Zentralblatt für Mathematik Mathematics Abstracts (c) FIZ Karlsruhe & Springer-Verlag Your query: au=(Wahba_G*) The database contains the following 1-40 best item(s) out of 73 relevant to your query. You may search for more documents relevant to your query at the end of this page. MATH Help We need YOUR feedback! Please send your comments to: math-db@zblmath.fiz-karlsruhe.de 786.62048 Gu, Chong; Wahba, Grace : Semiparametric analysis of variance with tensor product thin plate splines. ( English ) J. R. Stat. Soc., Ser. B 55, No.2, 353-368 (1993). Classification *62G07 Curve estimation 62J10 Analysis of variance, etc. 62M30 Statistics of spatial processes 46N30 Appl. of functional analysis in probability theory and statistics Keywords model selection; reproducing kernel; spatial data analysis; spline analysis-of-variance decomposition; tensor product smoothing splines; polynomial smoothing spline ANOVA models; thin plate smoothing splines We consider fitting multivariate models to response data based on analysis-of-variance (ANOVA)-like decompositions of functions of several variables, $$f(t\sb 1,\dots,t\sb d)=C+\Sigma\sb \alpha f\sb \alpha(t\sb \alpha)+\Sigma\sb{\alpha<\beta}f\sb{\alpha\beta}(t\sb \alpha,t\sb \beta)+ \dots.$$ A theory for fitting (some components of) these models with polynomial smoothing splines exists when each $t\sb \alpha$ is in a subset of the real line. In this case the various estimated components turn out to be certain tensor sums and products of polynomial splines. This approach may not be natural when one or more of the `variables' are geographic, in particular where nature does not know north from east. In this case splines of radial structure, such as thin plate splines, for the geographic component, are more natural.\par We extend this theory of polynomial smoothing spline ANOVA models, to include variables which take values in Euclidean $k$-space, and fits which turn out to include sums and products of both polynomial and thin plate smoothing splines. The cases of most interest would be $k=2$ and $k=3$. The formulation, interpretation and calculation of the models are discussed, and an application of the technique is illustrated. This work can be used to build predictive ANOVA-like models which describe a response as a function of spatial, temporal and other variables and to explore their interactions. The models can be fitted by using the existing publicly available code RKPACK. Publ. Year: 1993 Document Type: Journal Search for entries citing this one. View this entry as 930.07832 Gu, Chong; Heckman, Nancy; Wahba, Grace : A note on generalized cross-validation with replicates. ( English ) Stat. Probab. Lett. 14, No.4, 283-287 (1992). Classification *62G07 Curve estimation Publ. Year: 1992 Document Type: Journal Search for entries citing this one. View this entry as 734.41015 Wahba, Gace : Multivariate model building with additive interaction and tensor product thin plate splines. ( English ) Curves and surfaces, Pap. Int. Conf., Chamonix-Mont-Blanc/Fr. 1990, 491- 504 (1991). Classification *41A15 Spline approximation [For the entire collection see Zbl.729.00010.] \par Citations: Zbl729.00010 Publ. Year: 1991 Document Type: Conference article Search for entries citing this one. View this entry as 727.65009 Gu, Chong; Wahba, Grace : Minimizing GCV/GML scores with multiple smoothing parameters via the Newton method. ( English ) SIAM J. Sci. Stat. Comput. 12, No.2, 383-398 (1991). Classification *65D10 Smoothing 65K10 Optimization techniques (numerical methods) 65Y20 Complexity and performance of numerical algorithms Keywords variance estimation; Newton-type algorithm; multiple smoothing parameters; generalized cross validation; generalized maximum likelihood; speed; reliability; invariance; efficiency; complexity; starting values; fitting; additive/interaction spline models A (modified) Newton-type algorithm is presented to optimize the multiple smoothing parameters according to the generalized cross validation (GCV) and generalized maximum likelihood (GCV) criterion. The algorithm generalizes the single smoothing parameter method by the authors, {\it D. M. Bates}, and {\it Z. Chen} [SIAM J. Matrix Anal. Appl. 10, 457-480 (1989; Zbl.685.65134)] into the multiple smoothing parameters setting. The speed, reliability, invariance under transformations under which the problem is invariant, efficiency and complexity of the algorithm are discussed. \par Two sensible good starting values approaches reducing the execution loads are proposed. The algorithm's implementation is tested on fitting the additive/interaction spline models. Its applicability to the maximum likelihood and the threshold maximum likelihood estimation of the variance component models is noted. L.Bakule (Praha) Citations: Zbl685.65134 Publ. Year: 1991 Document Type: Journal Search for entries citing this one. View this entry as 813.62001 Wahba, Grace : Spline models for observational data. ( English ) CBMS-NSF Regional Conference Series in Applied Mathematics. 59. Philadelphia, PA: SIAM, (ISBN 0-89871-244-0). XII, 169 p. (1990). Classification *62-02 Research monographs (statistics) 62G07 Curve estimation 65U05 Numerical methods in probability and statistics 65D07 Splines (numerical methods) 65R20 Integral equations (numerical methods) 62G20 Nonparametric asymptotic efficiency 45B05 Fredholm integral equations Keywords equivalence; ordinary cross validation; leaving-out-one lemma; generalized cross validation; convergence rates; confidence intervals; quadrature formulae; regression splines; penalized GLIM models; linear inequality constraints; constrained nonlinear optimization; variational problems; optimization; reproducing kernel Hilbert spaces; derivatives; smoothing; stochastic processes; Bayes estimates; iterated Laplacian; thin-plate spline; perpendicularity of probability measures; kriging; generalized maximum likelihood estimate; bootstrapping; partial spline estimates; regularization; ill-posedness; nonlinear integral equations; loss functions; predictive mean-square error; nonlinear regression; log- likelihood ratio; identification; bibliography Statisticians are generally interested in smoothing data of the form $$y\sb i= f(x\sb i)+ \varepsilon\sb i, \qquad i=1,2,\dots, n,$$ where $\varepsilon\sb i$ are random disturbances and $f$ is only known to be ``smooth''. A formal statement of the problem may be formulated as follows: find $f$ in a given class of smooth functions $W$ on the interval $(a,b)$ to minimize (for some $\lambda>0$) $${1\over n} \sum\sb{i=1}\sp n (y\sb i- f(x\sb i))\sp 2+ \lambda \int\sb a\sp b (f\sp{(m)} (x))\sp 2 dx.$$ All splines considered in the book are solutions to variational problems. The variational problems are treated from a unified point of view as optimization problems in reproducing kernel Hilbert spaces and it is assumed that the reader has a knowledge of the basic properties of Hilbert spaces. \par Contents: 1. Background (1.1. Positive-definite functions, covariances, and reproducing kernels; 1.2. Reproducing kernel spaces on $[0,1]$ with norms involving derivatives; 1.3. The special and general spline smoothing problems; 1.4. The duality between reproducing kernel Hilbert spaces and stochastic processes; 1.5. The smoothing spline and the generalized smoothing spline as Bayes estimates).\par 2. More splines (2.1. Splines on the circle; 2.2. Splines on the sphere, the role of the iterated Laplacian; 2.3. Vector splines on the sphere; 2.4. The thin-plate spline on $E\sp d$; 2.5. Another look at the Bayes model behind the thin-plate spline). \par 3. Equivalence and perpendicularity, or What's So Special About Splines? (3.1. Equivalence and perpendicularity of probability measures; 3.2. Implications for kriging). \par 4. Estimating the smoothing parameter (4.1. The importance of a good choice of $\lambda$; 4.2. Ordinary cross validation and the ``leaving-out- one'' lemma; 4.3. Generalized cross validation (GCV); 4.4. Properties of the GCV estimate of $\lambda$; 4.5. Convergence rates with the optimal $\lambda$; 4.6. Other estimates of $\lambda$ similar to GCV; 4.7. More on other estimates; 4.8. The generalized maximum likelihood estimate of $\lambda$; 4.9. Limits of GCV). \par 5. Confidence intervals (5.1. Bayesian confidence intervals; 5.2. Estimate-based bootstrapping). \par 6. Partial spline models (6.1. Estimation; 6.2. Convergence of partial spline estimates; 6.3 Testing). \par 7. Finite-dimensional approximating subspaces (7.1. Quadrature formulae, computing with basis functions; 7.2. Regression splines). \par 8. Fredholm integral equations of the first kind (8.1. Existence of solutions, the method of regularization; 8.2. Further remarks on ill- posedness; 8.3. Mildly nonlinear integral equations; 8.4. The optimal $\lambda$ for loss functions other than predictive mean-square error). \par 9. Further nonlinear generalizations (9.1. Partial spline models in nonlinear regression; 9.2. Penalized GLIM models; 9.3. Estimation of the log-likelihood ratio; 9.4. Linear inequality constraints; 9.5. Inequality constraints in ill-posed problems; 9.6. Constrained nonlinear optimization with basis functions; 9.7. System identification). \par 10. Additive and interaction splines (10.1. Variational problems with multiple smoothing parameters; 10.2. Additive and interaction smoothing splines). \par 11. Numerical methods. \par 12. Special topics (12.1. The notion of ``high frequency'' in different spaces; 12.2. Optimal quadrature and experimental design). \par The bibliography contains more than three hundred items. The book is based on a series of 10 lectures at Ohio State University at Columbus in March 23-27, 1987. R.Zielinski (Warszawa) Publ. Year: 1990 Document Type: Book Search for entries