Support vector classification (SVC) and support vectors regression (SVR) are learning machines that have excellent generalization performance. The data which used by classical statistical learning theory is assumed precise. However, the data from real world sometimes low-quality or imprecise, the uncertainty theory and uncertain statistics are appropriate methods to process the imprecise observations. In this paper, the optimal hyperplane under the framework of uncertainty theory be put forward as the basis of SVR. Based on the definition of optimal hyperplane, the theorem of SVR with imprecise observation be proposed, this theorem obtains the basic ideology and dual problem, can solves the support vector problem under the uncertainty theory. Moreover, the cross-validation (CV) method and the average test error (ATE) be employed to evaluate the generalization performance of regression models. After evaluation of models, the forecast value can be computed and the root mean squared error (RMSE) be used to measure the effect of prediction. Finally, a numerical example be given to show the excellent performance of SVR, SVR has the minimum ATE and RMSE among some models, then the forecast value of the test set be calculated.

As an alternative model to the data envelopment analysis (DEA), the free disposal hull (FDH) model has excellent performance in measuring decision making unit (DMU) efficiency under the condition that the production possibilities do not satisfy the convexity assumption. However, in actual production and life, many data are difficult to collect and cannot obtain accurate values, such as carbon dioxide emissions. In the case of imprecise data, the FDH model cannot evaluate the efficiency of DMU. In this context, a new uncertain FDH model based on uncertainty theory is proposed. In this paper, the uncertain FDH model is solved precisely by means of the uncertain chance constraint method and the expected value method. Finally, an application example of measuring the efficiency of 30 banks in China in 2019 is given to documenting the feasibility of the proposed model.

Parameter estimation is widely used as an essential branch of uncertain time series, among which the least squares (LSE) estimation is the most representative. Since LSE estimation is ineffective in the presence of outliers, Hampel estimation that its stability solves this problem well. Therefore, in this paper, we use Hampel estimation to calculate the parameters of the uncertain autoregressive (UAR) model. The sum of sample errors (SSE) function is used to determine the parameters in Hampel estimation before fitting the UAR model. In addition, the residuals are analysed, and future trends in the data are predicted. Finally, numerical examples comparing the Hampel estimation with the LSE estimation, the least absolute deviation (LAD) estimation, and the Huber estimation illustrate the validity and stability of the Hampel estimation, as well as its applicability in predicting carbon dioxide emissions in China.

Support vector regression (SVR) has been widely used in academia and industry with excellent performance. Crisp data always be trained by classic SVR and its varieties. However, classic SVR is feeble if data are imprecise or low-quality. Hence, the uncertainty theory emerged as the times require, which can process the imprecise observations well. In this study, a novel SVR model be introduced into uncertainty theory, termed v-SVR with imprecise observations, designed to handle imprecise or low-quality data. Unlike the conventional [Formula: see text]-SVR with imprecise observations approach, v-SVR offers an automated computation of the accuracy parameter [Formula: see text], thereby eliminating the need for manual selection. This results in improved performance with simplified parameter tuning. The effectiveness of the approach in this paper be demonstrated through a numerical example.

The uncertain vector autoregressive model is able to model the interrelationships between different variables, which is more advantageous compared to the traditional autoregressive model, when modeling real-life objects and where the observed values are imprecise. In this paper, the parameters of the uncertain vector autoregressive model are estimated by using least absolute deviation estimation (LAD) to obtain a fitted uncertain vector autoregressive model, and residual analysis is performed to obtain estimates of expected values and variances of the residuals. In addition, future values are modeled by using forecasting methods, i.e., point estimation and interval estimation. The order of the uncertain vector autoregressive model is also determined by the indicator summation of test errors (STE) in the cross-validation, and we also analyze that the least absolute deviation estimation outperforms the least squares estimation method in the presence of outliers.

Uncertain differential equations are a type of differential equations driven by Liu processes. How to estimate the parameters in an uncertain differential equation based on the observed data is a crucial problem in the real applications of these equations. By means of the least squares estimation, this article proposes a principle of minimum noise as an approach to the problem. Following this principle, the estimates of the parameters in some special types of uncertain differential equations are derived, which are represented as functions of the observed data. In addition, some numerical experiments are performed to illustrate the principle.

Entropy is a measure of the uncertainty associated with a variable whose value cannot be exactly predicted. Based on the notion of chance measure, a concept of uncertain random entropy is introduced and used to provide a quantitative measurement of the uncertainty associated with uncertain random variables and its properties are studied in this paper. Relative entropy is a measure of the difference between two distribution functions. In order to deal with the divergence of uncertain random variables via chance distributions, this paper proposes also the relative entropy for uncertain random variables, as well as it investigates some mathematical properties of this concept. As an application, a model is presented to formulate a minimum spanning tree problem with uncertain random edge weights involving a relative entropy chance distribution. Finally, a numerical example of an uncertain random network is put forward to illustrate the effectiveness of the proposed model.