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.

摘要

我们提出了一种嵌套降秩回归 (NRRR) 方法，用于拟合具有多变量功能响应和预测变量的回归模型，以实现量身定制的降维并促进模型解释和可视化。我们的方法基于施加在功能回归表面上的两级低秩结构。全局低秩结构识别一小组潜在的主要功能响应和驱动潜在回归关联的预测因子。然后，局部低秩结构控制主要功能响应和预测变量之间关联的复杂性和平滑度。函数问题归结为通过基扩展的集成矩阵逼近任务，其中集成低秩矩阵的块共享一些公共行空间和/或列空间。这种嵌套的降秩结构还在多元时间序列建模和张量回归中找到了潜在的应用。开发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秩回归相当的错误率。模拟研究证明了 NRRR 的有效性。我们将所提出的方法应用于电力需求问题，以将每日用电量轨迹与每日温度联系起来。本文的补充文件可在线获取。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秩回归相当的错误率。模拟研究证明了 NRRR 的有效性。我们将所提出的方法应用于电力需求问题，以将每日用电量轨迹与每日温度联系起来。本文的补充文件可在线获取。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秩回归相当的错误率。模拟研究证明了 NRRR 的有效性。我们将所提出的方法应用于电力需求问题，以将每日用电量轨迹与每日温度联系起来。本文的补充文件可在线获取。