Spatially distributed functional data are prevalent in many statistical applications such as meteorology, energy forecasting, census data, disease mapping, and neurological studies. Given their complex and high-dimensional nature, functional data often require dimension reduction methods to extract meaningful information. Inverse regression is one such approach that has become very popular in the past two decades. We study the inverse regression in the framework of functional data observed at irregularly positioned spatial sites. The functional predictor is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect, while the relation between the scalar response and the functional predictor is modeled using the inverse regression framework. For estimation, we consider local linear smoothing with a general weighting scheme, which includes as special cases the schemes under which equal weights are assigned to each observation or to each subject. This framework enables us to present the asymptotic results for different types of sampling plans over time such as non-dense, dense, and ultra-dense. We discuss the domain-expanding infill (DEI) framework for spatial asymptotics, which is a mix of the traditional expanding domain and infill frameworks. The DEI framework overcomes the limitations of traditional spatial asymptotics in the existing literature. Under this unified framework, we develop asymptotic theory and identify conditions that are necessary for the estimated eigen-directions to achieve optimal rates of convergence. Our asymptotic results include pointwise and $L_2$ convergence rates. Simulation studies using synthetic data and an application to a real-world dataset confirm the effectiveness of our methods.

翻译：空间分布函数数据在气象学、能源预测、人口普查数据、疾病映射和神经科学研究等众多统计应用中普遍存在。由于这类数据具有复杂且高维的特性，通常需要借助降维方法提取有意义的信息。逆回归是过去二十年间广受关注的一种降维方法。本文在空间位置不规则分布的框架下研究函数型数据的逆回归问题。函数型预测变量由空间相关函数效应与空间独立函数金块效应共同构成，标量响应与函数型预测变量之间的关系采用逆回归框架进行建模。在估计过程中，我们采用具有一般加权方案的局部线性平滑方法，该方案涵盖了对每个观测值或每个实验对象赋予等权重的特殊情况。这一框架使我们能够针对不同时间采样方案（如非密集采样、密集采样和超密集采样）给出渐近结果。我们讨论用于空间渐近分析的域扩展-填充（DEI）框架，该框架融合了传统扩展域与填充域两种框架的特点。DEI框架克服了现有文献中传统空间渐近分析的局限性。在此统一框架下，我们建立了渐近理论，并识别了使估计特征方向达到最优收敛速率所需的必要条件。我们的渐近结果包括逐点收敛速率和$L_2$收敛速率。基于合成数据的模拟研究及真实数据应用验证了所提方法的有效性。