Set-Valued Approximation And Set-Valued Regression

Set-value theory emerged in the middle of the 20th century,it has extensive applications in many fields,such as optimal control,mathematical finance,economic fields.Set-valued stochastic integral equations and set-valued stochastic differential equations are important contents of set-value theory.They have important theoretical values and application prospects.In practical problems,the uncertainties of variables include not only randomness but also inaccuracy.This is the problem to be solved by set-valued random variables.It is a generalization of ordinary random variables.Firstly,this paper introduces the theory of set-valued and the research status of set-valued regression.Then,the relevant notation is given.The definition of the set-valued Lebesgue integral,the set-valued square integrable martingale and the related property theorems are introduced.The classical Ito-type set-valued stochastic differential equation is widely used in many fields.The solution of this equation is unique.Based on this,the paper makes innovations to change the Brownian motion in Ito type set-valued stochastic differential equations to the set-valued square integrable martingale.By using the famous inequalities and the idea of approximation,the existence and uniqueness of new set-valued stochastic differential equations is discussed.The traditional regression analysis uses point data as the research object,and the predicted results are point data naturally,but in reality,real data tends to change within a certain range.In order to solve this problem,this paper introduces the convex interval-valued autoregressive model,which can reflect the range of data changes,and more in line with the actual situation.This method is based on the regression analysis method of the interval data.In this paper,an convex interval-valued autoregressive model is established by generating random interval-valued data.It is found that as the amount of data generated by the simulation increases,the estimated parameters converge more and more to the true value.Using the convex interval-valued autoregressive model to conduct empirical research on the data of Shenzhen Stock Index and Shanghai Stock Index,the predicted trends of the futures are similar with the real values.In practical problems,when data lacks distribution information,convex interval-valued autoregressive model can be considered to predict the future development trend of interval-data.