**Wahba, G.**"*GCVPACK-Routines for Generalized Cross Validation*" TR 775(rev), D. Bates, M. Lindstrom, G. Wahba and B. Yandell. May 1987. Documentation for GCVPACK. Code available via netlib. An earlier version has appeared in Commun. Statist. Simul. Comput., 1987, v 16, pp263-297.

**Wahba, G.**"*Ill Posed Problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill Posed Problems With Noisy Data"*" TR 595. Prepared for the famous unpublished proceedings of the Delaware Conference on Ill Posed Inverse Problems

**Wahba, G.**"*Interpolating Surfaces: High Order Convergence Rates and Their Associated Designs, with Application to X-Ray Image Reconstruction*" TR 523, May 1978. The designs referred to where known then (in two dimensions) as blending function designs. Now they are mostly known as hyperbolic cross points. It will become evident why after you look at some of the plots. Modern work on these designs include work by Wasilkowski, Traub, Wasilewski, Novak and Ritter, among others. 44 page reportscanned into a pdf file.

**Wahba, G.**"*Estimating derivatives from outer space*" Mathematics Research Center TSR 989, May 1969. This paper solves a variational problem of the form: find $f$ to minimize a semi-norm in a reproducing kernel Hilbert space subject to the condition that the values of $f$ at specified points are within specified intervals. Can be viewed as a precursor to what is called "Support Vector Regression".

**Wahba, G.**"*Some tests for independence of stationary time series*" Stanford University Statistics Department TR 8, 1967. Expanded version published in 1971, see below.

**Bates, D., Reames, F. and Wahba, G.**"*Getting better contour plots with S and GCVPACK*" Computational Statistics & data Analysis 15 (1993), 329-342. Freeware code is presently available in R.

**Wahba, G.**"*Multivariate Function and Operator Estimation, Based on Smoothing Splines and Reproducing Kernels*" in Nonlinear Modeling and Forecasting, SFI Studies in the Sciences of Complexity, Proc. Vol. XII, Eds. M. Casdagli and S. Eubank, Addison-Wesley, 1992.

**C. Gu and G. Wahba**"*Minimizing GCV/GML Scores with Multiple Smoothing Parameters Via the Newton Method*" In SIAM J. Sci. Stat. Comput., 12(2), 383-398

**Wahba, G., Johnson, D. R, and Reames, F.**"*Multiple smoothing and weighting parameters in direct variational methods for objective analysis of meteorological information*" In "Assimilation of Observations in Meteorology and Oceanography", Proceedings of the WMO International Symposium, Clermont-Ferrand, 1990, F.- X. LeDimet and O. Talagrand, eds., WMO, 448-453 (1991).

**Wahba, G. and Wang, Y.**"*When is the optimal regularization parameter insensitive to the choice of the loss function?*" In Commun. Statist. - theory Meth., 19(5), 1685-1700

**Wahba, G.**"*On the dynamic estimtion of relative weights for observation and foreast in numerical weather prediction.*" This paper has appeared in "RSRM 87: Advances in Remote Sensing Retrieval Methods", A. Deepak, H. E. Fleming and J. S. Theon, eds., A. Deepak Publishing Co., Hampton, VA pp. 347-358 (1989)

**D. Cox, E.Koh, G. Wahba and B.Yandell**"*Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models.*" Ann. Statist. 16, 113-119 (1988).

**Wahba, G.**"*Three topics in Ill-posed Problems.*" This paper has appeared in "Inverse and Ill-Posed Problems, M. Engl and G. Groetsch, Eds., Academic Press 1987, pp 37-50. Among other things provides a discussion of how early stopping of iterative methods for solving large linear systems is a form of regularization.

**Wahba, G.**"*Partial spline models for the inclusion of tropopause and frontal boundary information in otherwise smooth two- and three-dimensional objective analysis.*" This paper appeared in J. Atmos. Ocean Tech, 3, 714-725.

**O'Sullivan, F. and Wahba, G.**"*A cross validated Bayesian retrieval algorithm for nonlinear remote sensing experiments.*" This paper has appeared in the Journal of Computational Physics. Provides a penalized likelihood algorithm for mildly nonlinear ill posed inverse problems, with the regularization parameter chosen by GCV. Applied to the retrieval of vertical temperature profiles from satellite radiance data.

