Multivariate uncertain calculus is a branch of mathematics that deals with differentiation and integration of uncertain fields based on uncertainty theory. This paper revises the definition of first partial derivatives of uncertain field and defines higher-order partial derivatives for the first time. Then some sufficient and necessary conditions for uncertain fields possessing partial derivatives are derived. Finally, some examples are documented to verify the solution of higher-order uncertain partial differential equation.

In uncertainty theory, there is a fundamental theorem of uncertain processes that Xt and X0 + t(X1-X0) share the same distribution if Xt is a stationary independent increment process. However, this conclusion fails in the presence of Hamel functions. To address this issue, a sample-continuity condition is imposed. With this additional condition, the fundamental theorem is reproved, and several significant consequences regarding the inverse uncertainty distribution, expected value, and variance are subsequently derived.

Leibniz rule is a method of differentiation with respect to a parameter under the integral sign in classical calculus. As the counterpart of Leibniz rule, the rule of parametric differentiation under the integral sign in uncertain calculus is deduced in this paper, which plays an important role in the research field of uncertain differential equations. Meanwhile, this paper also proves the existence of Liu integral and proposes the method of change of variables.

In order to estimate the unknown parameters in an uncertainty distribution function, this article uses the principle of least squares that minimizes the sum of the squared deviations between the uncertainty distribution and the empirical distribution of the observed data. After that, the principle of least squares is applied to determining the uncertain disturbance term of uncertain regression model and uncertain time series model, and estimating the unknown parameters in uncertain differential equation. Finally, in order to illustrate the proposed method, some real-world examples are provided, including PetroChina stock price, electricity price, grain yield, China's population, and beef price.

In uncertain statistics, the uncertain maximum likelihood estimation is a method of estimating the values of unknown parameters of an uncertain statistical model that make the observed data most likely. However, the observed data obtained in practice usually contain outliers. In order to eliminate the influence of outliers when estimating unknown parameters, this article modifies the uncertain maximum likelihood estimation. Following that, the modified uncertain maximum likelihood estimation is applied to uncertain regression analysis, uncertain time series analysis, and uncertain differential equation. Finally, some real-world examples are provided to illustrate the modified uncertain maximum likelihood estimation.

This paper initializes higher-order uncertain calculus that deals with higher-order differentiation and multiple integration of uncertain process based on uncertainty theory. Fubini theorem and fundamental theorem of higher-order uncertain calculus are derived. Finally, this paper rigorously defines higher-order uncertain differential equations and introduces some analytic methods for solving these equations.

This paper presents a statistical tool of uncertain significance test that uses uncertainty theory to test whether certain prespecified regression coefficients can be regarded as zero. A numerical example is given to illustrate how to test the significance of regression coefficients in an uncertain regression model. In order to compare uncertain significance test with stochastic significance test, both of these significance testing approaches are applied in studying the relationship between GDP and four indicators, including urban population scale, volume of foreign trade, fiscal expenditure, and water resource. The results show that uncertain significance test is more appropriate than stochastic significance test.

Uncertain hypothesis test is a statistical tool that uses uncertainty theory to determine whether some hypotheses are correct or not based on observed data. As an application of uncertain hypothesis test, this paper proposes a method to test whether an uncertain differential equation fits the observed data or not. In order to demonstrate the test method, some numerical examples are provided. Finally, both uncertain currency model and stochastic currency model are used to model US Dollar to Chinese Yuan (USD–CNY) exchange rates. As a result, it is shown that the uncertain currency model fits the exchange rates well, but the stochastic currency model does not.

All existing methods to estimate unknown parameters in uncertain differential equations are based on difference scheme, and do not work when the time intervals between observations are not short enough. In order to overcome this shortage, this paper presents a concept of residual. Afterwards, an algorithm is designed for calculating residuals of uncertain differential equation corresponding to observed data. In addition, this paper presents a method of moments based on residuals to estimate the unknown parameters in uncertain differential equations. Finally, some examples (including Alibaba stock price) are provided to illustrate the parameter estimation method.

Parameter estimation has become a crucial issue in the development of uncertain differential equation. This paper presents a new parameter estimation method in uncertain differential equation based on uncertain maximum likelihood estimation, and gives some analytical formulae of the uncertain maximum likelihood estimators in special linear uncertain differential equations. In addition, some numerical examples are provided to illustrate this parameter estimation method.

