G. Wahba, D.R. Johnson, F. Gao, and J. Gong. Adaptive tuning of numerical weather prediction models: Randomized gcv in three- and four-dimensional data assimilation. Monthly Weather Review, 11:3358--3369, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....estimation, penalized log density, and log hazard estimation are mentioned. Further references in the area of smoothing splines include [4, 5, 20, 22, 25, 28, 36, 37, 41, 42] The application of generalized cross validation to global scale numerical weather prediction models is described in [43]. Further applications can be found in image processing [2, 29, 35] astronomy [34] and chemistry [23] The reader is also referred to [8] and [18, p. 460] where several other methods for the evaluation of the GCV function are outlined. Our paper is organized as follows. The use of a stochastic ....

....X i=1 u T i (K T K I) Gamma1 u i ; m n; t nu ( nu nu X i=1 u T i (KK T I) Gamma1 u i ; m n: In [10, 11, 12] D. A. Girard gives a theoretical analysis of the accuracy of the stochastic trace estimator. Further numerical case studies are presented in [21] and in [43]. However the results in [1] indicate that only little is gained by a large value of nu . It appears that nu = 1 is the best compromise between accuracy and computational costs. We will come back to this issue in Section 10. We will use Hutchinson s trace estimator to compute an approximation of V ....

G. Wahba, D. R. Johnson, F. Gao and J. Gong, Adaptive Tuning of Numerical Weather Prediction Models: Randomized GCV in Three- and Four-Dimensional Data Assimilation, Monthly Weather Review, 123 (1995), pp. 3358--3369, also available by anonymous ftp from ftp.stat.wisc.edu in pub/wahba.

....Cross Validation (GCV) method (Wahba and Wendelberger 1980) The cross validation approach is based on maximizing the capability of a model to predict withheld data, and it does not require as many assumptions on the nature of the error distributions as does the maximum likelihood method. Wahba et al. 1995) show how GCV can be applied to the estimation of covariance parameters and possibly other tuning parameters of an atmospheric assimilation system. Using our notation, they specifically consider covariance models of the form S(ff) S(oe 1 ; oe 2 ; oe 2 1 S 1 oe 2 2 S 2 ( 2.43) where ....

....present in the data residuals. Estimates of the separate variances oe 1 and oe 2 are obtained as a by product of the GCV estimation procedure. Note that the identifiability requirement still holds; no method can produce meaningful estimates of poorly identifiable parameters. See Appendix A of Wahba et al. 1995) for a discussion of identifiability in the context of GCV. The GCV approach as formulated by Wahba et al. 1995) applies to the estimation of the parameters oe 1 ; oe 2 ; and based on a single vector of residuals valid at a fixed time t k . As in Dee (1995) their method was originally ....

[Article contains additional citation context not shown here]

Wahba, G., D. R. Johnson, F. Gao, and J. Gong, 1995: Adaptive tuning of numerical weather prediction models: Randomized GCV in three- and four-dimensional data assimilation. Mon. Wea. Rev., 123, 3358--3369.

.... method was used in oceanography application by ten Brummelhuis et al. 1990, 1993) Wahba(1990) and more recently Wahba et al. 1994, 1995) used more advanced statistical methods such as Gong et al. 1996) generalized the statistical parameter estimation using generalized cross validation(see also Wahba et al. 1995) as well as unbiased risk methods for adaptive tuning of parameters in one space and one time variables for the equivalent barotropic vorticity equation on a latitude circle, simultaneously tuning some smoothing, weighting and physical parameters. 4. Total variation as an L 1 regularization method ....

Wahba, G., D. R. Johnson, F. Gao, and J. Gong, 1995: Adaptive tuning of numerical weather prediction models, randomized GCV in three and four dimensional data assimilation, Mon. Wea. Rev., 123, 3358-3369.

