With the continuous development and advancement of neural networks, in the application of neural networks, users not only require neural networks to be able to complete a given task but also want to know when they can trust the network’s prediction results and when they need to be cautious about the prediction results. In response to the need for uncertainty estimation of neural networks, many researchers have invested in the study of uncertainty estimation. Existing uncertainty evaluation methods are difficult to apply to deep neural networks with large parameter scales, complex internal structures, and mappings between inputs and outputs that are hard to express. This paper proposes a key nodes based resampling uncertainty estimation method ((KN-RUE), which achieves uncertainty estimation of prediction results for arbitrarily given large-scale neural networks. In this method, the first step involves analyzing the differences in feature space between adversarial and clean samples, identifying the main nodes affected by adversarial samples, and determining the critical nodes within the network. Next, by resampling the parameters of key nodes, the model is extended while ensuring model performance as much as possible, thus completing the measurement of uncertainty in prediction results. Through experiments, the effectiveness of the extended model and the superiority of uncertainty estimation performance in KN-RUE have been verified.

Federated learning is a nascent distributed machine learning paradigm that enables multiple clients to collaborate in training a model for a specific task under the coordination of a central server, all while safeguarding the privacy of the user’s local data. Nevertheless, the constraint that distributed datasets must remain within local nodes introduces data heterogeneity in federated learning training. In this paper, we focus on how to mitigate the damage caused by the data heterogeneity of feature distribution skew in federated learning models during training. To achieve this goal, we propose a feature drift-corrected federated learning algorithm. We design a feature drift variable derived from the local models of clients and the global model of the server. This variable is incorporated into the client’s local loss function to rectify local model parameters. Additionally, we utilize the disparity between the global models before and after to regulate the local model. Validation experiments are conducted on multiple datasets exhibiting feature distribution skew. The implementation results demonstrate the efficacy of our approach in significantly enhancing the model performance of federated learning under feature distribution skew.

Uncertainty quantification is very important in many applications. As a generalization of Dempster-Shafer theory, the theory of D numbers is a new theoretical framework for uncertainty reasoning. Measuring the uncertainty of knowledge or information represented by D numbers is an unsolved issue in that theory. In this paper, inspired by distance-based uncertainty measures for Dempster-Shafer theory, a total uncertainty measure for a D number is proposed based on its belief intervals. The proposed total uncertainty measure can simultaneously capture the discord, and nonspecificity, and nonexclusiveness involved in D numbers. And some basic properties of this total uncertainty measure, including range, monotonicity, generalized set consistency, are also presented. At last, an illustrative application about feature evaluation is given to verify the effectiveness of the proposed uncertainty measure.

The Dempster-Shafer evidence theory is widely used in many fields of information fusion because of its advantage in handling uncertain information. One of the key issues in this theory is how to make decision based on a basic probability assignment (BPA). Currently, a feasible scheme is transforming a BPA to a distribution of probabilities. However, little attention was paid to the correlation between BPA and probability distribution. In this paper, a novel method about the probability transformation based on a correlation coefficient of belief functions is proposed. The correlation coefficient is a new measurement, which can effectively measure the correlation between BPAs. The proposed method aims at maximizing the correlation coefficient between the given BPA and the transformed probability distribution. On the basis of this idea, the corresponding probability distribution can be obtained and could reflect the original information of the given BPA to the maximum extent. It is valid to consider that the proposed probability transformation method is reasonable and effective. Numerical examples are given to show the effectiveness of the proposed method.

OWA operator is an effective aggregation method in multi-criteria decision making problem. However, in some multi-criteria decision making cases, the criteria satisfactions have some uncertainty, for instance, which is a set of interval values at a series of different levels. For multi-criteria decision making problem, it is necessary to aggregate criteria satisfactions. But the linear ordering of criteria satisfactions is unknown at a specific level in these cases. Therefore, OWA operator cannot be applied to aggregate the satisfactions directly. In this paper, a new method, named as the Interval Value Exceedance Method (IVEM), is proposed. By using the proposed method, the domination relationship of criteria satisfactions for each level can be obtained. Then OWA operator can be used to aggregate these satisfactions based on the domination relationship, even if the linear ordering of satisfaction is unknown.

