Functional Linear Regression with Mixed Predictors
Functional Linear Regression with Mixed Predictors
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发表时间：
2020-12
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通讯作者：
Daren Wang;Zifeng Zhao;Yi Yu;R. Willett
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作者：
Daren Wang;Zifeng Zhao;Yi Yu;R. Willett
我们考虑一个通用的功能线性回归模型，允许功能和高维矢量协变量。此外，提出的模型可以适应功能变量的离散观察结果以及用于功能回归系数的不同再现Hilbert Space（RKHS）。基于这种一般环境，我们建议在RKHS中采用惩罚的最小二乘方法，在该方法中，惩罚对功能估计器的平滑度和稀疏性都施加了。我们还表明，在此通用模型设置下，对估计量的过剩预测风险是最佳的。我们的分析揭示了一种有趣的相变现象，最佳的多余风险由功能回归系数的稀疏性和平滑度共同确定。我们设计了一种新颖的优化算法，同时处理平滑度和稀疏性惩罚。
We consider a general functional linear regression model, allowing for both functional and high-dimensional vector covariates. Furthermore, the proposed model can accommodate discretized observations of functional variables and different reproducing kernel Hilbert spaces (RKHS) for the functional regression coefficients. Based on this general setting, we propose a penalized least squares approach in RKHS, where the penalties enforce both smoothness and sparsity on the functional estimators. We also show that the excess prediction risk of our estimators is minimax optimal under this general model setting. Our analysis reveals an interesting phase transition phenomenon and the optimal excess risk is determined jointly by the sparsity and the smoothness of the functional regression coefficients. We devise a novel optimization algorithm, simultaneously handling the smoothness and sparsity penalization.