Sparse functional principal component analysis in a new regression framework

The functional principal component analysis is widely used to explore major sources of variation in a sample of random curves. These major sources of variation are represented by functional principal components (FPCs). The FPCs from the conventional FPCA method are often nonzero in the whole domain, and are hard to interpret in practice. The main focus is to estimate functional principal components (FPCs), which are only nonzero in subregions and are referred to as sparse FPCs. These sparse FPCs not only represent the major variation sources but also can be used to identify the subregions where those major variations exist. The current methods obtain sparse FPCs by adding a penalty term on the length of nonzero regions of FPCs in the conventional eigendecomposition framework. However, these methods become an NP-hard optimization problem. To overcome this issue, a novel regression framework is proposed to estimate FPCs and the corresponding optimization is not NP-hard. The FPCs estimated using the proposed sparse FPCA method is shown to be equivalent to the FPCs using the conventional FPCA method when the sparsity parameter is zero. Simulation studies illustrate that the proposed sparse FPCA method can provide more accurate estimates for FPCs than other available methods when those FPCs are only nonzero in subregions. The proposed method is demonstrated by exploring the major variations among the acceleration rate curves of 107 diesel trucks, where the nonzero regions of the estimated sparse FPCs are found well separated.