citing this one. View this entry as 724.62044 Wahba, Grace; Wang, Yonghua : When is the optimal regularization parameter insensitive to the choice of the loss function? ( English ) Commun. Stat., Theory Methods 19, No.5, 1685-1700 (1990). Classification *62G07 Curve estimation 62G99 Nonparametric inference 65R20 Integral equations (numerical methods) Keywords convergence rates; deconvolution; optimal smoothing parameter; Fredholm integral equation; optimal regularization parameter; first kind integral equations with noisy data; mean square error; penalty functional; generalized cross-validation We investigate the behavior of the optimal regularization parameter in the method of regularization for solving first kind integral equations with noisy data, under a range of definitions of ``optimal'', varying from mean square error in higher derivatives of the solution, to mean square error in the predicted data. We study how the optimal regularization parameter changes when the optimality criteria changes under a broad range of smoothness assumptions on the solution, the kernel of the integral operator, and the penalty functional. Although some of the calculations we present have been given elsewhere, we organize the results with a specific goal in mind. That is, we study a certain class of problems within which we can identify conditions on the solution, the kernel of the operator and the penalty functional for which the rate at which the optimal regularization parameter goes to zero is the same for both predictive mean square error and solution mean square error optimality criteria, and for which it is different. The former circumstances are of interest because then data based estimates of the regularization parameter such as generalized cross-validation, which are known to be optimal for predictive mean square error, will also go to zero at the optimal rate for solution mean square error. Publ. Year: 1990 Document Type: Journal Search for entries citing this one. View this entry as 704.62054 Wahba, G. : Partial and interaction spline models. ( English ) Bayesian statistics 3, Proc. 3rd Valencia Int. Meet., Altea/Spain 1987, 479-491 (1988). Classification *62J99 Linear statistical inference Keywords semiparametric regression; partial spline model; Gaussian errors; partially penalized GLIM models; Interaction splines; nonlinear interactions [For the entire collection see Zbl.702.00028.] \par A partial spline model is a model for a response as a function of variables, which is the sum of a ``smooth'' function of several variables and a parametric function of the same plus possibly some other variables. Partial spline models in one and several variables, with direct and indirect data, with Gaussian errors and as an extension of GLIM to partially penalized GLIM models are described. Application to the modelling of change of regime in several variables is described. Interaction splines are introduced and described and their potential use for modelling nonlinear interactions between variables by semiparametric methods is noted. Reference is made to recent work in efficient computational methods. Citations: Zbl702.00028 Publ. Year: 1988 Document Type: Conference article Search for entries citing this one. View this entry as 685.65134 Gu, Chong; Bates, Douglas M.; Chen, Zehua; Wahba, Grace : The computation of generalized cross-validation functions through Householder tridiagonalization with applications to the fitting of interaction spline models. ( English ) SIAM J. Matrix Anal. Appl. 10, No.4, 457-480 (1989). Classification *65U05 Numerical methods in probability and statistics 65D10 Smoothing 65F20 Overdetermined systems (numerical linear algebra) 65D07 Splines (numerical methods) 41A65 Abstract approximation theory 60G60 Random fields 62J05 Linear regression Keywords fitting; interaction splines; distributed truncation; algorithm; generalized cross-validation function; cross-validated regularization; Householder tridiagonalization; multivariate smoothing; remote sensing problems; Comparisons; numerical examples An algorithm for computing the generalized cross-validation function for the general cross-validated regularization or smoothing problem is provided. It is based on the Householder tridiagonalization of the matrix. A distributed truncation strategy is proposed that may speed up the tridiagonalization process. \par This algorithm is appropriate for certain multivariate smoothing problems with irregularly spaced data, and certain remote sensing problems, such as those that occur in meteorology. Comparisons with other algorithms are given. Applications and numerical examples illustrate the algorithm. Some remarks on other applications and further studies are offered. G.R.Grozev Publ. Year: 1989 Document Type: Journal Search for entries citing this one. View this entry as 695.62085 Shiau, Jyh-Jen Horng; Wahba Grace : Rates of convergence of some estimators for a semiparametric model. ( English ) Commun. Stat., Simulation Comput. 17, No.4, 1117-1133 (1988). Classification *62F10 Point estimation 62G05 Nonparametric estimation Keywords smoothing spline; generalized-cross-validation; rates of convergence; partial spline; semiparametric model We analyze the rates of convergence of the mean square error of a partial spline estimator and a Denby-Speckman-type estimator for the parametric component of a semiparametric model under a wide range of conditions. It is found that the Denby-Speckman-type estimator has a faster rate of convergence than the partial spline estimator for some cases. It is shown that the optimal rate of decrease for the smoothing parameter $\lambda$ under the criterion of minimizing MSE for the parametric component is not necessarily the same as the optimal rate for minimization of the predictive mean square error in the function estimate. Thus data based estimates for $\lambda$ optimal for predictive mean square error in the function, such as GCV, may not be optimal for mean square error in the parametric component, leaving open the question of a data based estimate for $\lambda$ in this context. Publ. Year: 1988 Document Type: Journal Search for entries citing this one. View this entry as 673.62017 Cox, Dennis; Koh, Eunmee; Wahba, Grace; Yandell, Brian S. : Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. ( English ) Ann. Stat. 16, No.1, 113-119 (1988). Classification *62F03 Parametric hypothesis testing 62G10 Nonparametric hypothesis testing 62H15 Multivariate hypothesis testing Keywords parametric versus smooth models; polynomial smoothing spline estimates; uniformly most powerful test; locally most powerful (LMP) test; generalized smoothing spline models; partial spline models; generalized cross validation; residual sum of squares {\it D. R. Cox} and {\it E. Koh} [A smoothing spline based test of model adequacy in polynomial regression. Tech. Rep. 787, Dept. Statistics, Univ. Wisconsin-Madison (1986)] considered the model $y\sb i=f(x(i))+\epsilon\sb i$, $\epsilon\sb i$ i.i.d. $N(0,\sigma\sp 2)$, with the (parametric) null hypothesis f(x), $x\in [0,1]$, a polynomial of degree m-1 or less, versus the alternative f is ``smooth'', based on the Bayesian model for f which leads to polynomial smoothing spline estimates for f. They showed that there was no uniformly most powerful test and found the locally most powerful (LMP) test. We extend their result to generalized smoothing spline models and to partial spline models. \par We also show that the test statistic has an intimate relationship with the behavior of the generalized cross validation (GCV) function at $\lambda =\infty$. If the GCV function has a minimum at $\lambda =\infty$, then GCV has chosen the (parametric) model corresponding to the null hypothesis; we show that if the LMP test statistic is no larger than a certain multiple of the residual sum of squares after (parametric) regression, then the GCV function will have a (possibly local) minimum at $\lambda =\infty$. Publ. Year: 1988 Document Type: Journal Search for entries citing this one. View this entry as 638.65048 Wahba, Grace : Three topics in ill-posed problems. ( English ) Inverse and ill-posed problems, Alpine-U.S. Semin. St. Wolfgang/Austria 1986, Notes Rep. Math. Sci. Eng. 4, 37-51 (1987). Classification *65J10 Equations with linear operators (numerical methods) 65F10 Iterative methods for linear systems 65R20 Integral equations (numerical methods) 65P05 Numerical methods for miscellaneous problems of PDE 45B05 Fredholm integral equations 45L10 Numerical approximation of solutions of integral equations 35R30 Inverse problems for PDE Keywords inverse problems; cross validation; ill-posed problems; Tikhonov regularization; Richardson/Landweber/Fridman/Picard/Cimmino iterative method(s); large ill-conditioned linear systems [For the entire collection see Zbl.623.00010.] \par Three topics in ill-posed problems are discussed. The first topic concerns the imposition of specified types of discontinuities on otherwise smooth two and three dimensional solutions of ill-posed problems. The work has application to the recovery of the three dimensional atmospheric temperature distribution from satellite observed radiances. It is based on the cross validated penalized likelihood approach to Tikhonov regularization by {\it F. O'Sullivan} and the author [J. Comput. Phys. 59, 441-455 (1985)]. The method proposed in this paper for imposing specified discontinuities on the solution given discrete, noisy values on functionals of the solution can be expected to have application in a number of problems in the atmospheric and geological sciences, for example, to the estimation of the three dimensional temperature distribution of the ocean, including the thermocline, and to the estimation of certain geological