Functional nonlinear principal component analysis

The widely adopted dimension reduction technique, functional principal component analysis (FPCA), typically represents functional data as a linear combination of functional principal components (FPCs) and their corresponding scores. However, this linear formulation is too restrictive to reflect reality because it fails to capture the nonlinear dependence of functional data when nonlinear features are present in the data. This study develops a novel FPCA model to uncover the nonlinear structures of functional data. The proposed method can accommodate multivariate functional data observed on different domains, and multidimensional functional data with gaps and holes. To navigate the complexities of spatial structure in multidimensional functional variables, tensor product smoothing and spline smoothing over triangulation are employed, providing precise tools for approximating nonparametric function. Furthermore, an efficient estimation approach and theory are developed when the number of FPCs diverges to infinity. To assess its performance comprehensively, extensive simulations are conducted, and the proposed method is applied to real data from the Alzheimer's Disease Neuroimaging Initiative study, affirming its practical efficacy in uncovering and interpreting nonlinear structures inherent in functional data.