Uncertain queueing system is a waiting line in which the interarrival times and service times are both modeled by uncertain variables. This paper investigates four important performance characteristics of uncertain queueing system, including waiting time, idle time, total idle time and total busy time. The analytical formulas of uncertainty distributions of those performance characteristics are also derived separately.

This paper first establishes uncertain hypothesis test as a mathematical tool that uses uncertainty theory to help people rationally judge whether some hypotheses are correct or not, according to observed data. As an application, uncertain hypothesis test is employed in uncertain regression analysis to test whether the estimated disturbance term and the fitted regression model are appropriate. In order to illustrate the test process, some numerical examples are documented.

Assume an uncertain process follows an uncertain differential equation, and some realizations of this process are observed. Parameter estimation for the uncertain differential equation that fits the observed data as much as possible is a core problem in practice. This paper first presents a problem of initial value estimation for uncertain differential equations and proposes an estimation method. In addition, the method of moments is recast for estimating the time-varying parameters in uncertain differential equations. Using those techniques, a COVID-19 spread model based on uncertain differential equation is derived, and the zero-day of COVID-19 spread in China is inferred.

Parameter estimation is a critical problem in the wide applications of uncertain differential equations. The method of moments is employed for the first time as an approach for estimating the parameters in uncertain differential equations. Based on the difference form of an uncertain differential equation, a function of the parameters is proved to follow a standard normal uncertainty distribution. Setting the empirical moments of the functions of the parameters and the observed data equal to the moments of the standard normal uncertainty distribution, a system of equations about the parameters is obtained whose solutions are the estimates of the parameters. Analytic examples and numerical examples are given to illustrate the proposed method of moments.

Regression analysis is a mathematical tool to estimate the relationship between explanatory variables and response variable. This paper defines a likelihood function in the sense of uncertain measure to represent the likelihood of unknown parameters. Furthermore, the method of maximum likelihood estimation is used for the parameter estimation of uncertain regression models, and the uncertainty distribution of the disturbance term is simultaneously calculated. Finally, some numerical examples are documented to illustrate the proposed method.

One type of production problem is concerned with a sequence of production cycles in which the cycle time (i.e., the length of each cycle) and reward (e.g., production amount or cost) are characterized in terms of uncertain variables (not random variables). In order to deal with this problem, this article proposes an uncertain production risk process. Moreover, the concepts of shortage index and shortage time of the uncertain production risk process are proposed, and the formulas for shortage index and the uncertainty distribution of shortage time are also derived. Finally, some numerical experiments are performed to illustrate the model and formulas.

Time series analysis is a method to predict future values based on previously observed values. Assuming the observed values are imprecise and described by uncertain variables, this paper proposes an approach of uncertain time series. By employing the principle of least squares, a minimization problem is derived to calculate the unknown parameters in the uncertain time series model. In addition, residual and confidence interval are also proposed. Finally, some numerical examples are given.

The basic urn problem is to determine the probability of drawing one colored ball from an urn with known composition of differently colored balls. If the composition is unknown, then it is called uncertain. This paper designs some uncertain urn problems in order to compare probability theory and uncertainty theory. It is concluded that uncertainty theory is better than probability theory to deal with uncertain urn problems. As a by-product, this paper also solves the choice problem in Ellsberg experiment by chance theory.

Data envelopment analysis (DEA) is a powerful analytical tool in operations research and management for measuring and estimating the efficiency of decision-making units. Both the inputs and the outputs are assumed to be known constants in the classical DEA models. However, in many cases, those data (e.g., carbon emissions and social benefit) cannot be measured in a precise way. Therefore, in this article, the inputs and outputs are considered as uncertain variables and a new uncertain DEA model is introduced. The sensitivity and stability of the new model are also analyzed. Finally, a numerical example of the new model is documented.

Regression analysis is a method to estimate the relationships among the response variable and the explanatory variables. Assuming the observations of the response variable are imprecise and modeling the observed data via uncertain variables, this paper explores an approach of uncertain regression analysis to estimating the relationships among the variables with imprecisely observed samples. On the principle of least squares, an optimization problem is derived to calculate the unknown parameters in the regression model. In particular, this paper investigates uncertain linear regression model and gives an analytic representation of the unknown parameters.