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 replication structure of functional data. Non-asymptotic bounds are provided for our algorithm, supplemented by numerical evidence from an extensive simulation study. Possible applications of the directional regularity approach are discussed, and we advocate its consideration as a standard pre-processing step in multivariate functional data analysis.

翻译：本文提出了方向正则性这一多元函数型数据各向异性的新定义。不同于传统上将各向异性视为沿维度平滑度的观点，方向正则性进一步从方向的视角审视各向异性。我们证明，通过适应多元过程的方向正则性进行基变换，可以获得更快的收敛速度。利用函数型数据的重复观测结构，我们构建了用于估计和识别基变换矩阵的算法。该算法提供了非渐近误差界，并通过大量模拟研究提供了数值证据支持。本文讨论了方向正则性方法的潜在应用场景，并建议将其作为多元函数型数据分析的标准预处理步骤加以考虑。