Due to their unique structure and powerful capability to handle uncertainty and partial reliability of information, Z-numbers have achieved significant success in various fields. Zadeh previously asserted that a Z-number can be regarded as a summary of probability distributions. Researchers have proposed various methods for determining the underlying probability distributions from a given Z-number. Conversely, can a Z-number be used to summarize a set of probability distributions? This problem remains unexplored. In this article, we propose a nonlinear model, termed Maximum Expected Minimum Entropy (MEME), for generating a Z-number from a set of probability distributions. Through this model, Z-numbers can be generated directly from data without requiring expert knowledge. Additionally, we applied the MEME model to classification problems, introducing a novel if-then rule form, termed Z-valuation if-then rules. These rules replace the deterministic consequent part of a fuzzy rule with an uncertain Z-valuation, thereby further summarizing the uncertain information in the rule's consequent. Based on the Z-valuation rules, we propose a Z-valuation rule-based (ZVRB) classification system, which aims to enhance decision-making processes in scenarios where uncertainty plays a key role. To validate the effectiveness of the ZVRB classification system, we conducted two experiments comparing it with both classic and advanced nonfuzzy classifiers as well as fuzzy classification systems. The results show that the ZVRB model is superior to the other comparative classifiers in terms of classification performance.

As a novel architecture of a fuzzy decision tree constructed on fuzzy rules, the fuzzy rule-based decision tree (FRDT) achieved better performance in terms of both classification accuracy and the size of the resulted decision tree than other classical decision trees such as C4.5, LADtree, BFtree, SimpleCart and NBTree. The concept of Z-number extends the classical fuzzy number to model both uncertain and partial reliable information. Z-numbers have significant potential in rule-based systems due to their strong representation capability. This paper designs a Z-number-valued rule-based decision tree (ZRDT) and provides the learning algorithm. Firstly, the information gain is used to replace the fuzzy confidence in FRDT to select features in each rule. Additionally, we use the negative samples to generate the second fuzzy numbers that adjust the first fuzzy numbers and improve the model's fit to the training data. The proposed ZRDT is compared with the FRDT with three different parameter values and two classical decision trees, PUBLIC and C4.5, and a decision tree ensemble method, AdaBoost.NC, in terms of classification effect and size of decision trees. Based on statistical tests, the proposed ZRDT has the highest classification performance with the smallest size for the produced decision tree.

Information that people rely on is often uncertain and partially reliable. Zadeh introduced the concept of Z-numbers as a more adequate formal construct for describing uncertain and partially reliable information. Most existing applications of Z-numbers involve discrete ones due to the high complexity of calculating continuous ones. However, the continuous form is the most common form of information in the real world. Simplifying continuous Z-number calculations is significant for practical applications. There are two reasons for the complexity of continuous Z-number calculations: the use of normal distributions and the inconsistency between the meaning and definition of Z-numbers. In this paper, we extend the triangular distribution as the hidden probability density function of triangular Z-numbers. We add a new parameter to the triangular distribution to influence its convexity and concavity, and then expand the value's domain of the probability measure. Finally, we implement the basic operations of triangular Z-numbers based on the extended triangular distribution. The suggested method is illustrated with numerical examples, and we compare its computational complexity and the entropy (uncertainty) of the resulting Z-number to the traditional method. The comparison shows that our method has lower computational complexity, higher precision and lower uncertainty in the results.

How to efficiently handle uncertain information is still an open issue. In this paper, a new method to deal with uncertain information, named as two-dimensional belief function (TDBF), is presented. A TDBF has two components, T = (), both and are classical belief functions, while is a measure of reliable of . The definition of TDBF and the discounting algorithm are proposed. Compared with the classical discounting model, the proposed TDBF is more flexible and reasonable. Numerical examples are used to show the efficiency and application of the proposed method.