We present a functional data analysis (FDA) framework based on explicit orthonormal basis expansion for modeling and denoising complex biomedical signals. Observed functional data are represented as smooth functions in a Hilbert space, and statistical inference is performed directly on their basis coefficients. This formulation provides a transparent and flexible approach to smoothing, regularization, and hypothesis testing. Applications to diffusion tensor imaging tract modeling and EEG denoising demonstrate the advantages of explicit basis representations for scalable and interpretable functional modeling.

翻译：我们提出了一种基于显式正交基展开的函数数据分析框架，用于建模和去噪复杂的生物医学信号。观测到的函数数据被表示为希尔伯特空间中的光滑函数，并直接在其基系数上进行统计推断。该公式为平滑、正则化和假设检验提供了一种透明且灵活的方法。在扩散张量成像纤维束建模和脑电图去噪中的应用，展示了显式基表示对于可扩展且可解释的函数建模的优势。