G. Wahba and J. Wendelberger. Some new mathematical methods for variational objective analysis using splines and cross-validation. Monthly Weather Rev., 108:1122--1145, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... and Wahba [39] The theoretical basis of the GCV (and other related algorithms, e.g. GACV Generalised Approximate Cross Validation) has been widely covered over several decades, as can be seen through Golub et al. 13] and most notably through the continuous work of Wahba and collaborators [36, 37, 38, 39, 40]. There are two major components of the surface fitting application, namely the fitting of a thin plate spline function to the observed data, and the calculation of the output surface by evaluating the spline function at each point on a regular grid. The section of the algorithm which constructs ....

G. Wahba and J. Wendelberger. Some new mathematical methods for variational objective analysis using splines and cross validation. Monthly Weather Rev., 108:1122--1145, 1980.

....is with Hiroshima City University, Hiroshima shi, 731 3194 Japan. The author is with the Graduate School of Information Science, Nara Institute of Science and Technology, Ikomashi, 630 0101 Japan. A similar means ofp.M setting wasemp5 yed in other fields of engineering at that time [2] The method of Isomichi [3] 4] which wasp.ME5 ted in the samep eriod, formulatespates.M to fit more general functions of one dimensional signals in quantified emp .HM4 data This method defines thepe.M 2 expM 2 itly, i e , the solution is a function with which the sum of the error between the ....

G. Wahba and J. Wendelberger, "Some new mathemati cal methods for variational objective analysis usingspng-C and cross validation," Monthly Weather Review, vol.108, p108,-CqM1L1980.

....curvature. Multivariate spline approximations have been proposed that retain a roughness minimizing property analogous to that of univariate cubic splines. Duchon [1] derived multivariate analogs of cubic splines where the locations of the data need not be regularly spaced. Wahba and Wendelberger [2] extended these results to generate smoothing splines, which allow for noisy data. This approach is discussed more fully in Wahba [3] Franke [4] studied multivariate thin plate splines with tension. 1 The above approximations fall into the more general category of radial basis functions ....

....spline. Naturally, the roughness measure depends on the choice of g. Section 2 gives the connection between radial basis function methods and roughness minimizing splines in the general context of smoothing splines. The formulation and proof is similar to that of Wahba and Wendelberger [2]. Radial basis functions may contain one or more free parameters, and smoothing splines have a smoothing parameter. Often there is no physical basis for choosing the values of these parameters. The cross validation method discussed in Section 3 provides an objective way to make these choices. 2 ....

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G. Wahba and J. Wendelberger, Some new mathematical methods for variational objective analysis using splines and cross validation, Monthly Weather Review, Vol. 108, pp. 1122-- 1143, 1980.

....65L10, 65L50, 65L60 1 Introduction Multivariate spline approximations have been studied intensively during the last decade. The use of multivariate analogs of cubic splines was derived successfully by Duchon [1] where the locations of the data need not be regularly spaced. Wahba and Wendelberger [14] extended these results to generate smoothing spline, which allow for noisy data. The use of multivariate thin plate splines with tension was reported by Franke [2] These approximations fall into the more general category of radial basis functions (RBFs) methods, which were described in details ....

Wahba G. and Wendelberger J., Some New Mathematical Methods for Variational Objective Analysis using Splines and Cross Validation, Monthly Weather Review, Vol. 108, pp. 1122--1143, 1980.

....model. In the early DA phase, the data and models both improve somewhat, and the first attempts are made at combining the two by direct insertion of the data into the models or nudging. During the mature DA phase, the data are more numerous and the models more skillful. Spline interpolation (Wahba and Wendelberger, 1980) and statistical regression, or optimal interpolation (Daley, 1991) methods become widespread. The present WMO Symposium illustrates the advanced DA phase in meteorology and oceanography. Plentiful data sets and truly fine models motivate the use of high performance DA methods, such as the ....

Wahba, G., and J. Wendelberger, 1980: Some new mathematical methods for variational objective analysis using splines and cross-validation. Mon. Wea. Rev., 108, 1122--1143.

....various candidate models based on independent data sets and on a variety of parametric and nonparameteric statistical tests. 2. 6 Generalized Cross Validation As a final experiment with the rawinsonde data we produced parameter estimates by means of the Generalized Cross Validation (GCV) method (Wahba and Wendelberger 1980), briefly described in Section I 4.5 and summarized in Appendix I B. Figure 7 shows the GCV estimates superimposed on the maximum likelihood estimates. The only difference between the two sets of estimates is that they are based on the minimization of two different cost functions. The estimates ....