properties, e.g. density, transmittivity, etc. \par The second topic concerns the Richardson/Landweber/Fridman/Picard/Cimmino iterative method(s) for iterative solutions of large linear systems. It has been observed by a number of workers that, when using certain iterative methods for solving large ill-conditioned linear systems, the approximation to the solution appears to improve with iteration up to a certain point, when further iteration begins to degrade the solution. A rough explanation for this phenomena is as follows: The first few iterations are recovering the projection of the solution onto singular vectors corresponding to large singular values. As the iteration proceeds, recovery of projections on singular vectors with small singular values begins, but recovery of these (high frequency) components is increasingly sensitive to noise in the data, either measurement error, or roundoff, as the singular values become smaller. A quantitative analysis of this phenomena shows that generalized cross validation (GCV) can be used to provide a data-based rule for choosing the point at which to stop the iteration, when the source of noise is random measurement error on the right hand side. \par The third topic concerns the parameter estimation problem for p.d.e.'s with particular application to the reservoir modelling problem. One has the equation (for example) $$ \frac{\partial u}{\partial t}=\sum\sp{3}\sb{l=1}\frac{\partial}{\partial x\sb l}(\alpha (x)\frac{\partial u}{\partial x\sb l})+q(x,t),\ x=(x\sb 1,x\sb 2,x\sb 3)\in \Omega,\ t\in [0,T], $$ where the forcing function q is known exactly and u is measured with noise at a finite set of points in x and t. The problem is to estimate the transmittivity $\alpha$, which is not necessarily a constant. This problem is both nonlinear and in general ill- posed. The degree of ill-posedness is a particular tricky question since it can be very sensitive to the ``true'' u. Conjectural remarks are made on the possibility of solving the penalized likelihood equations subject to linear inequality constraints based on physical information and using the GCV to choose the smoothing parameter in the resulting constrained implicit optimization problem. G.Wahba Citations: Zbl623.00010 Publ. Year: 1987 Document Type: Conference article Search for entries citing this one. View this entry as 618.62004 Bates, Douglas M.; Lindstrom, Mary J.; Wahba, Grace; Yandell, Brian S. : GCVPACK-routines for generalized cross validation. ( English ) Commun. Stat., Simulation Comput. 16, 263-297 (1987). Classification *62-04 Machine computation, programs (statistics) 65U05 Numerical methods in probability and statistics 62-07 Data analysis (statistics) 65-04 Machine computation, programs (numerical analysis) 65D10 Smoothing Keywords truncated singular value decomposition; semi-parametric models; Fortran-77 subroutines; generalized cross-validation; data smoothing; ridge regression; thin plate smoothing splines; deconvolution; smoothing of generalized linear models; ill-posed problems; GCV; regularization These Fortran-77 subroutines provide building blocks for generalized cross-validation (GCV) calculations in data analysis and data smoothing including ridge regression, thin plate smoothing splines, deconvolution, smoothing of generalized linear models, and ill-posed problems. We present some of the types of problems for which GCV is a useful method of choosing a smoothing or regularization parameter and we describe the structure of the subroutines. Publ. Year: 1987 Document Type: Journal Search for entries citing this one. View this entry as 614.62047 Villalobos, Miguel; Wahba, Grace : Inequality-constrained multivariate smoothing splines with application to the estimation of posterior probabilities. ( English ) J. Am. Stat. Assoc. 82, 239-248 (1987). Classification *62G05 Nonparametric estimation 65D07 Splines (numerical methods) 65U05 Numerical methods in probability and statistics 62H99 Multivariate analysis Keywords constrained surface estimation; estimating a smooth function; linear inequality constraints; thin-plate penalty functional; Hilbert space; nonlinear programming; thin-plate spline; generalized cross-validation; smoothing parameter; computational algorithm; estimation of posterior probabilities; classification problem We consider the problem of estimating a smooth function of several variables given discrete, scattered, noisy observations of its values, and given that it satisfies a family of linear inequality constraints. Constraints such as positivity over some region are included. The model is $z\sb i=f(y\sb 1(i),...,y\sb d(i))+\epsilon\sb i$ $(i=1,...,n)$, where the $\epsilon\sb i's$ are independent zero mean random variables, f is assumed to be smooth, with smoothness defined in terms of square integrability of certain derivatives, and f is known to satisfy a given set of linear inequality constraints. It is proposed that f be estimated as the minimizer, in an appropriate function space, of $$ (1)\ (1/n)\sum\sp{n}\sb{i=1}(z\sb i-f(y\sb 1(i),...,y\sb d(i)))\sp 2+\lambda J\sb m(f), $$ subject to f satisfying the constraints if they are a finite family, or satisfying a finite approximating set of constraints if they are not a finite family, where $J\sb m(f)$ is the thin-plate penalty functional defined by $$ J\sb m(f)=\sum\sb{\alpha\sb 1+...+\alpha\sb d=m}(m!/\alpha\sb 1!...\alpha\sb d!)\int...\int [\partial\sp mf/\partial y\sb 1\sp{\alpha\sb 1}...\partial y\sb d\sp{\alpha\sb d}]\sp 2dy\sb 1...dy\sb d. $$ More generally, the results apply to the model (2) $z\sb i=L\sb if+\epsilon\sb i$, $i=1,...,n$, where the $L\sb i's$ are bounded linear functionals on an appropriate Hilbert space, and the inequality constraints are of the form (3) $N\sb jf\le r\sb j$, $j=1,...,k$, with the $N\sb j$ also bounded linear functionals. Data and constraints involving function values, integrals, and some derivatives can be included in (2) and (3). Following {\it G. S. Kimeldorf} and the second author [J. Math. Anal. Appl. 33, 82-95 (1971; Zbl. 185, 309)], we characterize the minimizer of (1) when the constraints are finite in number, and we provide a representation for the minimizer that can be computed using available nonlinear programming algorithms. \par When the data as well as the constraints involve function values, the minimizer is shown to be a thin-plate spline. The method of generalized cross-validation for constrained problems can be used here for estimating a good value of the smoothing parameter $\lambda$, and an efficient computational algorithm for it is developed in this article. \par As an example the method is applied to the estimation of posterior probabilities in the classification problem. Numerical results and figures for both synthetic and experimental data are given. Citations: Zbl.185.309 Publ. Year: 1987 Document Type: Journal Search for entries citing this one. View this entry as 649.65013 Wahba, Grace : Multivariate thin plate spline smoothing with positivity and other linear inequality constraints. ( English ) Statistical image processing and graphics, Stat., Textb. Monogr. 72, 275- 289 (1986). Classification *65D10 Smoothing 65D07 Splines (numerical methods) 93E11 Filtering in stochastic control 41A15 Spline approximation Keywords multivariate thin plate spline; smoothing; linear inequality constraints; smoothing splines; butterworth filters; low pass filters; scattered noisy discrete non-Gaussian data Author's summary: The relationship between smoothing splines and butterworth filters, both on the line and on the plane, is reviewed. Splines may be thought of as the extension of low pass filters to noisy, unequally spaced data from non periodic functions. Furthermore, linear inequality constraints such as nonnegativity and monotonicity may be imposed on the spline. Numerical results in the smoothing of scattered noisy discrete non-Gaussian data on the plane subject to the constraint that the smoothed surface represent a probability (i.e., be between 0 and 1) are given. These results provide a method of estimating posterior probabilities in the classification problem, for bivariate data. W.Boehm Publ. Year: 1986 Document Type: Conference article Search for entries citing this one. View this entry as 614.76047 Wahba, Grace : Partial spline modelling of the tropopause and other discontinuities. ( English ) Function estimates, Proc. Conf., Arcata/Calif. 