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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 goodness-of-fit 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
Previous: General Smoothing Spline Regression
Yuedong Wang
2004-05-19