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The TPSPLINE Procedure |

Suppose that *H*_{m} is a space of functions whose partial derivatives
of total order *m* are in *L _{2}*(

Now, consider the data model

Using the notation from the section "The Penalized Least Squares Estimate", for a fixed ,estimate *f* by minimizing the penalized least squares function

There are several ways to define *J*_{m}(*f*). For the
thin-plate smoothing spline, with *x* of dimension *d*,
define *J*_{m}(*f*) as

When *d*=2 and *m*=2, *J*_{m}(*f*) is as follows:

In general, *m* and *d* must satisfy the condition that
2*m* - *d* > 0. For the sake of simplicity, the formulas and equations
that follow assume *m*=2.
Refer to Wahba (1990) and Bates
et al. (1987) for more details.

Duchon (1976) showed that can be represented as

where

If you
define **K** = (** K)_{ij} = E_{2}(x_{i}-x_{j})**
and

A unique solution is guaranteed if the matrix ** T** is of full rank
and .

If
and ** X = (T:Z)**, the expression for becomes

The coefficients and can be obtained by solving

To compute and , let the QR decomposition of ** X** be

where (** Q_{1}:Q_{2})** is an orthogonal matrix and

Since , must be in the column space of
** Q_{2}**. Therefore, can be expressed
as for
a vector . Substituting into
the preceding equation and multiplying through by

or

The coefficient can be obtained by solving

The influence matrix is defined as

Similar to the regression case, and if you consider the trace of as the degrees of freedom for the information signal and the trace of as the degrees of freedom for the noise component, the estimate can be represented as

where is the residual sum of squares. Theoretical properties of these estimates have not yet been published. However, good numerical results in simulation studies have been described by several authors. For more information, refer to O'Sullivan and Wong (1987), Nychka (1986a, 1986b, and 1988), and Hall and Titterington (1987).

where is the *i*th diagonal element of the
matrix and is the point of
the normal distribution. The confidence intervals are interpreted as
intervals "across the function" as opposed to point-wise
intervals.

Suppose that you fit a
spline estimate to experimental data that consists of a true function
*f* and a random error term, . In repeated experiments,
it is likely
that about of the confidence intervals cover the
corresponding true values, although some values are covered every time
and other values are not covered by the confidence intervals most of
the time. This effect is more pronounced when the true surface or
surface has small regions of particularly rapid change.

A large heavily penalizes the *m*th derivative of the
function, thus forcing *f*^{(m)} close to 0. The final
estimating function satisfies *f*^{(m)}(*x*) = 0.
A small places less of a penalty on rapid change in
*f*^{(m)}(*x*), resulting in an estimate that tends to interpolate
the data points.

The smoothing parameter greatly affects the analysis, and it should be selected with care. One method is to perform several analyses with different values for and compare the resulting final estimates.

A more objective way to select the smoothing parameter is to use the "leave-out-one" cross validation function, which is an approximation of the predicted mean squares error. A generalized version of the leave-out-one cross validation function is proposed by Wahba (1990) and is easy to calculate. This Generalized Cross Validation (GCV) function is defined as

The justification for using the GCV function to select relies on asymptotic theory. Thus, you cannot expect good results for
very small sample sizes or when there is not enough information in the
data to separate the information signal from the noise component.
Simulation studies suggest that for independent and identically
distributed Gaussian noise, you can obtain reliable estimates of
for *n* greater than 25 or 30. Note that, even for large
values of *n* (say ), in extreme Monte Carlo simulations
there may be a small percentage of unwarranted extreme estimates in
which or (Wahba 1983).
Generally, if is known to within an order of magnitude, the
occasional extreme case can be readily identified. As *n* gets larger,
the effect becomes weaker.

The GCV function is fairly robust against nonhomogeneity of variances and non-Gaussian errors (Villalobos and Wahba 1987). Andrews (1988) has provided favorable theoretical results when variances are unequal. However, this selection method is likely to give unsatisfactory results when the errors are highly correlated.

The GCV value may be suspect when is extremely small because computed values may become indistinguishable from zero. In practice, calculations with or near 0 can cause numerical instabilities resulting in an unsatisfactory solution. Simulation studies have shown that a with is small enough that the final estimate based on this almost interpolates the data points. A GCV value based on a may not be accurate.

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