变量投影稀疏函数主成分分析：可解释的函数降维及其在拉曼光谱数据中的应用
Variable-Projection Sparse Functional Principal Component Analysis: Interpretable Functional Dimensionality Reduction with Applications to Raman Spectral Data
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中文总结 AI 辅助
针对FPCA难以识别局部变异区域的问题，提出变量投影稀疏FPCA，通过稀疏权重函数实现可解释降维，在模拟和SERS数据中兼顾计算效率与重构精度。
中文摘要 AI 辅助
函数主成分分析（FPCA）为函数型数据提供低秩表示，但通常产生稠密成分，使得难以识别主导变异模式所贡献的局部区域。这一局限性在拉曼光谱中尤为相关，因为光谱是在有序域上观测的，且解释往往集中于具有化学意义的光谱区域。本研究提出变量投影稀疏FPCA（VP-SFPCA），通过促进局部化的稀疏权重函数实现可解释的函数降维。该方法将稀疏FPCA表述为正则化矩阵分解问题，该问题结合了函数内积几何，并区分了用于生成成分得分的稀疏权重函数与用于重构的正交载荷函数。变量投影对载荷函数进行条件最小化，将优化问题简化为稀疏权重。通过模拟研究和表面增强拉曼散射（SERS）光谱的实证分析评估了性能，以传统FPCA和SCAD-SFPCA分别作为稠密和稀疏函数基准。在模拟中，VP-SFPCA恢复了局部化函数结构，同时计算量远小于SCAD-SFPCA。在实证分析中，VP-SFPCA保持了这一计算优势，同时产生的留出重构误差接近传统FPCA。估计权重函数的几个显著特征也与已确立的腺嘌呤SERS谱带一致。总体而言，VP-SFPCA提供了一种计算上实用的方法，通过局部化提高主导函数模式的可解释性。
英文摘要
Functional principal component analysis (FPCA) provides low-rank representations of functional data but generally produces dense components, making it difficult to identify the localised regions contributing to dominant modes of variation. This limitation is particularly relevant in Raman spectroscopy, where spectra are observed over an ordered domain and interpretation often focuses on chemically meaningful spectral regions. This study proposes variable-projection sparse FPCA (VP-SFPCA) for interpretable functional dimensionality reduction through sparse weight functions that promote localisation. The method formulates sparse FPCA as a regularised matrix-factorisation problem that incorporates the functional inner-product geometry and distinguishes sparse weight functions used to generate component scores from orthonormal loading functions used for reconstruction. Variable projection conditionally minimises over the loading functions, reducing the optimisation problem to the sparse weights. Performance was evaluated through simulation studies and empirical analyses of surface-enhanced Raman scattering (SERS) spectra, with conventional FPCA and SCAD-SFPCA serving as dense and sparse functional benchmarks, respectively. In the simulations, VP-SFPCA recovered localised functional structure while requiring substantially less computation than SCAD-SFPCA. In the empirical analysis, VP-SFPCA retained this computational advantage while yielding held-out reconstruction error close to that of conventional FPCA. Several prominent features of the estimated weight functions also coincided with established adenine SERS bands. Overall, VP-SFPCA provides a computationally practical approach to improving the interpretability of dominant functional modes through localisation.
发表机构
- Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya（马来亚大学理学院数学科学研究所）
- Centre of Research for Statistical Modelling and Methodology, Faculty of Science, Universiti Malaya（马来亚大学理学院统计建模与方法研究中心）
- School of Mathematics and Statistics, University of Glasgow（格拉斯哥大学数学与统计学院）
- Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, Inria-Datavers（里尔大学、法国国家科学研究中心UMR 8524保罗·潘勒瓦实验室、Inria-Datavers）
机构由 AI 辅助整理，请以论文原文为准。