1985, Contemp. Math. 59, 125-135 (1986). Classification *76E20 Instability of flows in nature 76E15 Convective instability (fluid mechanics) 80A20 Heat and mass transfer 86A10 Meteorology 65D07 Splines (numerical methods) Keywords partial splines; three dimensional atmospheric temperature distribution; location of the tropopause; nonlinear inequality constraints [For the entire collection see Zbl.596.00015.] \par We show how surfaces (distributions) in two and three dimensions which are smooth except for specified types of discontinuities may be modelled with the use of partial splines. Using the partial spline model one may then estimate the distribution given scattered, noisy, direct or indirect observations, and the resulting estimate will possess the assumed type of discontinuity. \par The motivation for this work is the estimation of the three dimensional atmospheric temperature distribution given direct and indirect measurements of temperature and the location of the tropopause. The tropopause is modelled as jump of known location (but unknown size) in the vertical first derivative. Side information in the form of linear and nonlinear inequality constraints may be incorporated in the estimate. The approach described here should be applicable to other two and three dimensional imaging problems when the unknown distribution has discontinuities of known location but unknown magnitude. Citations: Zbl596.00015 Publ. Year: 1986 Document Type: Conference article Search for entries citing this one. View this entry as 626.65053 O'Sullivan, Finbarr; Wahba, Grace : A cross validated Bayesian retrieval algorithm for nonlinear remote sensing experiments. ( English ) J. Comput. Phys. 59, 441-455 (1985). Classification *65J15 Equations with nonlinear operators (numerical methods) 65D10 Smoothing 65U05 Numerical methods in probability and statistics 86A99 Miscellaneous topics in geophysics 86A10 Meteorology Keywords Gauss-Newton iteration; satellite meteorology; regularization; ill-posed problems; least squares; method of generalized cross-validation; remote sensing problem of atmospheric temperature profiles The paper extends the method of generalized cross-validation to nonlinear problems of the form $Y\sb i=N(\theta,x\sb i)+\epsilon\sb i,$ $i=1,...,n$. Here, $Y\sb i$ are the measurements, $x\sb i$ the design points, $\epsilon\sb i$ measurement errors, and $\theta$ the parameter to be estimated. An application to a remote sensing problem of atmospheric temperature profiles is described in detail. F.Natterer Publ. Year: 1985 Document Type: Journal Search for entries citing this one. View this entry as 596.65004 Wahba, Grace : A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. ( English ) Ann. Stat. 13, 1378-1402 (1985). Classification *65D10 Smoothing 65R20 Integral equations (numerical methods) Keywords maximum likelihood estimate; generalized cross validation estimate; spline smoothing; Sobolev space; smoothing parameter; Monte Carlo methods The special spline smoothing model to be considered is given by $y\sb i=f(t\sb i)+\epsilon\sb i$, $i=1,...,n$, $t\sb i\in [0,1]$, where $\epsilon =(\epsilon\sb 1,...,\epsilon\sb n)\sim N(0,\sigma\sp 2I\sb{n\times n})$, $\sigma\sp 2$ is unknown, and f($\cdot)$ is some function in the Sobolev space $W\sb 2\sp m[0,1]=\{f:$ $f,f\sp 1,...,f\sp{(m-1)}$ absolutely continuous, $f\sp{(m)}\in L\sb 2[0,1]\}$ and the smoothing spline estimate $f\sb{n,\lambda}$ of f is the minimizer in $W\sp m\sb 2[0,1]$ of $$ 1/n\sum\sp{n}\sb{i=1}(f(t\sb i)-y\sb i)\sp 2+\lambda \int\sp{1}\sb{0}(f\sp{(m)}(t))\sp 2dt. $$ A generalization of the maximum likelihood (GML) estimate for the smoothing parameter $\lambda$ is obtained, and this estimate is compared with the generalized cross validation (GCV) estimate both analytically and by Monte Carlo methods. The theoretical results are shown to extend to the generalized spline smoothing model, which includes the estimate of functions given noisy values of various integrals of them. Y.Sun Publ. Year: 1985 Document Type: Journal Search for entries citing this one. View this entry as 593.65013 Wahba, Grace : Cross validated spline methods for direct and indirect sensing experiments. ( English ) Statistical signal processing, Stat., Textbooks Monogr. 53, 179-190 (1984). Classification *65D10 Smoothing 65J05 General theory of numerical methods in abstract spaces 65U05 Numerical methods in probability and statistics 65-02 Research monographs (numerical analysis) Keywords cross validated spline methods; direct and indirect sensing; experiments; data analysis techniques; Hilbert space; data functionals; penalty function; smoothing parameter [For the entire collection see Zbl.547.00060.] \par The author has been developing data analysis techniques for a fairly general class of (direct and indirect) sensing experiments. The model is described as follows: f is some unknown function of one or more variables assumed to be in a Hilbert space H of "smooth" functions, and one observes $\{y\sb i\}$ $y\sb i=L\sb if+\epsilon\sb i$, $i=1,2,...,n$ where the data functionals $L\sb 1,...,L\sb n$ are n bounded linear functionals on H and $\epsilon\sb i$ are independent, zero mean measurement errors, with variances $w\sb i\sigma\sp 2$, $i=1,2,...,n$. The parameter $\sigma\sp 2$ may be unknown. \par The general approach to the estimation of f which is developed and tested goes as follows: the estimate, call it f, is the solution of the minimization problem: find $f\in H$ to minimize $$ 1/n\sum\sp{n}\sb{i=1}(y\sb i-L\sb if)\sp 2/w\sb i+\lambda J\sb{m,\sigma}(f), $$ subject to $f\in C$. Here $J\sb{m,\sigma}$ is a seminorm on H, or "penalty function" indexed by the parameters m and $\sigma$, and $\lambda$ is the bandwidth or smoothing parameter. \par The author refers to 31 papers which include related results, more exactly this work is a summary with some examples of the works listed in the references. A.Bleyer Citations: Zbl547.00060 Publ. Year: 1984 Document Type: Conference article Search for entries citing this one. View this entry as 572.62085 Nychka, Douglas; Wahba, Grace; Goldfarb, Stanley; Pugh, Thomas : Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections. ( English ) J. Am. Stat. Assoc. 79, 832-846 (1984). Classification *62P10 Appl. of statistics to biology 62G05 Nonparametric estimation 65U05 Numerical methods in probability and statistics 65R20 Integral equations (numerical methods) 65D07 Splines (numerical methods) 65C05 Monte Carlo methods Keywords random spheres model; stereology; three-dimensional sphere size distribution; two-dimensional cross sections; estimation of the tumor size distribution; mouse livers; cross-validated spline methods; ill-posed integral equations; eigensequence plots This paper is concerned with the problem of recreating the three- dimensional sphere size distribution from observations taken on two- dimensional cross sections. The problem arises in the estimation of the tumor size distribution of spherical micro tumors in mouse livers, from slices of liver tissue. \par In practice, since tumor cross-sections can only be observed if larger than some given radius $\epsilon$, the integral relationship between $F\sb 2$ (the cumulative distribution function of the two-dimensional cross sectional radii) and $f\sb 3$ (the probability density function of the three-dimensional distribution of sphere sizes) is made conditional on the radius of the tumor exceeding $\epsilon$. \par Recovering estimates of $f\sb 3$ from observations on $f\sb 2$ is ill- posed. The paper develops cross-validated spline methods to solve such ill-posed integral equations. \par The methods are tested on experimental data and the statistical properties of the estimate are explored via Monte Carlo simulation methods. Although the accuracy of the estimate is quite impressive, there are problems of estimation of $f\sb 3$ near the minimum radius $\epsilon$. \par The paper also briefly indicates ways of incorporating a priori information about the behaviour of $f\sb 3$ near $\epsilon$, into the estimate and shows how eigensequence plots can provide an insight into the ill-posedness of the problem. J.D.Wilson Publ. Year: 1984 Document Type: Journal Search for entries citing this one. View this entry as 565.65002 Wahba, Grace : Surface fitting with scattered noisy data on Euclidean D-space and on the sphere. ( English ) Rocky Mt. J. Math. 14, 281-299 (1984). Classification *65D10 Smoothing 65D07 Splines (numerical methods) 41A15 Spline approximation 41A63 Multidimensional approximation problems Keywords cross-validated thin plate smoothing splines; vector splines; smoothing noisy vector data An efficient numerical algorithm for computing the cross-validated thin plate smoothing splines is considered when several hundred data points are given. A theory of vector splines for smoothing noisy vector data on the sphere is given. The use of generalized cross-validation is described. A few interesting examples are given to test the theory. M.Sambandham Publ. Year: 1984 Document Type: Journal Search for entries citing this one. View this entry as 552.62023 Villalobos, Miguel A.; Wahba, Grace : Multivariate thin plate spline estimates for the posterior probabilities in the classification problem. ( English ) Commun. Stat., Theory Methods 12, 1449-1479 (1983). Classification *62G05 Nonparametric estimation 62H30 Statistical classification, etc. Keywords penalized log-likelihood; posterior probabilities; classification; multivariate thin plate splines; discrimination; degree of smoothness; generalized cross-validation A nonparametric estimate for the posterior probabilities in the classification problem using multivariate thin plate splines is proposed. This method presents a nonparametric alternative to logistic discrimination as well as to survival curve estimation. The degree of smoothness of the estimate is determined from the data using generalized cross-validation. Publ. Year: 1983 Document Type: Journal Search for entries citing this one. View this entry as 583.62066 Wahba, Grace : Constrained regularization for ill posed linear operator equations, with applications in meteorology and medicine. ( English ) Statistical decision theory and related topics III, Proc. 3rd Purdue Symp., West Lafayette/Indiana 1981, Vol. 2, 383-418 (1982). Classification *62J07 Ridge regression 62P99 Appl. of statistics 65U05 Numerical methods in probability and statistics 47A62 Equations involving linear operators, with operator unknowns Keywords ill posed linear operator equations; meteorology; stereology; computerized tomography; Hilbert space version of constrained ridge regression; extension of the generalized cross validation (GCV) method; first order approximation [For the entire collection see Zbl.564.00018.] \par We are interested in the Hilbert space version of constrained ridge regression, which we will show has many interesting applications. \par In this paper we propose an extension of the generalized cross validation (GCV) method, to the constrained case. The GCV estimate of $\lambda$ we propose in the constrained case can be