....of approximation of the ML method to parameter estimation of aquifer parameter is represented by Carrera and Neuman(1986) Dee(1995) used ML for covariance parameter estimation. The method was used in oceanography application by ten Brummelhuis et al. 1990, 1993) Wahba(1990) and more recently Wahba et al. 1994, 1995) used more advanced statistical methods such as Gong et al. 1996) generalized the statistical parameter estimation using generalized cross validation(see also Wahba et al. 1995) as well as unbiased risk methods for adaptive tuning of parameters in one space and one time variables for the ....

Wahba, G., D. R. Johnson, F. Gao, and J. Gong, 1994: Adaptive tuning of numerical weather prediction models: Part I: randomized GCV and related methods in three and four dimensional data assimilation, Tech Rep 920, Dept of statistics, University of Wisconsin, 1210 W. Dayton St., Madison, WI 53706, 34pp.

....is represented by Carrera and Neuman(1986) Dee(1995) used ML for covariance parameter estimation. The method was used in oceanography application by ten Brummelhuis et al. 1990, 1993) Wahba(1990) and more recently Wahba et al. 1994, 1995) used more advanced statistical methods such as Gong et al. 1996) generalized the statistical parameter estimation using generalized cross validation(see also Wahba et al. 1995) as well as unbiased risk methods for adaptive tuning of parameters in one space and one time variables for the equivalent barotropic vorticity equation on a latitude circle, ....

Gong, J., G. Wahba, D. R. Johnson, and J. Tribbia, 1996: Adaptive tuning of numerical weather prediction models: simultaneous estimation of weighting, smoothing and physical parameters, Submitted to Mon. Wea. Rev..

....estimation, penalized log density, and log hazard estimation are mentioned. Further references in the area of smoothing splines include [4, 5, 20, 22, 25, 28, 37, 38, 42, 43] The application of generalized cross validation to global scale numerical weather prediction models is described in [44]. Further applications can be found in image processing [2, 29, 36] astronomy [35] and chemistry [23] The reader is also referred to [8] and [18, p. 460] where several other methods for the evaluation of the GCV function are outlined. Our paper is organized as follows. The use of a stochastic ....

....i=1 u T i (K T K I) Gamma1 u i ; m n; t nu ( n u nu X i=1 u T i (KK T I) Gamma1 u i ; m n: In [10, 11, 12] D. A. Girard gives a theoretical analysis of the accuracy of the stochastic trace estimator. Further numerical case studies are presented in [21] and in [44]. However the results in [1] indicate that only little is gained by a large value of n u . It appears that n u = 1 is the best compromise between accuracy and computational costs. We will come back to this issue in Section 10. We will use Hutchinson s trace estimator to compute an approximation of ....

G. Wahba, D. R. Johnson, F. Gao and J. Gong, Adaptive Tuning of Numerical Weather Prediction Models: Randomized GCV in Three- and Four-Dimensional Data Assimilation, Monthly Weather Review, 123 (1995), pp. 3358--3369, also available by anonymous ftp from ftp.stat.wisc.edu in pub/wahba.

....Girard (1998) Randomized trace estimates are based on the fact that if A is any square matrix and ffl is a zero mean random n vector with 8 Bernoulli Observations and the ranGACV September 22, 1998 independent components with variance oe 2 ffl , then 1 oe 2 ffl E ffl 0 Affl = trA. See Gong, Wahba, Johnson Tribbia (1998) and references cited there for experimental results with multiple regularization and other parameters in the Gaussian case. In practice oe 2 ffl is replaced by 1 n P n i=1 ffl 2 i . Xiang and Wahba (1997) argued that the approximation f Y ffl Gamma f Y [W (f Y ) n Sigma ] ....

.... truth functions . 6.3 Minimizing ranGACV ( 1 ; 2 ) In order to implement the method in large data sets with multiple smoothing parameters a workable method for finding the minimizer of ranGACV which does not use derivatives is necessary. After a fair amount of experimentation (see, also Gong et al. 1998), Lin (1998a) we have found that the downhill simplex method works well for ranGACV functions encountered in the demographic data sets with Bernoulli data that we have analyzed. Starting guesses for the downhill simplex method may be obtained by trial and error, via a default (for example log fi ....

