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Дата изменения: Sat Apr 16 04:20:07 1994 Дата индексирования: Sun Dec 23 19:34:28 2007 Кодировка: Поисковые слова: cfht telescope |
Cross-validation divides the data into two sets. It uses one set to determine some characteristics of the data, and then uses the other set to verify the characteristics. Extending this concept, generalized cross-validation (GCV) uses each data point as the testing set, and then averages a weighted prediction error over the entire data set (Golub et al. 1979).
The GCV criterion can be stated as
where is the number of data points.
The matrix
is called the influence matrix at the
th iteration.
If we define
as the
th estimate for the original data
as determined by some iterative algorithm, then
is defined as
The main drawback of the GCV criterion comes from the denominator of .
Unlike the numerator, which can be easily computed at each iteration, the
denominator term, needs the equivalent of
restoration computations to be
computed explicitly. This computational expense is due to the trace term.
Therefore, we estimate the trace term using the following lemma.
Proof The proof is straightforward.
Several researchers independently observed that the denominator term can be estimated using this property (Girard 1989, Hutchinson 1990, Reeves and Mersereau 1991). So the denominator of V(n) can be computed as
and is now called the Randomized GCV (RGCV) criterion.
Still, we do not want to compute
. So using the relationship
, we define a new image
that satisfies
which means that is like a ``restored'' version of
at the
th iteration.
Therefore, instead of computing explicitly, we can compute
by restoring
in the same manner as we restore
. Thus,
now has the form
The computation of the RGCV criterion simply boils down to the computation
of . The factor we must deal with is ``restoring''
in the
same manner as
.
Fortunately, the influence matrix
can also be defined as (Wahba 1983)
where is the
th entry in the matrix
, and
Therefore, if one computes for each
iteration, and uses
as the update for , then the ``restoration'' on
proceeds
as the restoration on
.
Equivalently, from
,
Thus, if the derivative can be computed
analytically for all
, then the update for
can be established.