D numbers has been previously introduced in linguistic decision making due to its effectiveness and flexibility in dealing with uncertain information. The study applies the integration operator of D numbers to obtain the decision result by aggregating different opinions of experts which may be imprecise and uncertain. However, sometimes it would be more reasonable to consider the risk preference of the decision maker. In this paper, we proposed an improved aggregating method for linguistic information based on D numbers and OWAWA operator. The main advantage is that it can integrate the degree of importance that each experts has and the risk preference of the decision maker in the aggregation of linguistic information. An example is used to demonstrate the flexibility and reasonability of the proposed method.

Measuring the uncertainty of pieces of evidence is an open issue in belief function theory. A rational uncertainty measure for belief functions should meet some desirable properties, where monotonicity is a very important property. Recently, measuring the total uncertainty of a belief function based on its associated belief intervals becomes a new research idea and has attracted increasing interest. Several belief interval based uncertainty measures have been proposed for belief functions. In this paper, we summarize the properties of these uncertainty measures and especially investigate whether the monotonicity is satisfied by the measures. This study provide a comprehensive comparison to these belief interval based uncertainty measures and is very useful for choosing the appropriate uncertainty measure in the practical applications.

Axiomatic design (AD) provides a general theory for system and product development. In recent years, the principles of AD have been successfully applied to the decision-making field, and derived a fuzzy AD approach for fuzzy decision-making environment. In this work, the interest is paid on the theoretical developments and applications of AD in the uncertain environment expressed by Dempster–Shafer evidence theory. Based on the concept of belief structure satisfaction to uncertain target values, an evidential AD approach is proposed for decision making by combining the independence axiom and information axiom of AD with the framework of Dempster–Shafer theory. An illustrative example has demonstrated the effectiveness of the proposed approach. This work, on the one hand, has successfully generalized the principles of AD to the Dempster–Shafer uncertain environment; on the other hand, it has presented a successful application of the concept of belief structure satisfaction.

Regret decision theory is a classic theory for decision problem. Recently, Yager proposed a generalized regret based decision-making method, which calculates the effective regret associated with an alternative by aggregating this alternative's all regrets across all the possible states of nature. The generalized regret based decision-making method that can be applied in many fields is understandable and effective. However, as Yager pointed, an issue limits the application of this method, that is, the generalized regret based decision-making method is lack of indifference to irrelevant alternatives. In this paper, we analyze the cause of this issue, that is, unreasonable alternatives may change other alternatives' regrets by changing the maximal payoff under the occurrence of a state of nature. Furthermore, a new method based on original model is proposed to reduce the impact of unreasonable alternatives according to a parameter called impact factor defined to measure an alternative's quality. Finally, several numerical examples are illustrated to show this new method's effectiveness.

Dempster-Shafer theory of evidence is widely applied to uncertainty modelling and knowledge reasoning because of its advantages in dealing with uncertain information. But some conditions or requirements, such as exclusiveness hypothesis and completeness constraint, limit the development and application of that theory to a large extend. To overcome the shortcomings and enhance its capability of representing the uncertainty, a novel model, called D numbers, has been proposed recently. However, many key issues, for example how to implement the combination of D numbers, remain unsolved. In the paper, we have explored the combination of D Numbers from a perspective of conflict redistribution, and propose two combination rules being suitable for different situations for the fusion of two D numbers. The proposed combination rules can reduce to the classical Dempster's rule in Dempster-Shafer theory under certain conditions. Numerical examples and discussion about the proposed rules are also given in the paper.

Dempster-Shafer (D-S) evidence theory is widely used for information fusion field. However, one of the main issues of D-S evidence theory is that, when large amount of focal elements in Basic Probability Assignment (BPA) are available, the fusion of BPA requires high computational cost and long computing time. This problem greatly limits its application. In this paper, a novel method for approximating a BPA based on correlation coefficient is present, which can reduce the computational cost of evidence combination effectively. At last, several numerical examples are given to illustrate the superiority of the proposed method.