Grace Wahba

Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter

Consider the ridge estimate (λ) for β in the model unknown, (λ) = (X T X + nλI)−1 X T y. We study the method of generalized cross-validation (GCV) for choosing a good value for λ from the data. The estimate is the minimizer of V(λ) given by where A(λ) = X(X T X + nλI)−1 X T . This estimate is a rotation-invariant version of Allens PRESS, or ordinary cross-validation. This estimate behaves like a risk improvement estimator, but does not require an estimate of σ2, so can be used when n − p is small, or even if p ≥ 2 n in certain cases. The GCV method can also be used in subset selection and singular value truncation methods for regression, and even to choose from among mixtures of these methods.

SOME RESULTS ON TCHEBYCHEFFIAN SPLINE FUNCTIONS.

Abstract This report derives explicit solutions to problems involving Tchebycheffian spline functions. We use a reproducing kernel Hilbert space which depends on the smoothness criterion, but not on the form of the data, to solve explicitly Hermite-Birkhoff interpolation and smoothing problems. Sards best approximation to linear functionals and smoothing with respect to linear inequality constraints are also discussed. Some of the results are used to show that spline interpolation and smoothing is equivalent to prediction and filtering on realizations of certain stochastic processes.

Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy

We consider approximate solutions

Multicategory support vector machines: Theory and application to the classification of microarray data and satellite radiance data

f_{n,\lambda }

Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation

to linear operator equations

A completely automatic french curve: fitting spline functions by cross validation

\mathcal{K}f = g

Support Vector Machines for Classification in Nonstandard Situations

, of the form:

Spline Interpolation and Smoothing on the Sphere

f_{n,\lambda }

Automatic Smoothing of the Log Periodogram

is the minimizer in

Hybrid Adaptive Splines

\mathcal{H}