Wahba, G., and J. Wendelberger, 1980: Some new mathematical methods for variational objective analysis using splines and cross-validation. Mon.

....from two separate observing instruments. The heart of this paper is in Section 4, where we discuss in detail the application of the maximum likelihood method to the problem of estimating any unknown covariance parameters from the data. We briefly refer to the Generalized Cross Validation method (Wahba and Wendelberger 1980), which is reviewed in Appendix B. Section 5 contains concluding remarks. 2 Covariance models Suppose that the n vector w f k is a model forecast valid at time t k , and w t k is the unknown true state of the atmosphere at that time. It is convenient to define both quantities in terms of the ....

....However, if the underlying assumptions on the error distributions are wrong, one might legitimately ask whether there are any other criteria that lead to a more robust parameter estimation procedure. One candidate for such a criterion follows from the Generalized 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 ....

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Wahba, G., and J. Wendelberger, 1980: Some new mathematical methods for variational objective analysis using splines and cross-validation. Mon.

....ridge regression literature focuses on using (2.2.3) with C 0 C = I, so the penalty is based on the squared length of fi. However, there may be situations where the appropriate penalty is based on an assumption of spatial smoothness, typically of the form fi 0 C 0 Cfi. In spline smoothing (Wahba and Wendelberger 1980, Wahba 1983 plus many of the references therein) the matrix C is often used to penalize finite second differences of the parameter, which means using a (p Gamma 2) Theta p matrix C 25 of the form C = 2 6 6 6 6 6 6 6 4 1 Gamma2 1 0 : 0 0 1 Gamma2 1 : 0 . ....

....0 0 1 Gamma2 1 : 0 . 0 0 : 1 Gamma2 1 3 7 7 7 7 7 7 7 5 We can interpret C 0 C as penalizing mean square curvature of the solution, which is an intuitive penalty. However, little work has gone into finding more appropriate choices based on the data. Wahba and Wendelberger (1980) use GCV to estimate both the degree of derivative penalty and the smoothing parameter. A more natural specification of C arises if we use a Bayesian formulation, for in that case C 0 C corresponds to the a priori covariance matrix of fi. In the next chapter we will outline a procedure for ....

Wahba, G. and J. Wendelberger (1980). Some new mathematical methods for variational objective analysis using splines and cross validation. Monthly Weather Review 108, 1122--1143.

....H (ff) Omega H (fi) s is (in an obvious notation) R H (ff) s ff ; t ff )R H (fi) s (s fi ; t fi ) Of course any positive definite function may in principle play the role of a reproducing kernel here. Conditionally positive definite functions [32] as in thin plate splines [44] may also be used. The point evaluation functionals f f(t(i) may be replaced by bounded linear functionals on H, and other functions can be added to H 0 subject just to the uniqueness conditions, making this class of function estimates broadly useful in many applications. See [4] 5] 6] 7] ....

....functionals on H, and other functions can be added to H 0 subject just to the uniqueness conditions, making this class of function estimates broadly useful in many applications. See [4] 5] 6] 7] 15] 16] 17] 18] 19] 20] 21] 22] 23] 24] 25] 29] 33] 34] 35] 37] 38] 39] 40] 41] [44] . Wahba and Luo Smoothing Spline ANOVA for Large Data Sets 5 2 Choosing the Smoothing Parameters Probably the most popular method for choosing = 1 ; Delta Delta Delta ; p ) j ( Gamma1 1 ; Delta Delta Delta ; Gamma1 p ) is the GCV method, 7] 14] which chooses as the ....

[Article contains additional citation context not shown here]

G. Wahba and J. Wendelberger. Some new mathematical methods for variational objective analysis using splines and cross-validation. Monthly Weather Review, 108:1122--1145, 1980.

.... (s ff ; t ff )R H (fi) s (s fi ; t fi ) Of course any positive definite function may in principle play the role of a reproducing kernel here. Special properties of RK s related to splines are noted in Wahba (1990) Conditionally positive definite functions as occur in thin plate splines (Wahba and Wendelberger 1980) can be accomodated, see Gu and Wahba(1993a) and references cited there. Examples on the sphere can be found in Wahba(1981, 1982) Weber and Talkner (1993) and on a discrete index set, as might occur in large contingency tables, in Gu and Wahba (1991a) It is not hard to modify reproducing ....

Wahba, G. & Wendelberger, J. (1980), `Some new mathematical methods for variational objective analysis using splines and cross-validation', Monthly Weather Review 108, 1122--1145.

