Study Of Data-driven Uncertainty Set Based On Improved Robust Kernel Density Estimation And Experiment Simulations

In this thesis,Rubost Kernel Density Estimation(RKDE)in non-parametric estimation is used to estimate the density function with unknown real distribution.Firstly,three methods which include Kernel Density Estimation(KDE),Variable Kernel Density Estimation(VKDE)and RKDE are introduced in this thesis.When the kernel function is a gaussian function,the consistency of the three estimation functions is proved.The improved RKDE method is proposed by combining RKDE and VKDE,which not only assigns certain weight to the function corresponding to each observed value,but also adopts different smoothing parameters for each function.The effective estimation of the true density function is realized by fully considering the effect of each observed value and the corresponding smoothing parameter.In terms of solving methods,the Kernelized Iterative Reweighted Least Square(KIRWLS)method is used to allocate the weights of the estimated function,and the smoothing parameter is selected as the variable smoothing parameter in VKDE.Based on the historical data,the effective estimation of the true density function is obtained by the above method.And then the Data-driven uncertainty set is constructed by the distribution function of improved RKDE's function under the L1 distance,so that the uncertainty set fully contains the information of the distribution function corresponding to the random variable.In order to verify the effectiveness of the proposed method,three emulational experiments were carried out on MATLAB to compare the estimation effect of each kernel density estimation method under mixture normal distribution.Firstly,the importance of smoothing parameter to kernel density estimation is reflected through the performance of different smoothing parameters,and the relative optimal smoothing parameter was selected.The experiment 1 shows that when there is at least one normal distribution with a very low standard deviation in the mixture distribution,the estimation effect of KDE method is relatively poor,and the required estimation effect is not reached.Secondly,the advantages of VKDE over KDE method are demonstrated through the practical performance in the experiment 2.Finally,the proposed estimation method is compared with KDE,VKDE and RKDE method,and the experiment 3 show that the proposed method performs well in the mixture distribution.