Novel Correlation and Entropy Measures of Hesitant Fuzzy Sets
Correlation is one of the most widely used indices in data analysis, pattern recognition, machine learning, decision making, etc. It measures how well two variables move together in a linear fashion.
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Novel Correlation and Entropy Measures of Hesitant Fuzzy Sets
Correlation is one of the most widely used indices in data analysis, pattern recognition, machine learning, decision making, etc. It measures how well two variables move together in a linear fashion. The correlation coefficient, which was originally appeared in Karl Pearson’s proposal related to statistics, has been extended into different fuzzy circumstances. Different forms of fuzzy correlation coefficients have been proposed, such as the fuzzy correlation coefficients, the intuitionistic fuzzy correlation coefficients, and the hesitant fuzzy correlation coefficients. Xu and Xia (2011b) defined several correlation coefficients for HFEs. Afterwards, Chen et al. (2013a) proposed a formula to calculate the correlation coefficient between two HFSs. In this chapter, we first point out the weaknesses of the existing correlation coefficients between HFSs, and then introduce some novel correlation coefficient formulas for HFSs. Some new concepts, such as the mean of a HFS, the variance of a HFS and the correlation between two HFSs are defined. Based on these concepts, a novel correlation coefficient formula between two HFSs is introduced. Afterwards, the upper and lower bounds of the correlation coefficient are defined. A theorem is given to determine these two bounds. It is stated that the correlation coefficient between two HFSs should also be hesitant, and thus, the upper and lower bounds can further help to identify the correlation coefficient between HFSs. The significant characteristic of the introduced correlation coefficient is that it lies in the interval [−1, 1], which is in accordance with the classical correlation coefficient in statistics, whereas all the old correlation coefficients between HFSs in the literature are within the unit interval [0, 1]. The weighted correlation coefficient is also proposed to make it more applicable. In order to show the efficiency of the proposed correlation coefficients, they are implemented in medical diagnosis and cluster analysis. Some numerical examples are given in this chapter to illustrate the applicability and efficiency of the proposed correlation coefficient between HFSs. Entropy is another important index for fuzzy information, which measures the degree of uncertainty of a fuzzy set. Usually, there are two aspects of uncertainty associated with a fuzzy set. One is related to fuzziness, which results from the lack © Springer Nature Singapore Pte Ltd. 2017 H. Liao and Z. Xu, Hesitant Fuzzy Decision Making Methodologies and Applications, Uncertainty and Operations Research, DOI 10.1007/978-981-10-3265-3_2
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2 Novel Correlation and Entropy Measures of Hesitant Fuzzy Sets
of clear discrimination between the elements belonging or not belonging to a set. For classical fuzzy set, Zadeh (1965) first defined the entropy to measure the fuzziness of a fuzzy set and then many scholars developed different kinds of entropy formulas for fuzzy set (De Luca and Termini 1972; Kaufmann and Swanson 1975; Yager 1979; Parkash et al. 2008) and I
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