expensive to compute. Thus we propose a first order approximation to it which is very much cheaper to compute, and appears to be satisfactory in the examples we tried. Citations: Zbl564.00018 Publ. Year: 1982 Document Type: Conference article Search for entries citing this one. View this entry as 572.41016 Wahba Gobran, Aida : On cubature formulae of rectangles. ( English ) Proc. Math. Phys. Soc. Egypt 52, 19-27 (1981). Classification *41A55 Approximate quadratures 41A25 Degree of approximation, etc. Keywords cubature formula; double integrals Let $R\subset {\bbfR}\sp 2$ be a closed rectangular area. In this article we derive a cubature formula in R, which is exact for double integrals of all polynomials of degree $\le 9$ in $(x,y)\in {\bbfR}\sp 2$. This formula uses only 17 nodes, while the formulae known before use at least 21 nodes. Moreover, these nodes have a certain sort of symmetry, which reduces the number of coefficients only to 7. As stated before our formula is exact for all polynomials of degree $\le 9$. This formula in the case of general functions gives rise to some error, an estimate of this error is also derived. Publ. Year: 1981 Document Type: Journal Search for entries citing this one. View this entry as 544.65092 Wahba, Grace : A new approach to the numerical evaluation of the inverse Radon transform with discrete, noisy data. ( English ) Mathematical aspects of computerized tomography, Proc., Oberwolfach 1980, Lect. Notes Med. Inf. 8, 189-203 (1981). Classification *65R10 Integral transforms (numerical methods) 44A15 Special transforms 45H05 Integral equations with miscellaneous special kernels 45L10 Numerical approximation of solutions of integral equations Keywords Tikhonov regularization; inverse Radon transform; parallel beam computerized tomography; cubic polynomial spline; smoothing splines; projection data; regularized transform method; smoothing parameter [For the entire collection see Zbl.538.00034.] \par The inner (singular) integral in the inverse Radon transform for parallel beam computerized tomography devices can be integrated analytically if the Radon transform considered as a function of the ray position along the detector, is a cubic polynomial spline. Furthermore by using some spline identities, large terms that cancel can be eliminated analytically and the calculation of the resulting expression for the inner integral is done in a numerically stable fashion. We suggest using smoothing splines to smooth each set of projection data and by so doing obtain the Radon transform in the above spline form. The resulting analytic expression for the inner integral in the inverse transform is then readily evaluated, and the outer (periodic) integral is replaced by a sum. The work involved to obtain the inverse transform appears to be within the capability of existing computing equipment for typical large data sets. In this regularized transform method the regularization is controlled by the smoothing parameter in the splines. The regularization is directed against data errors and not to prevent unstable numerical operations. Strip integral as well as line integral data can be handled by this method. The method is shown to be closely related to the Tikhonov form of regularization. Citations: Zbl538.00034 Publ. Year: 1981 Document Type: Conference article Search for entries citing this one. View this entry as 538.65006 Wahba, Grace : Bayesian "confidence intervals" for the cross-validated smoothing spline. ( English ) J. R. Stat. Soc., Ser. B 45, 133-150 (1983). Classification *65D10 Smoothing 65D07 Splines (numerical methods) 41A15 Spline approximation 62F25 Parametric confidence regions, etc. 62A15 The Bayesian approach 65U05 Numerical methods in probability and statistics Keywords Bayes estimates; spline smoothing; cross-validation; confidence intervals The author considers the model $Y(t\sb i)=g(t\sb i)+\epsilon\sb i$, $i=1,...,n,t\sb i\in [0,1]$, where g(t) is a smooth function on [0,1] and $\epsilon\sb i$ are independent $N(0,\tau\sp 2)$-errors with $\tau\sp 2$ unknowns. She studies "confidence intervals" for the cross-validated smoothing spline estimate of g. The paper presents an affirmative answer (in the case of large n) to the question: Is there any reason to believe that the resulting 95 per cent confidence intervals will cover the true $g(t\sb i)$ about 95 per cent of the time? B.D.Bojanov Publ. Year: 1983 Document Type: Journal Search for entries citing this one. View this entry as 537.65008 Wahba, Grace : Spline interpolation and smoothing on the sphere. ( English ) SIAM J. Sci. Stat. Comput. 2, 5-16 (1981). Classification *65D10 Smoothing 41A15 Spline approximation 65A05 Tables 65D07 Splines (numerical methods) 86A10 Meteorology 33C35 Spherical functions, etc. Keywords spline interpolation; smoothing; sphere; periodic polynomial splines; reproducing kernel space; spherical harmonics; meteorological data Motivating his study by some need of the analysis of meteorological data the author solves the following extremal problems: A) minimize the functional $I\sb m(u)$ subject to $u(P\sb i)=z\sb i$, $i=1,...,n$, where $\{P\sb i\}$ are points on the sphere and $I\sb m$ is a natural analogue on the sphere of the functional $\int\sp{2\pi}\sb{0}[u\sp{(m)}(\theta)]\sp 2d\theta,$ appearing in the definition of periodic polynomial splines; B) minimize $1/n\sum\sp{n}\sb{j=1}[u(P\sb i)-z\sb i]\sp 2+\lambda I\sb m(u).$ \par [An erratum to this paper is published in ibid. 3, 385-386 (1982; reviewed below).] B.D.Bojanov Citations: Zbl537.65009 Publ. Year: 1981 Document Type: Journal Search for entries citing this one. View this entry as 537.65009 Wahba, Grace : Erratum: Spline interpolation and smoothing on the sphere. ( English ) SIAM J. Sci. Stat. Comput. 3, 385-386 (1982). Classification *65D10 Smoothing 65A05 Tables Keywords erratum; spline interpolation; smoothing; sphere This note contains a correct version of a table published in the author's paper, ibid. 2, 5-16 (1981; reviewed above). Citations: Zbl537.65008 Publ. Year: 1982 Document Type: Journal Search for entries citing this one. View this entry as 502.41004 Wahba, Grace : Vector splines on the sphere, with application to the estimation of vorticity and divergence fromdiscrete, noisy data. ( English ) Multivariate approximation theory II, Proc. Conf., Oberwolfach 1982, ISNM 61, 407-429 (1982). Classification *41A15 Spline approximation Keywords Hilbert space norms; meteorological application; Monte Carlo study; air vorticity; divergence Citations: Zbl485.00013 Publ. Year: 1982 Document Type: Conference article Search for entries citing this one. View this entry as 502.65007 Bates, Douglas M.; Wahba, Grace : Computational methods for generalized cross-validation with large data sets. ( English ) Treatment of integral equations by numerical methods, Proc. Symp., Durham 1982, 283-296 (1982). Classification *65D10 Smoothing 45L10 Numerical approximation of solutions of integral equations 65R20 Integral equations (numerical methods) 45B05 Fredholm integral equations 86A10 Meteorology Keywords generalized cross-validation; large data sets; smoothing or regularization parameters; ill-posed problems; computational cost; truncating; singular value decomposition; smoothing parameter; height and wind fields Citations: Zbl499.00015 Publ. Year: 1982 Document Type: Conference article Search for entries citing this one. View this entry as 488.65079 Dyn, Nira; Wahba, Grace : On the estimation of functions of several variables from aggregated data. ( English ) SIAM J. Math. Anal. 13, 134-152 (1982). Classification *65U05 Numerical methods in probability and statistics 65D10 Smoothing 62J02 General nonlinear regression 92C50 Medical appl. of mathematical biology 62P10 Appl. of statistics to biology Keywords roughness criteria; Laplacian smoothing histosplines; exact and inexact data; smoothing parameter; aggregated data; medical applications; incidence rates of cancer; population density Publ. Year: 1982 Document Type: Journal Search for entries citing this one. View this entry as 485.41012 Wahba, Grace : Spline bases, regularization, and generalized cross validation for solving approximation problems with large quantities of noisy data. ( English ) Approximation theory III, Proc.Conf. Hon. G. G. Lorentz, Austin/Tex. 1980, 905-912 (1980). Classification *41A15 Spline approximation Keywords minimization problem; thin plate splines Citations: Zbl468.00012 Publ. Year: 1980 Document Type: Conference article Search for entries citing this one. View this entry as 463.62034 Wahba, Grace : Data-based optimal smoothing of orthogonal series density estimates. ( English ) Ann. Stat. 9, 146-156 (1981). Classification *62G05 Nonparametric estimation Keywords optimal smoothing; orthogonal series density estimates; multivariate estimates; Fourier series expansion; two-step procedure; expected integrated mean square errors Citations: Zbl.188.506 Publ. Year: 1981 Document Type: Journal Search for entries citing this one. View this entry as 461.62059 Golub, Gene H.; Heath, Michael; Wahba, Grace : Generalized cross-validation as a method for choosing a good ridge parameter. ( English ) Technometrics 21, 215-225 (1979). Classification *62J07 Ridge regression 62J05 Linear regression Keywords generalized cross-validation; ridge parameter Publ. Year: 1979 Document Type: Journal Search for entries citing this one. View this entry as 449.65003 Wahba, Grace : Convergence rates of "thin plate" smoothing splines when the data are noisy. (Preliminary report). ( English ) Smoothing techniques for curve estimation, Proc. Workshop, Heidelberg 1979, Lect. Notes Math. 757, 233-245 (1979). Classification *65D10 Smoothing 65D07 Splines (numerical methods) 62J02 General nonlinear regression 65U05 Numerical methods in probability and statistics Keywords thin plate smoothing splines; noisy data; integrated mean square error; method of generalized cross-validation Citations: Zbl407.00010 Publ. Year: 1979 Document Type: Conference article Search for entries citing this one. View this entry as 442.62074 Wahba, Grace : Automatic smoothing of the log periodogram. ( English ) J. Am. Stat. Assoc. 