Principal Differential Analysis: Data Reduction by Differential Operators
针对光滑函数型数据，提出主微分分析方法，通过估计线性微分算子来近似消减样本函数，实现数据降维和探索性分析，并可用于建模和正则化。
SUMMARY Functional data are observations that are either themselves functions or are naturally representable as functions. When these functions can be considered smooth, it is natural to use their derivatives in exploring their variation. Principal differential analysis (PDA) identifies a linear differential operator L = w 0 I+w 1 D + . . . + w m-1 D m-1 + D m that comes as close as possible to annihilating a sample of functions. Convenient procedures for estimating the m weighting functions w j are developed. The estimated differential operator L is analogous to the projection operator used as the data annihilator in principal components analysis and thus can be viewed as a type of data reduction or exploration tool. The corresponding linear differential equation may also have a useful substantive interpretation. Modelling and regularization features can also be incorporated into PDA.