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.
Uncertain nonlinear time series analysis is a set of statistical techniques that use uncertainty theory to predict future values via nonlinear dynamics based on the previous observations. By assuming that the disturbance term is an uncertain variable, an uncertain nonlinear time series model is derived in this paper. In addition, this paper presents a method to estimate unknown parameters in an uncertain nonlinear time series model. Finally, some real examples (motion analysis and epidemic spreading) are provided to illustrate uncertain nonlinear time series analysis. As a result, it is shown that the uncertain nonlinear time series model may provide higher forecast accuracy than linear one.
In this paper, an optimization problem with uncertain constraint coefficients is considered. Possibility theory is used to model the uncertainty. Namely, a joint possibility distribution in constraint coefficient realizations, called scenarios, is specified. This possibility distribution induces a necessity measure in a scenario set, which in turn describes an ambiguity set of probability distributions in a scenario set. The distributionally robust approach is then used to convert the imprecise constraints into deterministic equivalents. Namely, the left-hand side of an imprecise constraint is evaluated by using a risk measure with respect to the worst probability distribution that can occur. In this paper, the Conditional Value at Risk is used as the risk measure, which generalizes the strict robust, and expected value approaches commonly used in literature. A general framework for solving such a class of problems is described. Some cases which can be solved in polynomial time are identified.
Multivariate uncertain calculus is a branch of mathematics that deals with differentiation and integration of uncertain fields based on uncertainty theory. This paper defines partial derivatives of uncertain fields for the first time by putting forward the concept of Liu field. Then the fundamental theorem, chain rule and integration by parts of multivariate uncertain calculus are derived. Finally, this paper presents an uncertain partial differential equation, and gives its integral form.
Uncertain partial differential equation (UPDE) was introduced in literature. But the solution of a UPDE was not defined well. In this article, we will rigorously give a suitable concept of a UPDE and define its solution by an integral equation. Then, some examples are given to show the rationality of the definition. Uncertain heat conduction equation is presented as an application of UPDE. For those UPDEs having no analytic solutions, (alpha)-path method is introduced to obtain the inverse uncertainty distributions of solutions to UPDEs.
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