.... (s ff ; t ff )R H (fi) s (s fi ; t fi ) Of course any positive definite function may in principle play the role of a reproducing kernel here. Special properties of RK s related to splines are noted in Wahba (1990) Conditionally positive definite functions as occur in thin plate splines (Wahba and Wendelberger 1980) can be accomodated, see Gu and Wahba(1993a) and references cited there. Examples on the sphere can be found in Wahba(1981, 1982) Weber and Talkner (1993) and on a discrete index set, as might occur in large contingency tables, in Gu and Wahba (1991a) It is not hard to modify reproducing ....

....estimation to much larger data sets is an area of active research. One tool to consider is to approximate the span of the (n M) basis functions in (2.10) by a carefully chosen subset. The variational problem of (3.2) is then solved in this lower dimensional subspace. This approach was proposed in Wahba(1980) for the special case of thin plate splines, and has been developed and implemented by Hutchinson (1984) in ANUSPLIN(1984) Hutchinson and Gessler (1994) O Sullivan (1990) and others. See also the discussion in Section 7 of Nychka, Wahba, Goldfarb and Pugh (1984) and references cited there. ....

Wahba, G. & Wendelberger, J. (1980), `Some new mathematical methods for variational objective analysis using splines and cross-validation', Monthly Weather Review 108, 1122--1145.

....observational data, alternatively, forecast data may be incorporated into x . Formally, the matrices S and Sigma may be derived as covariance matrices under certain statistical assumptions. See Parrish and Derber(1992) Lorenc, Bell and McPherson(1991) Lorenc(1986) Wahba( 1990, 1985, 1982) Wahba and Wendelberger(1980), Kimeldorf and Wahba(1971) Four dimensional assimilation with the model as a strong constraint can be put in this framework by, for example, letting x be the state at an initial time and including model integrations in K. Physically based penalties, for example, energy in gravity waves, may be ....

.... a good estimate of the which minimizes R( under fairly general conditions on x true , irrespective of whether x true is considered to be a fixed vector satisfying certain conditions, or whether it is considered as a random vector with covariance a multiple of Sigma, see Craven and Wahba(1979) Wahba and Wendelberger(1980), Speckman(1985) Li(1986) Generally is also a good estimate of the minimizer of D( kx Gamma x true k 2 under some fairly but not completely general conditions, see Wahba and Wang(1990) A cross validation based estimate for oe 2 that has been shown to work well in examples is oe 2 ....

[Article contains additional citation context not shown here]

Wahba, G. & Wendelberger, J. (1980), `Some new mathematical methods for variational objective analysis using splines and cross-validation', Monthly Weather Review 108, 1122--1145.

....H (ff) Omega H (fi) s is (in an obvious notation) R H (ff) s ff ; t ff )R H (fi) s (s fi ; t fi ) Of course any positive definite function may in principle play the role of a reproducing kernel here. Conditionally positive definite functions [32] as in thin plate splines [44] may also be used. The point evaluation functionals f f(t(i) may be replaced by bounded linear functionals on H, and other functions can be added to H 0 subject just to the uniqueness conditions, making this class of function estimates broadly useful in many applications. See [4] 5] 6] 7] ....

....functionals on H, and other functions can be added to H 0 subject just to the uniqueness conditions, making this class of function estimates broadly useful in many applications. See [4] 5] 6] 7] 15] 16] 17] 18] 19] 20] 21] 22] 23] 24] 25] 29] 33] 34] 35] 37] 38] 39] 40] 41] [44] . 2 Backfitting Hastie and Tibshirani [26] Section 5.2.3, discuss the backfitting (a. k. a. GaussSeidel) algorithm in the context of the general setup of SS ANOVA problems Wahba and Luo Smoothing Spline ANOVA for Large Data Sets 5 as was described by [6] Further discussion of the ....

[Article contains additional citation context not shown here]

G. Wahba and J. Wendelberger. Some new mathematical methods for variational objective analysis using splines and cross-validation. Monthly Weather Review, 108:1122--1145, 1980.

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G. Wahba and J. Wendelberger. Some new mathematical methods for variational objective analysis using splines and cross-validation. Monthly Weather Rev., 108:1122--1145, 1980.

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Wahba, G., Wendelberger, J. (1980): Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation. Mon. Wea. Rev., 108, 1122-1143

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G. Wahba and J. Wendelberger, Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation, Monthly Weather Review, 108 (1980), pp. 1122--1143.

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G. Wahba, J. Wendelberger,(1980) "Some new mathematical methods for variational objective analysis using splines and cross-validation", Monthly Weather Review, pp.1122-1143.

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