Variable-Projection Sparse Functional Principal Component Analysis: Interpretable Functional Dimensionality Reduction with Applications to Raman Spectral Data

RESUMO

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.