A Metric On Uncertain Variables

When studying the distance between two uncertain variables,we need to define the concrete form of the distance.It is a classic way that we can use the concept of expected value to define the distance.However,there is a coefficient 2 in the right side of the triangle inequality of the distance which is not consistent with the triangle inequality of the traditional distance.Another way is to use the concept of moment for uncertain variables.Although this d:istance satisfies the triangle inequality of the traditional distance,it does not satisfy the homogeneity.As a result,it cannot obtain the corresponding norm.In this paper,a new metric satisfying the necessary conditions of metric is proposed.It represents the biggest numerical gap of the absolute value of the difference between the two uncertain variables on an event with uncertain measure 1.Then,some formulas of calculating the metric in some cases are presented.After that,the convergence of uncertain sequences in metric is defined and then we study the relationships among convergence in metric,convergence in measure,convergence in mean,convergence in distribution and uniform convergence almost surely.Thus we can consider the convergence in metric or the stability in metric when discussing the convergence of uncertain sequences or the stability of uncertain differential systems.Afterwards,we can obtain a new norm for uncertain variables based on the metric with the help of the functional analysis.In the sense of the norm,the form of the Young inequality and Minkowski inequality about uncertain variables can be inferred.Finally,the normed vector space consisting of uncertain variables can be proved to be a complete space.