Research On The Method Of Selecting The Number Of Basis Functions Based On Functional Data Analysis

Regarding the selection of the number of basis functions,from two perspectives,one is to directly truncate the number of basis functions according to the principle of truncation,and the other is to use the method of variable selection to realize the selection of basis functions.The former proposed an adaptive weighted truncation method to determining the number of principal components in the basis function expansion,and the method is applied to linear regression model of univariate function and functional quantile regression model with covariate being functional and response variable being scalar.In the first model,firstly,two methods respectively based on percentage of variance explained(PVE)and percentage of association-variation explained(PAVE)criteria were used to truncate the number of functional principal components;secondly,the weighted sum of the two numbers was calculated and the optimal weight was obtained by minimizing the estimation error via optimization algorithm;finally,the final items of principal components were selected.In the second model,using functional quantile regression to estimate unknown parameters to improve the percentage of associated variation explained is the first step,then following the steps in the first model can obtain the last number of truncations.Through the adaptive selection of weights,this method considers the eigenvalues and the correlation between the covariates and the response variables into the selection of the truncation number,and the estimation of least square method and functional quantile regression can better describe the characteristics of distribution and obtain more robust regression coefficients.For the above two models,the results of Monte Carlo simulation show that the new method is superior to PVE-based and PAVE-based methods.The analysis of diffusion tensor imaging data and the spectral data also indicate that the proposed method can effectively improve the accuracy of prediction.The latter is based on the expanded form of the reproducing kernel hilbert space,which is used in the multivariate functional linear regression model where covariate being functional and response variable being scalar,and the basis function is selected by combining with the variable selection method.First,the model is transformed into a structured form by using of Taylor expansion with integral residual and inner product property of the reproducing kernel hilbert space,then the within and between group coefficients were shrunk by Adaptive Elastic Net penalty simultaneously,finally determine the number of basis functions.This paper proves that the compressed estimation has oracle property,and monte carlo simulation results also show that the variable selection method based on the reproducing kernel hilbert space expansion is superior to the variable selection method based on ordinary basis function expansion under different sample size,different noise and variable correlation interference,and it is especially suitable for the scenario when the covariates are highly correlated.Finally,the paper demonstrates the application of the new method by analyzing important influencing factors of the average selling price of commercial housing.