Multivariate Functional Regression Via Nested Reduced-Rank Regularization
Multivariate Functional Regression Via Nested Reduced-Rank Regularization
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DOI：
10.1080/10618600.2021.1960850
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发表时间：
2020-03
影响因子：
2.4
通讯作者：
Xiaokang Liu;Shujie Ma;Kun Chen
中科院分区：
文献类型：
--
作者：
Xiaokang Liu;Shujie Ma;Kun Chen
摘要：本文提出了一种嵌套降阶回归（NRRR）方法，用于拟合具有多元函数响应和预测因子的回归模型，以实现有针对性的降维，促进模型的解释和可视化。我们的方法是基于在功能回归表面上施加的两级低秩结构。全局低秩结构确定了一小组潜在的主要功能响应和预测因子，这些响应和预测因子驱动潜在的回归关联。然后，局部低秩结构控制主要功能响应和预测因子之间关联的复杂性和平滑性。泛函问题可以归结为一个通过基展开的积分矩阵逼近任务，其中积分低秩矩阵的块共享一些公共的行空间和/或列空间。这种嵌套的降阶结构在多元时间序列建模和张量回归中也有潜在的应用。提出了一种块坐标下降算法。我们建立了NRRR的一致性，并通过非渐近分析表明它至少可以达到与降阶回归相当的错误率。仿真研究证明了NRRR算法的有效性。我们将提出的方法应用于电力需求问题，将每日电力消耗轨迹与每日温度联系起来。本文的补充文件可在网上获得。
Abstract We propose a nested reduced-rank regression (NRRR) approach in fitting a regression model with multivariate functional responses and predictors to achieve tailored dimension reduction and facilitate model interpretation and visualization. Our approach is based on a two-level low-rank structure imposed on the functional regression surfaces. A global low-rank structure identifies a small set of latent principal functional responses and predictors that drives the underlying regression association. A local low-rank structure then controls the complexity and smoothness of the association between the principal functional responses and predictors. The functional problem boils down to an integrated matrix approximation task through basis expansion, where the blocks of an integrated low-rank matrix share some common row space and/or column space. This nested reduced-rank structure also finds potential applications in multivariate time series modeling and tensor regression. A blockwise coordinate descent algorithm is developed. We establish the consistency of NRRR and show through nonasymptotic analysis that it can achieve at least a comparable error rate to that of the reduced-rank regression. Simulation studies demonstrate the effectiveness of NRRR. We apply the proposed methods in an electricity demand problem to relate daily electricity consumption trajectories with daily temperatures. Supplementary files for this article are available online.