M. F. Hutchinson and F. R. de Hoog, Smoothing noisy data with spline functions, Numer. Math., 1985, no. 1, 99--106.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Department of Primary Industries (QDPI) for interpolating weather data such as rainfall and temperature readings, is due to M. F. Hutchinson (of the Centre for Resource and Environmental Studies, Australian National University ANU) The algorithm is described in detail in Hutchinson and DeHoog [20]. The work is also detailed in Gu et al. 17] 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 ....

....to its context. 3 Direct versus iterative methods The matrix inverses which appear in the GCV function are not explicitly calculated and stored, but are instead obtained by solving an associated matrix system, or system of linear equations. In the initial implementation of Hutchinson and DeHoog [20], the Householder reduction was used to calculate Q = UTU T , where U is orthogonal and T is tridiagonal. Then writing w = U T z, GCV( n w T (T I) Gamma2 w [tr( T I) Gamma1 ) 2 (2) can be evaluated quite economically using the Cholesky LL T factorisation of the ....

M. F. Hutchinson and F. DeHoog. Smoothing noisy data with spline functions. Numer. Math., 47:99--106, 1985.

....However, our method is not e#ected by unequal argument spacing. In defense of the Kalman filter approach, however, Kohn Ansley (1993) found that, for the L = D 2 case, the approach is at least competitive with the widely available cubic smoothing spline method of Reinsch (1967, 1971) and Hutchison de Hoog (1985). Moreover, the use of the Kalman filter brings other useful things, such as confidence and prediction intervals in O(n) calculations, and bandwidth choice by maximum likelihood. We hope that the state space approach continues to evolve, and that these disparate strands in the literature converge ....

M. F. Hutchison & F. R. de Hoog (1985). Smoothing noisy data with spline functions. Numerische Mathematik, 47, 99--106.

....of the solution as measured by the second term. In [73] there is proposed a method to estimate a good value for the smoothing parameter ffi . This simple method leads [47] and [103] to propose the method of Generalized Cross Validation (GCV) The GCV method was successfully improved by [10] [58], 94] 95] and more recently by [45] and others. The theoretical aspects of GCV were analyzed by [47] 70] and [96] 14 Earth s Gravity Field Determination Spherical harmonic analysis T (n; j) T; Y n;j ) L 2( Omega Gamma = Z Omega T (j)Y n;j (j)d (j) 14.1) and spherical ....

Hutchinson, M.F., deHoog, F.R. (1987): Smoothing Noisy Data with Spline Functions. Numer. Math., 50, 311-319

....However, our method is not effected by unequal argument spacing. In defense of the Kalman Filter approach, however, Kohn Ansley (1993) found that, for the L = D 2 case, the approach is at least competitive with the widely available cubic smoothing spline method of Reinsch (1967, 1971) and Hutchison de Hoog (1985). Moreover, the use of the Kalman Filter brings other useful things, such as confidence and prediction intervals in O(n) calculations, and bandwidth choice by maximum likelihood. We hope that the state space approach continues to evolve, and that these disparate strands in the literature converge ....

M. F. Hutchison & F. R. de Hoog (1985). Smoothing noisy data with spline functions.

....is also O(n) is essentially as fast for the variable w j case as for the constant. It should be noted, however, that in the constant coefficient case, Kohn and Ansley (1993) find that the Kalman Filter approach is at least competitive with the widely available method of Reinsch (1967,1971) and Hutchison and de Hoog (1985). Moreover, the use of the Kalman Filter brings other useful things, such as confidence and prediction intervals in O(n) calculations, and bandwidth choice by maximum likelihood. Wang and Brown (1996) use a modified spline to fit a family of curves, each of which has period of one day. They ....

Hutchison, M. F. and de Hoog, F. R. (1985) Smoothing noisy data with spline functions. Numerische Mathematik, 47, 99--106.

