Subsections:

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); Seaman and Hutchinson (1985).

Suppose that is a space of functions whose partial derivatives of total order m are in , where is the domain of .

Now, consider the data model

where .

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

is the penalty term to enforce smoothness on f. There are several ways to define . For the thin-plate smoothing spline, with of dimension d, define as

where . Under this definition, gives zero penalty to some functions. The space that is spanned by the set of polynomials that contribute zero penalty is called the polynomial space. The dimension of the polynomial space M is a function of dimension d and order m of the smoothing penalty, .

Given the condition that , the function that minimizes the penalized least squares criterion has the form

where and are vectors of coefficients to be estimated. The M functions are linearly independent polynomials that span the space of functions for which is zero. The basis functions are defined as

When d = 2 and m = 2, then , , , and . is as follows:

For the sake of simplicity, the formulas and equations that follow assume m = 2. See Wahba (1990) and Bates et al. (1987) for more details.

Duchon (1976) showed that can be represented as

where for d = 2. For derivations of for other values of d, see Villalobos and Wahba (1987).

If you define with elements and with elements , the goal is to find vectors of coefficients and that minimize

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

If and , the expression for becomes

The coefficients and can be obtained by solving

To compute and , let the QR decomposition of be

where is an orthogonal matrix and is an upper triangular, with (Dongarra et al., 1979).

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

or

The coefficient can be obtained by solving

The influence matrix is defined as

and has the form

Similar to the regression case, if you consider the trace of as the degrees of freedom for the model and the trace of as the degrees of freedom for the error, 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, see O’Sullivan and Wong (1987); Nychka (1986a, 1986b, 1988); Hall and Titterington (1987).

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

where is the ith diagonal element of the matrix and is the quantile of the standard normal distribution. The confidence intervals are interpreted as intervals “across the function” as opposed to pointwise intervals.

For SCORE data sets, the hat matrix is not available. To compute the Bayesian confidence interval for a new point , let

and let be an vector with ith entry

When d = 2 and m = 2, is computed with

is a vector of evaluations of by the polynomials that span the functional space where is zero. The details for , , and are discussed in the previous section. Wahba (1983) showed that the Bayesian posterior variance of satisfies

where

Suppose that you fit a spline estimate that consists of a true function f and a random error term to experimental data. 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.

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 close to 0. A small places less of a penalty on rapid change in , 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 model from the error 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 might 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 might be suspect when is extremely small because computed values might become indistinguishable from zero. In practice, calculations with or near 0 can cause numerical instabilities that result 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 might not be accurate.