Orthogonal Series Estimate of Additive Model.

Orthogonal series estimate of additive model in which independent variables are represented as distributions is proposed. Two methods to realize this methodology are developed: (1)Orthogonal functions are given beforehand. Regressed functions are derived as a linear combination of the functions. This method ends up with ridge regression. Hence, hat matrix and related statistics such as GCV are derived easily. Smoothing splines are considered special cases of this method. Further, neural network is also associated with this method. (2) Orthogonal functions are created from data using Gram-Schmidt’s orthogonalization process. Regressed curves are obtained without procedure of least squares. Linear filters are used to smooth the curves. This regression needs a smaller amount of computational cost than generalized smoothing splines.