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Model and Estimation
The general smoothing spline regression (SSR) model with one variable
assumes that (Wahba, 1990)



(1) 
where 's are univariate responses; is an unknown
function of an independent variable with belonging
to an arbitrary domain and , a given
Reproducing Kernel Hilbert Space (RKHS); are bounded
linear functionals on ; and 's are random errors with
. Note that
may be a vector. For most applications, 's are evaluation
functionals at design points: .
Suppose that



(2) 
where is a finite dimensional space with basis
functions
, and is a RKHS with
reproducing kernel . See Aronszajn (1950) and
Wahba (1990) for more information about RKHS. The estimate
of ,
, is the minimizer of the following
penalized least squares



(3) 
where is the orthogonal projection operator of onto in ,
and is a smoothing parameter controlling the balance
between goodnessoffit measured by the least squares and departure
from the null space measured by . Note that
functions in are not penalized.
Let
.
Define
,
and
. Given , the solution to ()
has the form (Wahba, 1990)



(4) 
where the coefficients
and
are solutions to
The Fortran subroutine dsidr.r in RKPACK was developed to
solve equations () (Gu, 1989). In our ASSIST package, the
S function dsidr serves as an intermediate interface between
S and the driver dsidr.r.
Next: The ssr Function
Up: General Smoothing Spline Regression
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Yuedong Wang
20040519