In this work we propose a generalized additive functional regression model for partially observed functional data. Our approach accommodates functional predictors of varying dimensions without requiring imputation of missing observations. Both the functional coefficients and covariates are represented using basis function expansions, with B-splines used in this study, though the method is not restricted to any specific basis choice. Model coefficients are estimated via penalized likelihood, leveraging the mixed model representation of penalized splines for efficient computation and smoothing parameter estimation.The performance of the proposed approach is assessed through two simulation studies: one involving two one-dimensional functional covariates, and another using a two-dimensional functional covariate. Finally, we demonstrate the practical utility of our method in an application to air-pollution classification in Dimapur, India, where images are treated as observations of a two-dimensional functional variable. This case study highlights the models ability to effectively handle incomplete functional data and to accurately discriminate between pollution levels.

翻译：本文提出了一种针对部分观测函数数据的广义加性函数回归模型。该方法能够处理不同维度的函数预测变量，无需对缺失观测进行填补。函数系数与协变量均通过基函数展开表示，本研究采用B样条基，但该方法不限于任何特定的基函数选择。模型系数通过惩罚似然法估计，利用惩罚样条的混合模型表示实现高效计算与平滑参数估计。通过两项模拟研究评估所提方法的性能：一项涉及两个一维函数协变量，另一项使用二维函数协变量。最后，我们将该方法应用于印度迪马普尔的空气污染分类问题，其中图像被视为二维函数变量的观测值。该案例研究凸显了模型有效处理不完整函数数据并准确区分污染水平的能力。