Journal of Intelligent & Fuzzy Systems 33 (2017) 105–112 DOI:10.3233/JIFS-161161 IOS Press 105 Partial entropy of uncertain random variables Hamed Ahmadzade a , Rong Gao b,∗ , Mohammad Hossein Dehghan a and Yuhong Sheng c a Department of Mathematical Sciences, University of Sistan and Baluchestan, Zahedan, Iran b Department of Mathematical Sciences, Tsinghua University, Beijing, China c College of Mathematical and System Sciences, Xinjiang University, Urumqi, China Abstract. In this paper, we mainly propose a definition of partial entropy for uncertain random variables. In fact, partial entropy is a tool to characterize how much of entropy of an uncertain random variable belongs to the uncertain variable. Furthermore, some properties of partial quadratic entropy are investigated such as positive linearity. Finally, some other types of partial entropies are studied. Keywords: Chance theory, uncertain random variable, entropy, partial entropy 1. Introduction In real life, we are always around indeterminacy which means that the outcomes of events cannot be exactly predicted in advance, such as rolling a die, stock price, coal reserve, and strength of bridge. For describing indeterminate phenomena, we introduce probability theory as a commonly used tool. How- ever, probability theory is valid under the assumption that the estimated probability distribution is close enough to the real frequency. For obtaining esti- mated probability distribution by statistic method, we should possess lots of observed data. However, it is not easy for us to obtain the observed data due to eco- nomic, technical or some other reasons. In this case, we have to invite some domain experts to estimate their belief degrees that possible events may occur. While Nobelist Kahneman and Tversky [15] asserted that human beings tend to overweight the unlikely events. That is, belief degrees has a larger range of ∗ Corresponding author. Rong Gao, Department of Mathemat- ical Science, Tsinghua University, Beijing 100190, China. Tel.: +86 13041229049; E-mail: gaor14@mails.tsinghua.edu.cn. values than true frequencies. In this circumstance, probability theory is not enough to model human beings’ belief degrees. Then Zadeh [36] founded fuzzy set theory to model fuzziness, however a series of paradoxes presented by Liu [21], which implies that fuzzy set theory is not suitable to model this type of uncertain phenomena. In order to model human uncertainty, another type of indeterminacy, uncertainty theory was introduced by Liu [18] and refined by Liu [19], which is a branch of mathematics on the basis of the normal- ity, duality, subadditivity, and product axioms. Then some concepts were proposed in [18], such as uncer- tain measure for modeling belief degrees, uncertain variable for describing uncertain quantities, uncer- tainty distribution for describing uncertain variables and expected value for ranking uncertain variables. Nowadays uncertainty theory is almost complete in theoretical aspect. In fact, it is also well developed in many fields and many scholars have done lost of work such as [5, 8, 10, 11, 20, 31, 35]. Entropy is a commonly used tool to measure the degree of uncertainty in information sciences. It was first proposed by Shannon [29] for random variables 1064-1246/17/$35.00 © 2017 – IOS Press and the authors. All rights reserved