Gong, J., Wahba, G., Johnson, D. & Tribbia, J. (1998), `Adaptive tuning of numerical weather prediction models: simultaneous estimation of weighting, smoothing and physical parameters', Monthly Weather Review 125, 210--231.

....known as the influence matrix, such that y = A( y quantities independent of y: 2.10) The GCV (generalized cross validation) estimate of is the minimizer of V ( where V ( 1 n dat RSS( 1 n dat tr(I Gamma A( 2 (2. 11) 2 This may be a subset of the entire data set, see Wahba, Johnson, Gao and Gong (1994). GONG, WAHBA, JOHNSON and TRIBBIA July 22, 1996 where n dat is the number of data points (dimension of y) and RSS( ky Gamma yk 2 . Here and elsewhere, if there is no subscript on k Delta k, then the Euclidean norm is intended. The unbiased risk estimate of , which may be used when oe ....

....in the present experiment (164) is probably not be enough to estimate all five parameters in a reproducible fashion. The GML (generalized maximum likelihood) estimate for the parametrization employed here, that is, based on factoring out oe 2 o , may be derived following Wahba (1990b) see also Wahba, Johnson, Gao and Gong (1994), Wahba (1985) Calculations with the GML in the present context appear to be more costly than the randomized trace version of the GCV , although this may change as more advanced numerical methods become available. Other parametrizations for maximum likelihood estimates are also available, see, ....

Wahba, G., Johnson, D., Gao, F. & Gong, J. (1994), Adaptive tuning of numerical weather prediction models: Part I, randomized GCV and related methods in three and four dimensional data assimilation, Technical Report 920, Department of Statistics, University of Wisconsin, Madison, WI.

....the influence matrix, such that y = A( y quantities independent of y: 2.10) The GCV (generalized cross validation) estimate of is given by the minimizer of V ( where V ( 1 n dat RSS( 1 n dat tr(I Gamma A( 2 (2. 11) 2 This may be a subset of the entire data set, see Wahba, Johnson, Gao and Gong (1994). GONG, WAHBA, JOHNSON and TRIBBIA January 17, 1997 where n dat is the number of data points (dimension of y) and RSS( ky Gamma yk 2 . Here and elsewhere, if there is no subscript on k Delta k, then the Euclidean norm is intended. The UBR (unbiased risk) estimate of , which may be used ....

....T in the present experiment (164) is probably not enough to estimate all five parameters in a reproducible fashion. The GML (generalized maximum likelihood) estimate for the parametrization employed here, that is, based on factoring out oe 2 o , may be derived following Wahba (1990b) see also Wahba, Johnson, Gao and Gong (1994), Wahba (1985) Calculations with the GML in the present context appear to be more costly than the randomized trace version of the GCV , although this may change as more advanced numerical methods become available. Other parametrizations for maximum likelihood estimates are also available, see, ....

Wahba, G., Johnson, D., Gao, F. & Gong, J. (1994), Adaptive tuning of numerical weather prediction models: Part I, randomized GCV and related methods in three and four dimensional data assimilation, Technical Report 920, Department of Statistics, University of Wisconsin, Madison, WI.

....as good as exact calculations for large n, see for example Girard(1998) Randomized trace estimates are based on the fact that if A is any square matrix and ffi is a zero mean random n vector with independent components with variance oe 2 ffi , then E ffi 0 Affi = 1 oe 2 ffi trA. See Gong et al.(1998) and references cited there for experimental results with multiple regularization parameters. Returning to the 0 1 data case, it is easy to see that the minimizer f ( Delta) of I is continuous in y, not withstanding the fact that in our training set the y i take on only values 0 or 1. Letting ....

Gong, J., Wahba, G., Johnson, D. & Tribbia, J. (1998), `Adaptive tuning of numerical weather prediction models: simultaneous estimation of weighting, smoothing and physical parameters', Monthly Weather Review 125, 210--231.