75, 122-132 (1980). Classification *62M10 Time series, etc. (statistics) 62M15 Spectral analysis of processes 62G05 Nonparametric estimation 65C05 Monte Carlo methods 93E10 Estimation and detection in stochastic control 65D10 Smoothing Keywords automatic smoothing of log periodogram; log spectral density estimate; optimal choice of bandwidth parameter; spline spectral density estimate; stationary Gaussian time series; empirical Bayes method; nonparametric density estimation Publ. Year: 1980 Document Type: Journal Search for entries citing this one. View this entry as 442.93053 Wahba, Grace : Parameter estimation in linear dynamic systems. ( English ) IEEE Trans. Autom. Control AC-25, 235-238 (1980). Classification *93E10 Estimation and detection in stochastic control 93C05 Linear control systems Keywords linear dynamic systems; Kiefer-Wolfowitz experimental design problem Publ. Year: 1980 Document Type: Journal Search for entries citing this one. View this entry as 436.65049 Miranker, W.L.; Veldhuizen, M.van; Wahba, G. : Two methods for the stiff highy oscillatory problem. ( English ) Topics in numerical analysis III, Proc. R. Irish Acad. Conf., Dublin 1976, 257-273 (1977). Classification *65L05 Initial value problems for ODE (numerical methods) 65L07 Numerical investigation of stability of solutions of ODE Keywords stiff highly oscillatory initial value problems; averaging technique; method of envelopes Citations: Zbl416.00016 Publ. Year: 1977 Document Type: Conference article Search for entries citing this one. View this entry as 434.65104 Wahba, G. : Smoothing and ill-posed problems. ( English ) Solution methods for integral equations, theory and applications, Math. Concepts Meth. Sci. Eng., Vol. 18, 183-194 (1979). Classification *65R20 Integral equations (numerical methods) 45L05 Theoretical approximation of solutions of integral equations 45L10 Numerical approximation of solutions of integral equations 45K05 Integro-partial differential equations Keywords method of weighted cross-validation; noisy data; numerical results; Fujita's equation for equilibrium sedimentation Citations: Zbl424.00015 Publ. Year: 1979 Document Type: Conference article Search for entries citing this one. View this entry as 413.65008 Craven, Peter; Wahba, Grace : Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross-validation. ( English ) Numer. Math. 31, 377-403 (1979). Classification *65D10 Smoothing 65D07 Splines (numerical methods) 41A15 Spline approximation 65K05 Mathematical programming (numerical methods) 65C05 Monte Carlo methods Keywords smoothing noisy data; spline functions; correct degree of smoothing; method of generalized cross-validation; estimates; Monte-Carlo experiment Citations: Zbl377.65007 Publ. Year: 1979 Document Type: Journal Search for entries citing this one. View this entry as 409.65036 Athavale, Manohar L.; Wahba, Grace : Determination of an optimal mesh for a collocation-projection method for solving two-point boundary value problems. ( English ) J. Approximation Theory 25, 38-49 (1979). Classification *65L10 Boundary value problems for ODE (numerical methods) 34B05 Linear boundary value problems of ODE 65J10 Equations with linear operators (numerical methods) 62C12 Empirical statistical decision procedures 62L20 Stochastic approximation Keywords MESH-POINT OPTIMIZATION; COLLOCATION-PROJECTION METHODS; BOUNDARY VALUE PROBLEMS; ORDINARY DIFFERENTIAL EQUATIONS; REPRODUCING KERNEL; HILBERT SPACE; OPTIMAL MESH; APPROXIMATE SOLUTION; SEQUENTIAL DESIGN OF AN EXPERIMENT; LINEAR OPERATOR EQUATIONS Citations: Zbl.152.175; Zbl221.62027; Zbl291.62091; Zbl329.65040 Publ. Year: 1979 Document Type: Journal Search for entries citing this one. View this entry as Search for next 40 documents MATH Help New query form. We need YOUR feedback! Please send your comments to: math-db@zblmath.fiz-karlsruhe.de Zentralblatt für Mathematik Mathematics Abstracts (c) FIZ Karlsruhe & Springer-Verlag Your query: au=(Wahba_G*) The database contains the following 41-73 best item(s) out of 73 relevant to your query. MATH Help We need YOUR feedback! Please send your comments to: math-db@zblmath.fiz-karlsruhe.de 407.62048 Wahba, Grace : Improper priors, spline smoothing and the problem of guarding against model errors in regression. ( English ) J. R. Stat. Soc., Ser. B 40, 364-372 (1978). Classification *62J99 Linear statistical inference 62G05 Nonparametric estimation 65D10 Smoothing 62F15 Bayesian inference 65D07 Splines (numerical methods) Keywords SPLINE SMOOTHING; NONPARAMETRIC REGRESSION; IMPROPER PRIORS; MODEL ERRORS Publ. Year: 1978 Document Type: Journal Search for entries citing this one. View this entry as 402.65032 Wahba, Grace : Practical approximate solutions to linear operator equations when the data are noisy. ( English ) SIAM J. Numer. Anal. 14, 651-667 (1977). Classification *65J10 Equations with linear operators (numerical methods) 65R20 Integral equations (numerical methods) Keywords APPROXIMATE SOLUTIONS; LINEAR OPERATOR EQUATIONS; INTEGRAL EQUATIONS; NUMERICAL METHODS; REGULARIZED SOLUTION; REGULARIZATION PARAMETER; METHOD OF WEIGHTED CROSS-VALIDATION; CONVERGENCE RATES Publ. Year: 1977 Document Type: Journal Search for entries citing this one. View this entry as 377.65007 Craven, Peter; Wahba, Grace : Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross-validation. ( English ) Numerische Math. (to appear). Classification *65D10 Smoothing 41A15 Spline approximation 65K05 Mathematical programming (numerical methods) 65C05 Monte Carlo methods Publ. Year: 1979 Document Type: Journal Search for entries citing this one. View this entry as 374.41009 Wahba, Grace : A survey of some smoothing problems and the method of generalized cross- validation for solving them. ( English ) Appl. of Stat., Proc. Symp., Dayton 1976, 507-523 (1977). Classification *41A25 Degree of approximation, etc. 41A15 Spline approximation Publ. Year: 1977 Document Type: Conference article Search for entries citing this one. View this entry as 347.49029 Wahba, Grace; Nashed, M.Z. : The approximate solution of a class of constrained control problems. ( English ) Proc. 6th Hawaii int. Conf. Syst. Sci., Hawaii 1973, 112-114 (1973). Classification *49M99 Methods of successive approximations 49K27 Optimal control problems in abstract spaces (nec./ suff.) Publ. Year: 1973 Document Type: Conference article Search for entries citing this one. View this entry as 336.65039 Miranker, W.L.; Wahba, G. : An averaging method for the stiff highly oscillatory problem. ( English ) Math. Comput. 30, 383-399 (1976). Classification *65L05 Initial value problems for ODE (numerical methods) Publ. Year: 1976 Document Type: Journal Search for entries citing this one. View this entry as 329.62034 Wahba, Grace : Histosplines with knots which are order statistics. ( English ) J. roy. statist. Soc., Ser. B. 38, 140-151 (1976). Classification *62G05 Nonparametric estimation 65D10 Smoothing Publ. Year: 1976 Document Type: Journal Search for entries citing this one. View this entry as 329.65040 Wahba, Grace : On the optimal choice of nodes in the collocation-projection method for solving linear operator equations. ( English ) J. Approximation Theory 16, 175-186 (1976). Classification *65J05 General theory of numerical methods in abstract spaces 65L10 Boundary value problems for ODE (numerical methods) 47A50 Equations and inequalities involving linear operators 34B05 Linear boundary value problems of ODE Publ. Year: 1976 Document Type: Journal Search for entries citing this one. View this entry as 324.65060 Nashed, M.Z.; Wahba, Grace : Some exponentially decreasing error bounds for a numerical inversion of the Laplace transform. ( English ) J. math. Analysis Appl. 52, 660-668 (1975). Classification *65R20 Integral equations (numerical methods) 44A10 Laplace transform Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 311.65007 Wahba, Grace : Smoothing noisy data with spline functions. ( English ) Numerische Math. 24, 383-393 (1975). Classification *65D10 Smoothing 41A05 Interpolation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 309.47012 Nashed, M.Z.; Wahba, Grace : Regularization and approximation of linear operator equations in reproducing kernel spaces. ( English ) Bull. Amer. math. Soc. 80, 1213-1218 (1974). Classification *47A50 Equations and inequalities involving linear operators 45L10 Numerical approximation of solutions of integral equations 65F20 Overdetermined systems (numerical linear algebra) 65D99 Numerical approximation 65J05 General theory of numerical methods in abstract spaces Publ. Year: 1974 Document Type: Journal Search for entries citing this one. View this entry as 305.62021 Wahba, Grace : Optimal convergence properties of variable knot, kernel, and orthogonal series methods for density estimation. ( English ) Ann. of Statist. 3, 15-29 (1975). Classification *62G05 Nonparametric estimation 65D99 Numerical approximation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 305.62022 Wahba, Grace : Interpolating spline methods for density estimation. I: Equi-spaced knots. ( English ) Ann. of Statist. 3, 30-48 (1975). Classification *62G05 Nonparametric estimation 65D99 Numerical approximation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 305.62043 Wahba, G.; Wold, S. : A completely automatic French curve: Fitting spline functions by cross validation. ( English ) Commun. Statist. 