Multi-dimensional Functional Principal Component Analysis

Lu-Hung Chen, Ci-Ren Jiang
Institute of Statistics, National Chung-Hsing University, Taichung 402, Taiwan
arXiv:1510.04439 [stat.ME], (15 Oct 2015)

   title={Multi-dimensional Functional Principal Component Analysis},

   author={Chen, Lu-Hung and Jiang, Ci-Ren},






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Functional principal component analysis is one the most commonly employed approaches in functional/longitudinal data analysis and we extend it to conduct $d$-dimensional functional/longitudinal data analysis. The computational issues emerging in the extension are fully addressed with our proposed solutions. The local linear smoothing technique is employed to perform estimation because of its capabilities of performing large-scale smoothing and of handling data with different sampling schemes (possibly on irregular domain) in addition to its nice theoretical properties. Besides taking the fast Fourier transform strategy in smoothing, the modern GPGPU (general-purpose computing on graphics processing units) architecture is applied to perform parallel computation to save computation time. To resolve the out-of-memory issue due to large-scale data, the random projection procedure is applied in the eigen-decomposition step. We show that the proposed estimators can achieve the classical nonparametric rates for longitudinal data and the optimal convergence rates for functional data if the number of observations per sample is of the order $(n/ log n)^{d/4}$. Finally, the performance of our approach is demonstrated with simulation studies and the fine particulate matter (PM 2.5) data measured in Taiwan.
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