Feature Screening Of Ultrahigh Dimensional Partial Functional Linear Model

In recent years,with the rapid development of information technology,complex data has become more and more common.In complex data,functional data and ultrahigh-dimensional data have appeared in psychology,economics,meteorology,medicine,biology and many other fields.Functional data is the data that varies with a continuous set,such as time,space.It can take the form of a curve,a plane,or a three-dimensional space and so on.The analysis method of functional data usually map the data into a space with finite bases to reduce the dimension,such as functional principal component analysis,or expand the functional data using spline base.On the hand,in statistical studies,we call the data as ultrahigh-dimensional when the dimension of variablediverges with the power-function of sample size9),i.e.log??=?9?),>0.The handling methods of ultrahigh-dimensional data usually screen important variable first,and establish further model next.Although functional data and ultrahigh-dimensional data have drawn some attention,due to the complexity of the two kinds of data,there is not much research on the mixed data containing both functional data and ultrahigh-dimensional data.In this thesis,we consider a linear model that the response variable is scalar,and the pre-dictor variable include both functional data and ultrahigh-dimensional data,assuming the true model is sparse.We combine forward regression and group variable selection and propose a FR-gSCAD method to select important variables that include both functional predictors and scalar predictors,and estimate their coefficient simultaneously.In the second chapter,we intro-duce the functional component analysis,some criteria of model identification and the method of group variable selection.In the third chapter,we introduce our method of variable selection.Under some common conditions,we prove that forward regression can screen all the important predictors with probability tending to 1 when the model contain both functional and ultrahigh-dimensional scalar predictors.Because of getting a nested model from froward regression algo-rithm,we use EBIC criteria to select the optimal model,and obtain that the optimal model also has consistent screening property.In the fourth chapter,we present the statistical simulations which show the proposed method performs well.