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Wahba, G., Johnson, D., Gao, F. & Gong, J. (1995a), `Adaptive tuning of numerical weather prediction models: randomized GCV in three and four dimensional data assimilation', Mon. Wea. Rev. 123, 3358--3369.

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Gong, J., Wahba, G., Johnson, D. & Tribbia, J. (1998), `Adaptive tuning of numerical weather prediction models: simultaneous estimation of weighting, smoothing and physical parameters', Monthly Weather Review 125, 210--231.

....assumptions about K; Sigma and x. We remark that both the GCV and GML estimate can be defined when K is nonlinear. In either the linear or nonlinear case the trace of A may be estimated by the randomized trace method without having A explicitly, given a black box which produces Kx given y, see Wahba, Johnson, Gao Gong (1995). Dee daSilva (1998) and Dee, Gaspari, Redder, Rukhovets daSilva (1998) have used maximum likelihood methods to estimate parameters in forecast error covariances in several practical examples, and have compared some of the results with GCV estimates, obtaining generally similar results in the ....

Wahba, G., Johnson, D., Gao, F. & Gong, J. (1995), `Adaptive tuning of numerical weather prediction models: randomized GCV in three and four dimensional data assimilation', Mon. Wea.

....S (i.e. o ) since it is theoretically based on assuming that the problem is being scaled so that S Gamma1=2 ffl N (0; oe 2 0 I) where S is assumed reasonably correct. Subsets of observations where this is not true, for example radiance data) may be excluded from V by partial GCV, see Wahba, Johnson, Gao Gong (1994), there (I Gamma A) is replaced by E(I Gamma A) where E is a possibly weighted indicator matrix for the observations to be included. However the GCV estimate is robust to various assumptions about K; Sigma and x. We remark that both the GCV and GML estimate can be defined when K is nonlinear. ....

Wahba, G., Johnson, D., Gao, F. & Gong, J. (1994), Adaptive tuning of numerical weather prediction models: Part I, randomized GCV and related methods in three and four dimensional data assimilation, Technical Report 920, Department of Statistics, University of Wisconsin, Madison, WI.

....of the atmosphere. Research supported in part by NASA Grant NAG5 3769 and NSF Grant DMS9121003. ADAPTIVE TUNING, 4D VAR AND REPRESENTERS IN RKHS Grace Wahba 1 Department of Statistics University of Wisconsin Madison WI USA Summary: We (abstractly) generalize the toy weak 4D Var model in Gong, Wahba, Johnson Tribbia (1998) to include adaptive tuning of a variety of parameters throughout the 4D Var variational problem, and note issues of sensitivity and identifiability. We discuss models for model errors which include systematic, short memory and long memory errors. Finally we remark on the role of the theory of ....

....We discuss models for model errors which include systematic, short memory and long memory errors. Finally we remark on the role of the theory of representers in reproducing kernel Hilbert spaces in the weak 4D Var setting. 1 INTRODUCTION We first consider the general setup in the experiment in Gong et al. 1998), which is a toy weak 4D Var model (actually one time and one space variable) with five unknown smoothing, weighting and distributed parameters, which were simultaneously adaptively tuned using generalized cross validation (GCV) calculated via the randomized trace technique. In that setup model ....

[Article contains additional citation context not shown here]

Gong, J., Wahba, G., Johnson, D. & Tribbia, J. (1998), `Adaptive tuning of numerical weather prediction models: simultaneous estimation of weighting, smoothing and physical parameters', Monthly Weather Review 125, 210--231.

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G. Wahba, D.R. Johnson, F. Gao, and J. Gong. Adaptive tuning of numerical weather prediction models: Randomized gcv in three- and four-dimensional data assimilation. Monthly Weather Review, 11:3358--3369, 1995.

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Wea. Rev., 108, 1122--1145. Wahba, G., D. R. Johnson, F. Gao, and J. Gong, 1995: Adaptive tuning of numerical weather prediction models: Randomized GCV in three- and four-dimensional data assimilation. Mon. Wea. Rev., 123, 3358--3369.

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