We introduce directional regularity, a new definition of anisotropy for multivariate functional data. Instead of taking the conventional view which determines anisotropy as a notion of smoothness along a dimension, directional regularity additionally views anisotropy through the lens of directions. We show that faster rates of convergence can be obtained through a change-of-basis by adapting to the directional regularity of a multivariate process. An algorithm for the estimation and identification of the change-of-basis matrix is constructed, made possible due to the unique replication structure of functional data. Non-asymptotic bounds are provided for our algorithm, supplemented by numerical evidence from an extensive simulation study. We discuss two possible applications of the directional regularity approach, and advocate its consideration as a standard pre-processing step in multivariate functional data analysis.

翻译：本文提出了一种针对多元函数型数据的新型各向异性定义——方向正则性。不同于传统上将各向异性视为沿维度方向平滑度的观念，方向正则性进一步从方向视角审视各向异性。我们证明，通过基变换适应多元过程的方向正则性，可以获得更快的收敛速率。借助函数型数据特有的复制结构，我们构建了用于估计和识别基变换矩阵的算法。该算法的非渐近界被严格推导，并通过大量模拟研究的数值证据加以验证。我们探讨了方向正则性方法的两种潜在应用场景，并主张将其作为多元函数型数据分析的标准预处理步骤予以推广。