再生核Hilbert空间展开的函数型回归模型变量选择

Variable Selection of Functional Regression Model for Reproducing Kernel Hilbert Space Expansion

针对协变量是函数型、响应变量是标量的多元函数型回归模型,文章提出了函数系数基于再生核Hilbert空间展开的变量选择方法。首先,利用带积分余项的泰勒展开式和再生核Hilbert空间内积性质将模型转化为结构化形式,其次,通过自适应弹性网惩罚对结构化模型中的组间和组内系数同时进行压缩。结果证明了这种压缩估计具有Oracle性质,蒙特卡罗模拟结果也显示新方法在不同样本量、不同噪声和变量相关性干扰下均优于基于普通基函数展开的变量选择方法,且尤其适用于原始协变量高度相关的情形。最后,通过分析一个商品房平均销售价格影响因素数据演示了新方法的应用。

For the multivariate functional regression model in which the covariable is functional and the response variable is scalar, this paper proposes a variable selection method of function coefficients based on Hilbert space expansion of regenerative kernel. First, the model is transformed into a structured form by using Taylor expansion with integral residual and inner product property of the reproducing kernel Hilbert space, and then the inter-group and intra-group coefficients in the structured model are compressed by Adaptive Elastic Net penalty simultaneously. The results show that the compressed estimation has oracle properties, and the results of Monte Carlo simulation also show that the new method is better than traditional methods under different sample size, different noise and variable correlation interference, especially suitable for the scenario when the covariates are highly correlated. Finally, the application of the new method is demonstrated by analyzing influencing factors of the average selling price of commercial housing.