**Wahba, G.**"*Design Criteria and Eigensequence Plots for Satellite-Comuted Tomography.*" This paper has appeared in the Journal of Atmospheric and Oceanic Technology, June 1985, pp 125-132. Proposes use of "degrees of freedom for signal" as a design criteria for comparing different designs, and describes the use of eigensequence plots as a tool in this process.

**Wahba, G.**"*A Comparison of GCV and GML for Choosing the Smoothing Parameter in the Generalized Spline Smoothing Problem*", Ann. Statist. 1985, 13, 4, 1378-1402

**Wahba, G.**"*Variational methods for multidimensional inverse problems.*" This paper has appeared in "Remote Sensing Retrieval Methods", A. Deepak, H. E. Fleming and M. T. Chahine, eds. A. Deepak publishing Co., pp. 385-408 (1985)

**D. Nychka, G. Wahba, S.Goldfarb and T. Pugh**"*Cross-Validated Spline Methods for the Estimation of Three-Dimensional Tumor Size Distributions From Two Dimensional Cross Sections.*" JASA 1984, 79, 388, pp 832-846. A Mildly Ill-posed Inverse Problem. Includes advice about computing the representers and their inner products.

**Wahba, G.**"*Surface Fitting with Scattered Noisy Data on Euclidean D-Space and on the Sphere*" Rocky Mountain Journal of Mathematics 14, 1, (1984) pp 261-299. Estimates divergence and vorticity and fits wind fields on the sphere

**Wahba, G.**"*Bayesian "Confidence Intervals" for the Cross-validated Smoothing Spline*" This paper has appeared in the Journal of the Royal Statistical Society, Series B (1983) pp133-150.

**Wahba, G.**"*Vector Splines on the Sphere, With Application to the Estimation of Vorticity and Divergence from Discrete, Noisy Data*" These vector splines may be used to estimate wind fields, and also to fit smooth transformations of the sphere onto itself, of possible application to the fitting of non-isotropic covariances. In 'Multivariate Approximation Theory II, W. Schempp and K. Zeller, eds. Birkhauser Verlag 1982. [This paper was scanned and then `Capture' was used to translate the scanned pdf into letters. It does not do equations, and occasionally it could not interpret a word. In that case the original scanned word is retained, and it looks funny. Please ignore.]

**Wahba, G.**"*Variational Methods in Simultaneous Optimum Interpolation and Initialization*" This paper has appeared in `The Interaction Between Objective Analysis and Initialization', Proceedings of the Fourteenth Stanstead Seminar, 1982. D. Williamson, Editor. It is primarily of historical interest, since the results are by now widely understood. It may be one of the earliest papers in the meteorological literature to relate the relationshop between variational methods and Bayes estimates.

**Wahba, G.**"*Constrained Regularization For Ill Posed Linear Operator Equations, With Applications in Meteorology and Medicine*" This paper appeared in "Statistical Decision Theory and Related Topics III", v2, 1982. Shanti S. Gupta and James O. Berger, Eds.

**Wahba, G.**"*Numerical experiments with the thin plate histospline*" This paper has appeared in Commun. Statist. Theor. Meth., A10(24), 2475-2514 (1981). It has application to the problem of fitting models for data that has been aggregated over irregular geographical regions such as counties or states. You will need acrobat to read it.

**Wahba, G.**"*Some new techniques for variational objective analysis on the sphere using splines, Hough functions, and sample spectral data.*" This paper has appeared in Preprints of the Seventh Conference on Probability and Statistics in the Atmospheric Sciences, American Meteorological Society, November 1981.

**Wahba, G.**"*Spline Interpolation and Smoothing on the Sphere*" This paper has appeared in SIAM J. Sci. Stat. Comput, (2), 5-16(1981) and an important errata sheet [(3),385-6,(1982)] is attached at the end of the scanned pdf file. The splines on the sphere have power law decay in their energy distributions.

**Wahba, G.**"*A New Approach to the Numerical Evaluation of the Inverse Radon Transform with Discrete, Noisy Data,*" TR 612, July 1980. Has appeared in Mathematical Aspects of Computerized Tomograpy, G. T. Herman and F. Natterer, eds. Lecture notes in Medical Informatics, 189-203, Springer-Verlag (1981) Typo: The first line on p 7 should begin "where k_\epsilon(u) = 1 if" ...

