Sparse principal component analysis for functional data

Principal component analysis (PCA) has been widely used for data analysis. However, it tends to provide excessively rough estimate if the data are contaminated by noise. To address this problem, Ramsay and Silverman (1997) suggested smoothed PCA by considering a penalty term, defined by integrated squared second derivative, into the standard PCA. The smoothness of the estimate can be controlled by a smoothing parameter. Nevertheless, smoothed PCA is not able to estimate both peaks and flat regions of the signal well simultaneously. We propose a sparse PCA technique to overcome this shortcoming. If the signal has a sparse representation in a certain basis, we can effectively reduce the ratio of signal dimension to sample size, which is an indicator of the perturbation amplitude, by choosing a sub-basis in a much smaller dimension. Sparse PCA is the most efficient algorithm among the three PCA algorithms and it also provides consistent sample estimate. The theoretical basis of the sparse PCA algorithm will be discussed. Numerical studies demonstrate that the sparse PCA can be significantly superior to the other PCA methods, in terms of efficiency and accuracy.