Dimension Reduction Of Partial Linear Multi-Index Additive Model And Its Application

In this paper,we mainly study the dimensionality reduction problem of partial linear multi-index additive model based on high-dimensional longitudinal data and the application of the model in medical costs.In the observation of the collected data,the same subject or unit under test,in the chronological order or spatial order,is repeatedly tracked of.We call the data for the longitudinal data.Longitudinal data often appear in the fields of medicine,biology,psychology,sociology,economics and insurance.The study of longitudinal data not only can understand the trend of individual or unit changes over time,but also can understand the trend of overall change over time.Its essential feature is the combination of time series data and cross-sectional data,the data within the group related and the data between the groups independent.There are two difficulties in analyzing longitudinal data.The first difficulty lies in the need to consider the correlation between the individual subjects or units under different observations.The second difficulty lies in how to improve and innovate the existing statistical theories and methods in the context of data dependency,so that it can be applied to the processing of high-dimensional longitudinal data.This paper studies the dimensionality reduction problem of partial linear multi-index additive model.The innovation of this paper is as follows: Firstly,some linear multi-index additive models are proposed based on high-dimensional longitudinal data.The advantage of this model is that the traditional statistical model of the covariate is a low-dimensional case,which is improved to allow statistical models with unknown connection functions with high dimensional covariates.Secondly,the partial linear multiindex additive model under high-dimensional longitudinal data is dimensioned by using some theories and methods of full dimensionality reduction.Thirdly,the theory and method of minimum mean variance estimation are used to reduce the dimension of the linear multi-index model under high-dimensional longitudinal data.Based on the existing statistical theories and methods,it has been improved and innovated to apply to the analysis of high dimensional longitudinal data.For the estimation of the parameters and nonparametric parts in the partial linear multi-index additive model,two estimation methods are given in this paper:Firstly,we use the partially sufficient dimension reduction in the multi-index part of the model to reduce the covariate to low dimension,and obtain the basis of the partial central subspace and its structural dimension.After using the partial dimensionality reduction method,the multi-index additive model is transformed into a standard partial linear additive model,and then we use the kernel estimation method proposed by Manzan and Zerom(2005)to estimate the nonparametric part of the model.Thus,we can achieve the dimensionality reduction of some linear multi-index additive model and the nonparametric estimation of the unknown connection function in the model.The second method is using the average of MAVE(Xia(2002)),which is used to improve the complexity of high dimensional data "dimension disaster" and high dimensional longitudinal data structure.At the same time,the unknown connection function is estimated.Thus,we can achieve the dimensionality reduction of some linear multi-index additive model and the nonparametric estimation of the unknown connection function in the model.At the same time,this paper also gives the numerical simulation and asymptotic properties of the two methods respectively.According to the numerical simulation,we find that the two methods have a good estimate of the model.Finally,we used a set of chronic heart failure patients with medical costs data for a case study.