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computes thin-plate smoothing splines

CALL TPSPLINE( fitted, coeff, adiag, gcv, x, y<, lambda> );

The TSPLINE subroutine computes thin-plate smoothing spline (TPSS) fits to approximate smooth multivariate functions that are observed with noise. The generalized cross validation (GCV) function is used to select the smoothing parameter.

The TPSPLINE subroutine returns the following values:
is an n×1 vector of fitted values of the TPSS fit evaluated at the design points x. The n is the number of observations. The final TPSS fit depends on the optional lambda.

is a vector of spline coefficients. The vector contains the coefficients for basis functions in the null space and the representer of evaluation functions at unique design points. (Refer to Wahba 1990 for more detail on reproducing kernel Hilbert space and representer of evaluation functions.) The length of coeff vector depends on the number of unique design points and the number of variables in the spline model. In general, let nuobs and k be the number of unique rows and the number of columns of x respectively. The length of coeff equals to k+ nuobs+1. The coeff vector can be used as an input of TPSPLNEV to evaluate the resulting TPSS fit at new data points.

is an n ×1 vector of diagonal elements of the "hat" matrix. See the "Details" section.

If lambda is not specified, then gcv is the minimum value of the GCV function. If lambda is specified, then gcv is a vector (or scalar if lambda is a scalar) of GCV values evaluated at the lambda points. It provides users both with the ability to study the GCV curves by plotting gcv against lambda, and with the chance to identify a possible local minimum.

The inputs to the TPSPLINE subroutine are as follows:
is an n ×k matrix of design points on which the TPSS is to be fit. The k is the number of variables in the spline model. The columns of x need to be linearly independent and contain no constant column.

is the n ×1 vector of observations.

is a optional q ×1 vector containing \lambda values in log_{10}(n\lambda) scale. This option gives users the power to control how they want the TPSPLINE subroutine to function. If lambda is not specified (or lambda is specified and q > 1) the GCV function is used to choose the "best" \lambda and the returning fitted values are based on the \lambda that minimizes the GCV function. If lambda is specified and q=1, no minimization of the GCV function is involved and the fitted, coeff and adiag values are all based on the TPSS fit using this particular lambda. This gives users the freedom to choose the \lambda which they think appropriate.

Aside from the values returned, the TPSPLINE subroutine also prints other useful information such as the number of unique observations, the dimensions of the null space, the number of parameters in the model, a GCV estimate of \lambda, the smoothing penalty, the residual sum of square, the trace of (I-A(\lambda)), an estimate of \sigma^2, and the sum of squares for replication.

Note: No missing values are allowed within the input arguments. Also, you should use caution if you want to specify small lambda values. Since the true \lambda=(10^{\log_{10} {lambda}})/n, a very small value for lambda can cause \lambda to be smaller than the magnitude of machine error and usually the returned gcv values from such a \lambda cannot be trusted. Finally, when using TPSPLINE be aware that TPSS is a computationally intensive method. Therefore a large data set (that is, a large number of unique design points) will take a lot of computer memory and time.

For convenience, we illustrate the TPSS method with a two-dimensional independent variable X = (x1,x2). More details can be found in Wahba (1990), or in Bates, et al. (1987).

Assume that the data is from the model
y_i = f({x}_i) + \epsilon_i,
where (xi, yi), i = 1, ... ,n are the observations. The function f is unknown and you assume that it is reasonably smooth. The error terms \epsilon_i, i=1, ... ,nare independent zero-mean random variables.

You will measure the smoothness of f by the integral over the entire plane of the square of the partial derivatives of f of total order 2, that is
J_2(f) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}
 [ \frac{\partial^2 f}{...
 ...rtial {x_2}} 
 + [ \frac{\partial^2 f}{\partial {x_2}^2} ]^2 
  dx_1 dx_2.
Using this as a smoothness penalty, the thin-plate smoothing spline estimate f_{\lambda} of f is the minimizer of
S_\lambda(f) = \frac{1}n \sum^n_{i=1} (y_i-f(x_i))^2 +
 \lambda J_2(f).
Duchon (1976) derived that the minimizer f_{\lambda} can be represented as
f_\lambda(x) = \sum_{i=1}^3 \beta_i \phi_i(x) +
 \sum_{i=1}^n \delta_i E_2(x-x_i),
where (\phi_1(x),\phi_2(x),\phi_3(x)) = (1,x^1,x^2) and E_2(\bs)=\frac{1}{2^3 \pi} \Vert\bs\Vert^2ln(\Vert\bs\Vert).

Let matrix K have entries (K)ij = E2(xi-xj) and matrix T have entries (T)_{ij} = \phi_j(x_i). Then the minimization problem can be rewritten as finding coefficients \beta and \delta to minimize
S_\lambda(\beta, \delta) = 
\frac{1}n \Vert y- T\beta - K\delta \Vert^2 + 
\lambda \delta^T K\delta.
The final TPSS fits can be viewed as a type of generalized ridge regression estimator. The \lambda is called the smoothing parameter, which controls the balance between the goodness of fit and the smoothness of the final estimate. The smoothing parameter can be chosen by minimizing the Generalized Cross Validation function (GCV). If you write
\hat{y} = A(\lambda) y,
and call the A(\lambda) as the "hat" matrix, the GCV function V(\lambda) is defined as
V(\lambda) = \frac{(1/n) \Vert(I- A(\lambda) y\Vert^2}
 {[(1/n) {tr}(I- A(\lambda))]^2}.
The returned values from this function call will provide the \hat{y} as fitted, the (\beta,\delta) as coeff, and diag(A(\lambda)) as adiag.

To evaluate the TPSS fit f_\lambda(x) at new data points, you can use the TPSPLNEV call.

Suppose X new, a m ×k matrix, contains the m new data points at which you want to evaluate f_{\lambda}. Let (T^{new}_{ij}) = \phi_j(x^{new}_i) and (K newij) = E2(x newi-xj) be the ijth elements of T new and K new respectively. The prediction at new data points X new is
y_{pred}=T^{new} \beta + K^{new} \delta.
Therefore, using the coefficient (\beta,\delta) obtained from TPSPLINE call, the y pred can be easily evaluated.

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