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 The TPSPLINE Procedure

## Computational Formulas

The theoretical foundations for the thin-plate smoothing spline are described in Duchon (1976, 1977) and Meinguet (1979). Further results and applications are given in Wahba and Wendelberger (1980), Hutchinson and Bischof (1983), and Seaman and Hutchinson (1985).

Suppose that Hm is a space of functions whose partial derivatives of total order m are in L2(Ed) where Ed is the domain of x.

Now, consider the data model

where .

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 Jm(f). For the thin-plate smoothing spline, with x of dimension d, define Jm(f) as

where .

When d=2 and m=2, Jm(f) is as follows:

In general, m and d must satisfy the condition that 2m - 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 = E2(xi-xj) and T = (T)ij = (xij), the goal is to find coefficients and that minimize

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 (Q1:Q2) is an orthogonal matrix and R is upper triangular, with XT Q2 = 0 (Dongarra et al. 1979).

Since , must be in the column space of Q2. Therefore, can be expressed as for a vector . Substituting into the preceding equation and multiplying through by Q2 T gives

or

The coefficient can be obtained by solving

The influence matrix is defined as

and has the form

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).

### Confidence Intervals

Viewing the spline model as a Bayesian model, Wahba (1983) proposed Bayesian confidence intervals for smoothing spline estimates as follows:

where is the ith 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.

### Smoothing Parameter

The quantity is called the smoothing parameter, which controls the balance between the goodness of fit and the smoothness of the final estimate.

A large heavily penalizes the mth 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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