**Wahba, G., and Wendelberger, J.**"*Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation*" This paper has appeared in the Monthly Weather Review, (108), 1122-1143 (1980). Proposes the cross validated thin plate spline for smoothing noisy data on the plane.

**Wahba, G.**"*Automatic Smoothing of the Log Periodogram*" This paper has appeared in J. Amer. Statist. Assoc. 75, March 1980, 122-132.

**Golub, G., and Heath, M. and Wahba, G.**"*Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter.*" This paper has appeared in Technometrics, 21, 215-223 (1979).

**Craven, P. and Wahba, G.**"*Smoothing Noisy Data with Spline Functions*" This paper has appeared in Numer. Math 31, 377-403 (1979).

**Athavale, M. and Wahba, G.**"*Determination of an optimal mesh for a collocation-projection method for solving two-poit boundary value problems*", J. Approx. Theory 25,1 (1979) 38-49.

**Wahba, G.**"*Improproper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression*", J. Roy. stat. Soc.B, 1978 40, 3, 364-372

**Wahba, G.**"*Practical Approximate Solutions to Linear Operator Equations When the Data Are Noisy*", SIAM J. Numer. Anal., 14, 4 (1977) 651-667. This very early paper describes the key role of reproducing kernels in the regularized solution of ill-posed inverse problems. What amounts to the GCV estimate of the smoothing parameter lambda is discussed (but called "weighted cross validation" there), and various convergence rates when using this lambda are given. It is shown how the rates are related to the rate of decay of the eigenvalues of the reproducing kernel.

**Nashed, Z. and Wahba, G.**"*Regularization and Approximation of Linear Operator Equations in Reproducing Kernel Spaces*", Bull. Amer. Math. Soc., 80, 6 (1974) 1213-1218.

**Nashed, Z. and Wahba, G.**"*Generalized Inverses in in Reproducing Kernel Spaces: An approach to regularization of linear operator equations.*", SIAM J. Math. Anal. 5,6, 1974.

**Wahba, G.**"*Regression Design for Some Equivalence Classes of Kernels*", Ann. Stat. 1974, 2, 925-934.

**Wahba, G.**"*Convergence rates of certain approximate solutions to linear operator equations*", J. Approximation Theory 7, 167-185(1973) .

**Wahba, G.**"*A class of approximate solutions to linear operator equations*", J. Approximation Theory 9, 61-77 (1973).

**Kimeldorf, G. and Wahba, G.**"*Some results on Tchebycheffian Spline Functions*", J. Mathematical Analysis and Applications, 33, 1 (1971) 82-95. This paper describes the form of the solution of a class of variational problems in RKHS when the observables are related to bounded linear functionals in the space (including evaluation functionals)! [The proof of the "representer theorem".] A connection with Bayes estimates and variational problems in RKHS is also noted.

**Wahba, G.**"*On the Regression Design Problem of Sacks and Ylvisaker*", Ann. Math Stat 1971, 42, 1035-1053.

**Wahba, G.**"*Some Tests of Independence for Stationary Multivariate Time Series*", J. Roy. Statist. Soc. 1971, 33, 1, 153-166

**Kimeldorf, G. and Wahba, G.**"*A Correspondance Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines.*", Ann. Math Stat 1970, 41, 2,495-502.

**Wahba, G.**"*Estimation of the coefficients in a distributed lag model*", Econometrica 37, 398-407 (1969). "*Printer friendly copy.*",

**Wahba, G.**"*On the Distribution of Some Statistics Useful in the Analysis of Jointly Stationary Time Series*", Ann. Math. Stat. 1968, 39, 6, 1849-1862

**Wahba, G.**"*A Least Squares Estimate of Satellite Attitude*", SIAM Review, 8, 3, (1966) 384-386. This problem, with solution, was proposed to the SIAM Review by G. Wahba. In keeping with their custom, several solutions other than that by the proposer were printed, beginning with one by Farrell and Stuelpnagel. It is known in the aeronautics literature as "Wahba's Problem".