the measures comparing them with the highly reputed wrapper approach. Consistency measures achieve similar results to those of the wrapper approach with much better efficiency.

A cohort of 300 women with breast cancer who were submitted for surgery is analysed by using a non-homogeneous Markov process. Three states are considered: no relapse, relapse and death. As relapse times change over time, we have extended previous approaches for a time homogeneous model to a non-homogeneous multistate process. The trends of the hazard rate functions of transitions between states increase and then decrease, showing that a changepoint can be considered. Piecewise Weibull distributions are introduced as transition intensity functions. Covariates corresponding to treatments are incorporated in the model multiplicatively via these functions. The likelihood function is built for a general model with k changepoints and applied to the data set, the parameters are estimated and life-table and transition probabilities for treatments in different periods of time are given. The survival probability functions for different treatments are plotted and compared with the corresponding function for the homogeneous model. The survival functions for the various cohorts submitted for treatment are ®tted to the empirical survival functions.