Design and Development of an Algorithm for Fuzzy Entropy

Abstract

A measure is developed for measuring the amount of information given when the characterizing function of a fuzzy set is only partly specified. Its modification is considered when an aprior characterizing function for the set is also given. For a fuzzy set, we may not given the values of all of mA(X1), mA(X2)……. mA(Xn), but we may give some partial information about these in the form of equality or inequality relation between the values of these. We have given a method for measuring the information provided by each of these pieces of knowledge. This knowledge will change if some prior information based on intuition or experience is available about the possible values of these membership functions. We have considered here how this information is modified in this case. Finally we have taken a general situation when we have measured some partial knowledge given about n positive real numbers and we have evaluated the information contained in this partial knowledge. This thesis deals with probabilistic measures of information. A large number of measures of probabilistic information have been developed during the last five decades. Probabilistic measures of fuzzy information include fuzzy entropy, fuzzy directed divergence, fuzzy distance, fuzzy total ambiguity etc. Fuzzy uncertainty is different from probabilistic uncertainty. Fuzzy entropy measures uncertainty due to fuzziness of information, while probabilistic entropy measures uncertainty due to the information being available in terms of a probability distribution only. A close link has been established between measure of information for probabilities and fuzzy set cases. This a step in the direction of integrating these two approaches to understand uncertainty. In this thesis incomplete quantitative data has been dealt by using the concept of fuzzy entropy. Genetic programming has been used to classify the incomplete data. Certain attributes related to the data have been considered. Test data used in this knowledge discovery algorithm knows the entire attribute clearly. The developed algorithm is very effective and can be used in the various application related to knowledge discovery and machine learning. The developed knowledge discovery algorithm using fuzzy entropy has been tested for verity of incomplete data sets pertain to various application and it is found that the error level is merely ± 4.40%, which is far better than other available knowledge discovery algorithms.