We propose a supervised principal component regression method for relating functional responses with high dimensional covariates. Unlike the conventional principal component analysis, the proposed method builds on a newly defined expected integrated residual sum of squares, which directly makes use of the association between functional response and predictors. Minimizing the integrated residual sum of squares gives the supervised principal components, which is equivalent to solving a sequence of nonconvex generalized Rayleigh quotient optimization problems and thus is computationally intractable. To overcome this computational challenge, we reformulate the nonconvex optimization problems into a simultaneous linear regression, with a sparse penalty added to deal with high dimensional predictors. Theoretically, we show that the reformulated regression problem recovers the same supervised principal subspace under suitable conditions. Statistically, we establish non-asymptotic error bounds for the proposed estimators. Numerical studies and an application to the Human Connectome Project lend further support.
翻译:我们建议了一种受监督的主要部分回归法,用于将功能反应与高维共差联系起来。与常规主要部分分析不同,拟议方法以新定义的预期综合残余方和正方和正方和正方之间直接利用功能反应和预测器之间的联系。最小化综合残余方和正方和受监督的主要部分提供了受监督的主要部分,这相当于解决一系列非混凝土普遍雷利商商价优化问题,因此在计算上是难以解决的。为了克服这一计算挑战,我们重新将非对流优化问题改成一个同时线性回归,加上微量的罚款,以处理高维预测器。理论上,我们表明重订回归问题在适当条件下回收了同样的受监督的主要次空间。从统计学上讲,我们为拟议的估算器设定了非被动误差界限。数字研究和人类连接器项目应用程序提供了进一步的支持。