The Research Of Functional Dimension Reduction Based On Fuzzy Partition And Transformation

Functional Sliced Inverse Regression(FSIR),which is an important method in full functional dimension reduction,is widely used in the estimation of functional regression models.However,FSIR has two shortcomings.On the one hand,it is necessary to ensure that n/S(n is sample size)is not too small during the selection process of the slice S.That is,to ensure that there is enough sample size in each slice,which makes FSIR not suitable for small sample data,and is easily affected by the number of slices when the observation size is small.On the other hand,when the contact function is an even function,FSIR cannot estimate the dimension reduction direction.That has E({x,?>|y)=0,then the estimation direction ? is 0,which affects the estimation effect of FSIRIn view of the above two problems,this paper proposes new estimation methods Firstly,it is proposed to construct slices based on fuzzy partitioning instead of the slice method which can be regarded as hard partitioning,so that the sample size in each slice is almost equal to the total sample size.Therefore,FSIR can be applied to small sample data.Secondly,a functional sliced inverse regression method based on response variable or prediction curve transformation is proposed.That is,by transforming the response variable or the prediction curve,the contact function is no longer an even function,so that FSIR can estimate the dimension reduction direction that is not 0.Next,the corresponding numerical simulations are performed on the proposed new method.It shows that the inverse regression method based on fuzzy partitioning is more significant when the sample size is not large.Under the condition of symmetric dependence,the method based on response variable or predictive curve transformation is superior to the original method in estimation accuracy and prediction ability.Further,this paper concludes from the actual data analysis results whether the stability or accuracy of the estimation results are well improved.