4, 1-17 (1975). Classification *62J05 Linear regression 62M10 Time series, etc. (statistics) 65D10 Smoothing 65D25 Numerical differentiation 65R20 Integral equations (numerical methods) 41A15 Spline approximation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 305.62060 Wahba, G.; Wold, S. : Periodic splines for spectral density estimation: The use of cross validation for determining the degree of smoothing. ( English ) Commun. Statist. 4, 125-141 (1975). Classification *62M15 Spectral analysis of processes 65D10 Smoothing 41A15 Spline approximation 15A09 Matrix inversion 62J05 Linear regression Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 299.65008 Wahba, Grace : Smoothing noisy data with spline functions. ( English ) Numerische Math. (to appear) Classification *65D10 Smoothing 41A05 Interpolation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 291.62091 Wahba, Grace : Regression design for some equivalence classes of kernels. ( English ) Ann. of Statist. 2, 925-934 (1974). Classification *62K05 Optimal statistical designs 60G15 Gaussian processes 62M10 Time series, etc. (statistics) 62J05 Linear regression 46C99 Inner product spaces, Hilbert spaces Publ. Year: 1974 Document Type: Journal Search for entries citing this one. View this entry as 287.47009 Nashed, M.Z.; Wahba, Grace : Generalized inverses in reproducing Kernel spaces: An approach to regularization of linear operator equations. ( English ) SIAM J. math. Analysis 5(1974), 974-987 (1975). Classification *47A50 Equations and inequalities involving linear operators Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 279.62027 Wahba, Grace : On the distribution of some statistics useful in the analysis of jointly stationary time series. ( English ) Ann. math. Statistics 39, 1849-1862 (1968). Classification *62M15 Spectral analysis of processes 62E15 Exact distribution theory in statistics 62M10 Time series, etc. (statistics) Publ. Year: 1968 Document Type: Journal Search for entries citing this one. View this entry as 273.45012 Nashed, M.Z.; Wahba, Grace : Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind. ( English ) Math. Comput. 28, 69-80 (1974). Classification *45L10 Numerical approximation of solutions of integral equations 45C05 Eigenvalue problems (integral equations) 65F20 Overdetermined systems (numerical linear algebra) 65J05 General theory of numerical methods in abstract spaces 45B05 Fredholm integral equations 65R20 Integral equations (numerical methods) 47A50 Equations and inequalities involving linear operators Publ. Year: 1974 Document Type: Journal Search for entries citing this one. View this entry as 267.45015 Wahba, Grace : A class of approximate solutions to linear operator equations. ( English ) J. Approximation Theory 9, 61-77 (1973). Classification *45J05 Integro-ordinary differential equations 65R20 Integral equations (numerical methods) 34A45 Theoretical approximation of solutions of ODE 45L10 Numerical approximation of solutions of integral equations Publ. Year: 1973 Document Type: Journal Search for entries citing this one. View this entry as 267.47009 Nashed, M.Z.; Wahba, Grace : Generalized inverses in reproducing Kernel spaces: An approach to regularization of linear operator equations. ( English ) SIAM J. math. Analysis (to appear) Classification *47A50 Equations and inequalities involving linear operators Publ. Year: 1974 Document Type: Journal Search for entries citing this one. View this entry as 254.49037 Wahba, Grace : On the minimization of a quadratic functional subject to a continuous family of linear inequality constraints. ( English ) SIAM J. Control 11, 64-79 (1973). Classification *49M40 Methods of quadratic programming type Publ. Year: 1973 Document Type: Journal Search for entries citing this one. View this entry as 252.65100 Wahba, Grace : Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind. ( English ) J. Approximation Theory 7, 167-185 (1973). Classification *65R20 Integral equations (numerical methods) 45B05 Fredholm integral equations Publ. Year: 1973 Document Type: Journal Search for entries citing this one. View this entry as 227.49011 Wahba, Grace : On the minimization of a quadratic functional subject to a continuous family of linear inequality constraints. ( English ) SIAM J. Control (1972). Classification *49M40 Methods of quadratic programming type Publ. Year: 1972 Document Type: Journal Search for entries citing this one. View this entry as 226.60064 Kimeldorf, George S.; Wahba, Grace. : Spline functions and stochastic processes. ( English ) Sankhya, Ser. A 32, 173-180 (1970). Classification *60G35 Appl. of stochastic processes 41A15 Spline approximation 60G05 Foundations of stochastic processes Publ. Year: 1970 Document Type: Journal Search for entries citing this one. View this entry as 226.62038 Wahba, Grace : A polynomial algorithm for density estimation. ( English ) Ann. Math. Statistics 42, 1870-1886 (1971). Classification *62G07 Curve estimation Publ. Year: 1971 Document Type: Journal Search for entries citing this one. View this entry as 221.62027 Wahba, Grace : On the regression design problem of Sacks and Ylvisaker. ( English ) Ann. Math. Statistics 42, 1035-1053 (1971). Classification *62K05 Optimal statistical designs 41A15 Spline approximation 60H10 Stochastic ordinary differential equations Publ. Year: 1971 Document Type: Journal Search for entries citing this one. View this entry as 219.62017 Wahba, G. : Some tests of independence for stationary multivariate time series ( English ) J. R. Stat. Soc., Ser. B 33, 153-166 (1971). Classification *62M10 Time series, etc. (statistics) Publ. Year: 1971 Document Type: Journal Search for entries citing this one. View this entry as 213.20904 Wahba, G. : Estimation of the coefficients in a multidimensional distributed lag model ( English ) Econometrica 37, 398-407 (1969). Classification *62M15 Spectral analysis of processes 62M09 Non-Markovian processes: estimation Publ. Year: 1969 Document Type: Journal Search for entries citing this one. View this entry as 201.39702 Kimeldorf, G.S.; Wahba, G. : Some results on Tchebycheffian spline functions and stochastic processes ( English ) J. Math. Anal. Appl. 33, 82-95 (1971). Publ. Year: 1971 Document Type: Journal Search for entries citing this one. View this entry as 193.45201 Kimeldorf, G.S.; Wahba, G. : A correspondence between Bayesian estimation on stochastic processes and smoothing by splines ( English ) Ann. Math. Stat. 41, 495-502 (1970). Keywords probability theory Publ. Year: 1970 Search for entries citing this one. View this entry as 185.30902 Kimeldorf, G.S.; Wahba, G. : Some results on Tchebycheffian spline functions and stochastic processes ( English ) J. Math. Anal. Appl. (to appear) Keywords approximation and series expansions Search for entries citing this one. View this entry as MATH Help New query form. We need YOUR feedback! Please send your comments to: math-db@zblmath.fiz-karlsruhe.de ........... ........... Zentralblatt für Mathematik Mathematics Abstracts (c) FIZ Karlsruhe & Springer-Verlag Your query: au=(Wahba_G*) The database contains the following 41-73 best item(s) out of 73 relevant to your query. MATH Help We need YOUR feedback! Please send your comments to: math-db@zblmath.fiz-karlsruhe.de 407.62048 Wahba, Grace : Improper priors, spline smoothing and the problem of guarding against model errors in regression. ( English ) J. R. Stat. Soc., Ser. B 40, 364-372 (1978). Classification *62J99 Linear statistical inference 62G05 Nonparametric estimation 65D10 Smoothing 62F15 Bayesian inference 65D07 Splines (numerical methods) Keywords SPLINE SMOOTHING; NONPARAMETRIC REGRESSION; IMPROPER PRIORS; MODEL ERRORS Publ. Year: 1978 Document Type: Journal Search for entries citing this one. View this entry as 402.65032 Wahba, Grace : Practical approximate solutions to linear operator equations when the data are noisy. ( English ) SIAM J. Numer. Anal. 14, 651-667 (1977). Classification *65J10 Equations with linear operators (numerical methods) 65R20 Integral equations (numerical methods) Keywords APPROXIMATE SOLUTIONS; LINEAR OPERATOR EQUATIONS; INTEGRAL EQUATIONS; NUMERICAL METHODS; REGULARIZED SOLUTION; REGULARIZATION PARAMETER; METHOD OF WEIGHTED CROSS-VALIDATION; CONVERGENCE RATES Publ. Year: 1977 Document Type: Journal Search for entries citing this one. View this entry as 377.65007 Craven, Peter; Wahba, Grace : Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross-validation. ( English ) Numerische Math. (to appear). Classification *65D10 Smoothing 41A15 Spline approximation 65K05 Mathematical programming (numerical methods) 65C05 Monte Carlo methods Publ. Year: 1979 Document Type: Journal Search for entries citing this one. View this entry as 374.41009 Wahba, Grace : A survey of some smoothing problems and the method of generalized cross- validation for solving them. ( English ) Appl. of Stat., Proc. Symp., Dayton 1976, 507-523 (1977). Classification *41A25 Degree of approximation, etc. 41A15 Spline approximation Publ. Year: 1977 Document Type: Conference article Search for entries citing this one. View this entry as 347.49029 Wahba, Grace; Nashed, M.Z. : The approximate solution of a class of constrained control problems. ( English ) Proc. 6th Hawaii int. Conf. Syst. Sci., Hawaii 1973, 112-114 (1973). Classification *49M99 Methods of successive approximations 49K27 Optimal control problems in abstract spaces (nec./ suff.) Publ. Year: 1973 Document Type: Conference article Search for entries citing this one. View this entry as 336.65039 Miranker, W.L.; Wahba, G. : An averaging method for the stiff highly oscillatory problem. ( English ) Math. Comput. 