....ridge regression, univariate and multivariate smoothing spline regression, partial spline models, penalized GLIM (generalized linear models) penalized likelihood 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 ....

M. F. Hutchinson and F. R. de Hoog, Smoothing Noisy Data with Spline Functions, Numerische Mathematik, 47 (1985), pp. 99--106.

....and Laurent (1968) drawing on earlier unpublished work by Atteia (1966) and Greville (1964) dealt with spline smoothing in a very general and abstract context, and pointed specifically to the possibility of an O(n) algorithm. The papers by Reinsch (1967, 1970) and subsequent development by Hutchison and de Hoog (1985) enabled the development of O(n) smoothing software in the cubic polynomial spline case, that is, when the penalty on g is R (D 2 g) 2 . We want to put the results of Anselone and Laurent (1968) to work in the more general context of L spline smoothing, where the penalty R (D 2 g) 2 ....

Hutchison, M. F. and de Hoog, F. R.(1985) Smoothing noisy data with spline functions. Numerische Mathematik, 47, 99-106.

....= i=n and the test function and the random errors are chosen according to each of the cases presented in Section 2.2. In our simulation, the cubic smoothing procedure in [13] and [15] was applied. For computational aspects of the sum of residuals and the two traces, one can refer to [13] 17] and [18]. The notation of Section 2.2 is used in this section but the set of knots is replaced by a single parameter p. For each replicate the following criteria were used to select p: CV; GCV; AIC; AICC; and GAIC (c) equation (3.1) with c = 1=2, 3 4, 7 8, and 15 16. we obtain, from our simulation ....

Hutchinson, M. F. and de Hoog, F. R. (1985), Smoothing noisy data with spline functions, Numer. Math. 47, 99-106.

.... we derive a computational algorithm completely different from that of Ng (1994) It has the advantage of avoiding the ill conditioning problem in computing the smoothing spline by a standard method such as Reinsch s (1967) which is used in Ng (1994) As has been noted by Reinsch (1971) and Hutchinson and de Hogg (1985), such method can be quite ill conditioned when the data are very unequally spaced. In particular, it breakdowns if ties are present in the data (because of rounding) The problem rearely asizes in the regression context since the design points are often regular or nearly so. In density ....

Hutchinson, M. F., de Hogg, F. R. (1985) Smoothing noisy data with spline functions.

....H( XA Gamma1 X T Gamma Gamma1 : 42) We have mentioned before that the matrix A is positive definite and banded with the bandwidth (4m Gamma 2)D 1 and that the matrix X has at most 2m Gamma 1 non zero elements in each row. Employing the method proposed by Hutchinson and de Hoog [6, 15], the diagonal blocks of the matrix H( Gamma1 and consequently the cross validation score can be calculated with O( mD) 2 M) arithmetic operations, once the solution trajectory is determined. It follows that the number of operations needed for the calculation of the cross validation ....

M. F. Hutchinson and F. R. de Hoog. Smoothing noisy data with spline functions. Numer. Math., 47:99--106, 1985.

....and the results achieved. 1 Introduction The original surface fitting program used for this application is due to M. F. Hutchinson (of the Centre for Resourse and Environmental Studies, Australian National University (ANU) and the algorithm is described in detail in Hutchinson and DeHoog [7]. The work is also detailed in Gu et al. 5] and Wahba [9] 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 ....

M. F. Hutchinson and F. DeHoog. Smoothing noisy data with spline functions. Numer. Math., 47:99--106, 1985.

....ridge regression, univariate and multivariate smoothing spline regression, partial spline models, penalized GLIM (generalized linear models) penalized likelihood 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 ....

M. F. Hutchinson and F. R. de Hoog, Smoothing Noisy Data with Spline Functions, Numerische Mathematik, 47 (1985), pp. 99--106.

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M. F. Hutchinson and F. R. de Hoog, Smoothing noisy data with spline functions, Numer. Math., 1985, no. 1, 99--106.

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