30, 383-399 (1976). Classification *65L05 Initial value problems for ODE (numerical methods) Publ. Year: 1976 Document Type: Journal Search for entries citing this one. View this entry as 329.62034 Wahba, Grace : Histosplines with knots which are order statistics. ( English ) J. roy. statist. Soc., Ser. B. 38, 140-151 (1976). Classification *62G05 Nonparametric estimation 65D10 Smoothing Publ. Year: 1976 Document Type: Journal Search for entries citing this one. View this entry as 329.65040 Wahba, Grace : On the optimal choice of nodes in the collocation-projection method for solving linear operator equations. ( English ) J. Approximation Theory 16, 175-186 (1976). Classification *65J05 General theory of numerical methods in abstract spaces 65L10 Boundary value problems for ODE (numerical methods) 47A50 Equations and inequalities involving linear operators 34B05 Linear boundary value problems of ODE Publ. Year: 1976 Document Type: Journal Search for entries citing this one. View this entry as 324.65060 Nashed, M.Z.; Wahba, Grace : Some exponentially decreasing error bounds for a numerical inversion of the Laplace transform. ( English ) J. math. Analysis Appl. 52, 660-668 (1975). Classification *65R20 Integral equations (numerical methods) 44A10 Laplace transform Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 311.65007 Wahba, Grace : Smoothing noisy data with spline functions. ( English ) Numerische Math. 24, 383-393 (1975). Classification *65D10 Smoothing 41A05 Interpolation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 309.47012 Nashed, M.Z.; Wahba, Grace : Regularization and approximation of linear operator equations in reproducing kernel spaces. ( English ) Bull. Amer. math. Soc. 80, 1213-1218 (1974). Classification *47A50 Equations and inequalities involving linear operators 45L10 Numerical approximation of solutions of integral equations 65F20 Overdetermined systems (numerical linear algebra) 65D99 Numerical approximation 65J05 General theory of numerical methods in abstract spaces Publ. Year: 1974 Document Type: Journal Search for entries citing this one. View this entry as 305.62021 Wahba, Grace : Optimal convergence properties of variable knot, kernel, and orthogonal series methods for density estimation. ( English ) Ann. of Statist. 3, 15-29 (1975). Classification *62G05 Nonparametric estimation 65D99 Numerical approximation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 305.62022 Wahba, Grace : Interpolating spline methods for density estimation. I: Equi-spaced knots. ( English ) Ann. of Statist. 3, 30-48 (1975). Classification *62G05 Nonparametric estimation 65D99 Numerical approximation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 305.62043 Wahba, G.; Wold, S. : A completely automatic French curve: Fitting spline functions by cross validation. ( English ) Commun. Statist. 4, 1-17 (1975). Classification *62J05 Linear regression 62M10 Time series, etc. (statistics) 65D10 Smoothing 65D25 Numerical differentiation 65R20 Integral equations (numerical methods) 41A15 Spline approximation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 305.62060 Wahba, G.; Wold, S. : Periodic splines for spectral density estimation: The use of cross validation for determining the degree of smoothing. ( English ) Commun. Statist. 4, 125-141 (1975). Classification *62M15 Spectral analysis of processes 65D10 Smoothing 41A15 Spline approximation 15A09 Matrix inversion 62J05 Linear regression Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 299.65008 Wahba, Grace : Smoothing noisy data with spline functions. ( English ) Numerische Math. (to appear) Classification *65D10 Smoothing 41A05 Interpolation Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 291.62091 Wahba, Grace : Regression design for some equivalence classes of kernels. ( English ) Ann. of Statist. 2, 925-934 (1974). Classification *62K05 Optimal statistical designs 60G15 Gaussian processes 62M10 Time series, etc. (statistics) 62J05 Linear regression 46C99 Inner product spaces, Hilbert spaces Publ. Year: 1974 Document Type: Journal Search for entries citing this one. View this entry as 287.47009 Nashed, M.Z.; Wahba, Grace : Generalized inverses in reproducing Kernel spaces: An approach to regularization of linear operator equations. ( English ) SIAM J. math. Analysis 5(1974), 974-987 (1975). Classification *47A50 Equations and inequalities involving linear operators Publ. Year: 1975 Document Type: Journal Search for entries citing this one. View this entry as 279.62027 Wahba, Grace : On the distribution of some statistics useful in the analysis of jointly stationary time series. ( English ) Ann. math. Statistics 39, 1849-1862 (1968). Classification *62M15 Spectral analysis of processes 62E15 Exact distribution theory in statistics 62M10 Time series, etc. (statistics) Publ. Year: 1968 Document Type: Journal Search for entries citing this one. View this entry as 273.45012 Nashed, M.Z.; Wahba, Grace : Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind. ( English ) Math. Comput. 28, 69-80 (1974). Classification *45L10 Numerical approximation of solutions of integral equations 45C05 Eigenvalue problems (integral equations) 65F20 Overdetermined systems (numerical linear algebra) 65J05 General theory of numerical methods in abstract spaces 45B05 Fredholm integral equations 65R20 Integral equations (numerical methods) 47A50 Equations and inequalities involving linear operators Publ. Year: 1974 Document Type: Journal Search for entries citing this one. View this entry as 267.45015 Wahba, Grace : A class of approximate solutions to linear operator equations. ( English ) J. Approximation Theory 9, 61-77 (1973). Classification *45J05 Integro-ordinary differential equations 65R20 Integral equations (numerical methods) 34A45 Theoretical approximation of solutions of ODE 45L10 Numerical approximation of solutions of integral equations Publ. Year: 1973 Document Type: Journal Search for entries citing this one. View this entry as 267.47009 Nashed, M.Z.; Wahba, Grace : Generalized inverses in reproducing Kernel spaces: An approach to regularization of linear operator equations. ( English ) SIAM J. math. Analysis (to appear) Classification *47A50 Equations and inequalities involving linear operators Publ. Year: 1974 Document Type: Journal Search for entries citing this one. View this entry as 254.49037 Wahba, Grace : On the minimization of a quadratic functional subject to a continuous family of linear inequality constraints. ( English ) SIAM J. Control 11, 64-79 (1973). Classification *49M40 Methods of quadratic programming type Publ. Year: 1973 Document Type: Journal Search for entries citing this one. View this entry as 252.65100 Wahba, Grace : Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind. ( English ) J. Approximation Theory 7, 167-185 (1973). Classification *65R20 Integral equations (numerical methods) 45B05 Fredholm integral equations Publ. Year: 1973 Document Type: Journal Search for entries citing this one. View this entry as 227.49011 Wahba, Grace : On the minimization of a quadratic functional subject to a continuous family of linear inequality constraints. ( English ) SIAM J. Control (1972). Classification *49M40 Methods of quadratic programming type Publ. Year: 1972 Document Type: Journal Search for entries citing this one. View this entry as 226.60064 Kimeldorf, George S.; Wahba, Grace. : Spline functions and stochastic processes. ( English ) Sankhya, Ser. A 32, 173-180 (1970). Classification *60G35 Appl. of stochastic processes 41A15 Spline approximation 60G05 Foundations of stochastic processes Publ. Year: 1970 Document Type: Journal Search for entries citing this one. View this entry as 226.62038 Wahba, Grace : A polynomial algorithm for density estimation. ( English ) Ann. Math. Statistics 42, 1870-1886 (1971). Classification *62G07 Curve estimation Publ. Year: 1971 Document Type: Journal Search for entries citing this one. View this entry as 221.62027 Wahba, Grace : On the regression design problem of Sacks and Ylvisaker. ( English ) Ann. Math. Statistics 42, 1035-1053 (1971). Classification *62K05 Optimal statistical designs 41A15 Spline approximation 60H10 Stochastic ordinary differential equations Publ. Year: 1971 Document Type: Journal Search for entries citing this one. View this entry as 219.62017 Wahba, G. : Some tests of independence for stationary multivariate time series ( English ) J. R. Stat. Soc., Ser. B 33, 153-166 (1971). Classification *62M10 Time series, etc. (statistics) Publ. Year: 1971 Document Type: Journal Search for entries citing this one. View this entry as 213.20904 Wahba, G. : Estimation of the coefficients in a multidimensional distributed lag model ( English ) Econometrica 37, 398-407 (1969). Classification *62M15 Spectral analysis of processes 62M09 Non-Markovian processes: estimation Publ. Year: 1969 Document Type: Journal Search for entries citing this one. View this entry as 201.39702 Kimeldorf, G.S.; Wahba, G. : Some results on Tchebycheffian spline functions and stochastic processes ( English ) J. Math. Anal. Appl. 33, 82-95 (1971). Publ. Year: 1971 Document Type: Journal Search for entries citing this one. View this entry as 193.45201 Kimeldorf, G.S.; Wahba, G. : A correspondence between Bayesian estimation on stochastic processes and smoothing by splines ( English ) Ann. Math. Stat. 41, 495-502 (1970). Keywords probability theory Publ. Year: 1970 Search for entries citing this one. View this entry as 185.30902 Kimeldorf, G.S.; Wahba, G. : Some results on Tchebycheffian spline functions and stochastic processes ( English ) J. Math. Anal. Appl. (to appear) Keywords approximation and series expansions Search for entries citing this one. View this entry as MATH Help New query form. We need YOUR feedback! Please send your comments to: math-db@zblmath.fiz-karlsruhe.de