Generalized Cross-validation Technique Research Articles

An adaptive total generalized variational model for speckle reduction in ultrasound images

In this paper, we propose an adaptive total generalized variation model for removing the speckle noise in ultrasound image. The existence and uniqueness of the minimizers to the proposed model are established. Furthermore, we design an alternating minimization algorithm to solve numerically the model. By incorporating the generalized cross validation technique, the regularization parameter of the model can be updated during the iterations of the designed algorithm. Finally, experimental results are reported to demonstrate the effectiveness of the proposed method, and its performance is competitive with that of the other testing methods.

Semi-parametric models of spatial market integration

We develop fully flexible, semi-parametric models of price linkages in spatially distinct regional US markets for plywood and lumber products. The models are developed within the framework of generalized additive models (GAM) which are estimated using penalized maximum likelihood estimation methods and generalized cross-validation techniques. Such nonlinear models are a natural extension to an extensive literature that has developed increasingly flexible models of the Law of One Price and spatial market integration. The GAM estimates and marginal responses to price changes exhibit substantial nonlinearities. Dynamic impulse responses demonstrate significant asymmetries in price responses to exogenous shocks. The results are largely consistent with efficiently linked regional markets for lumber and plywood products in that price transmission elasticities are generally close to unity and localized market shocks evoke equilibrating adjustments in regional markets.

Penalized spline smoothing using Kaplan-Meier weights with censored data.

In this paper, we consider the problem of nonparametric curve fitting in the specific context of censored data. We propose an extension of the penalized splines approach using Kaplan-Meier weights to take into account the effect of censorship and generalized cross-validation techniques to choose the smoothing parameter adapted to the case of censored samples. Using various simulation studies, we analyze the effectiveness of the censored penalized splines method proposed and show that the performance is quite satisfactory. We have extended this proposal to a generalized additive models (GAM) framework introducing a correction of the censorship effect, thus enabling more complex models to be estimated immediately. A real dataset from Stanford Heart Transplant data is also used to illustrate the methodology proposed, which is shown to be a good alternative when the probability distribution for the response variable and the functional form are not known in censored regression models.

An estimate of the inflation factor and analysis sensitivity in the ensemble Kalman filter

Abstract. The ensemble Kalman filter (EnKF) is a widely used ensemble-based assimilation method, which estimates the forecast error covariance matrix using a Monte Carlo approach that involves an ensemble of short-term forecasts. While the accuracy of the forecast error covariance matrix is crucial for achieving accurate forecasts, the estimate given by the EnKF needs to be improved using inflation techniques. Otherwise, the sampling covariance matrix of perturbed forecast states will underestimate the true forecast error covariance matrix because of the limited ensemble size and large model errors, which may eventually result in the divergence of the filter. In this study, the forecast error covariance inflation factor is estimated using a generalized cross-validation technique. The improved EnKF assimilation scheme is tested on the atmosphere-like Lorenz-96 model with spatially correlated observations, and is shown to reduce the analysis error and increase its sensitivity to the observations.

Automatic regularization parameter selection by generalized cross-validation for total variational Poisson noise removal.

In this paper, we propose an alternating minimization algorithm with an automatic selection of the regularization parameter for image reconstruction of photon-counted images. By using the generalized cross-validation technique, the regularization parameter can be updated in the iterations of the alternating minimization algorithm. Experimental results show that our proposed algorithm outperforms the two existing methods, the maximum likelihood expectation maximization estimator with total variation regularization and the primal dual method, where the parameters must be set in advance.

Implementation and evaluation of data analysis strategies for time-resolved optical spectroscopy.

Time-resolved optical spectroscopy plays a key role in illuminating the mechanisms of many fundamental processes in physics, chemistry, and biology. However, to extract the essential information from the highly complex time-resolved data, advanced data analysis techniques are required. Here we present the implementation strategies and the evaluation of the familiar global lifetime and target analysis as well as the not so widely adopted lifetime distribution analysis (LDA). Furthermore, we demonstrate the implementation of analysis strategies dealing with a number of artifacts inherently present in data from ultrafast optical experiments. The focus of the work is placed on LDA as it allows invaluable exploration depth of the kinetic information contained in the experimental data. We establish a clear regularization procedure for the use of LDA in ultrafast optical spectroscopy and evaluate the performance of a number of factors that play a role in the reliable reconstruction of lifetime distributions. Our results show that the optimal regularization factor can be determined well with the L-curve and the generalized cross-validation techniques. Moreover, the performance evaluations indicate that the most efficient regularization norm is the identity matrix. The analytical procedures described in this work can be readily implemented and used for the analysis of any time-resolved data.

Automatic estimation of the regularization parameter in 2D focusing gravity inversion: application of the method to the Safo manganese mine in the northwest of Iran

We investigate the use of Tikhonov regularization with the minimum support stabilizer for underdetermined 2-D inversion of gravity data. This stabilizer produces models with non-smooth properties which is useful for identifying geologic structures with sharp boundaries. A very important aspect of using Tikhonov regularization is the choice of the regularization parameter that controls the trade off between the data fidelity and the stabilizing functional. The L-curve and generalized cross validation techniques, which only require the relative sizes of the uncertainties in the observations are considered. Both criteria are applied in an iterative process for which at each iteration a value for regularization parameter is estimated. Suitable values for the regularization parameter are successfully determined in both cases for synthetic but practically relevant examples. Whenever the geologic situation permits, it is easier and more efficient to model the subsurface with a 2-D algorithm, rather than to apply a full 3-D approach. Then, because the problem is not large it is appropriate to use the generalized singular value decomposition for solving the problem efficiently. The method is applied on a profile of gravity data acquired over the Safo mining camp in Maku-Iran, which is well known for manganese ores. The presented results demonstrate success in reconstructing the geometry and density distribution of the subsurface source.

Challenging the curse of dimensionality in multivariate local linear regression

Local polynomial fitting for univariate data has been widely studied and discussed, but up until now the multivariate equivalent has often been deemed impractical, due to the so-called curse of dimensionality. Here, rather than discounting it completely, we use density as a threshold to determine where over a data range reliable multivariate smoothing is possible, whilst accepting that in large areas it is not. The merits of a density threshold derived from the asymptotic influence function are shown using both real and simulated data sets. Further, the challenging issue of multivariate bandwidth selection, which is known to be affected detrimentally by sparse data which inevitably arise in higher dimensions, is considered. In an effort to alleviate this problem, two adaptations to generalized cross-validation are implemented, and a simulation study is presented to support the proposed method. It is also discussed how the density threshold and the adapted generalized cross-validation technique introduced herein work neatly together.

Smoothing a Satellite Orbit on the Basis of B‐Spline and Regularization

AbstractA new algorithm is developed to compute a smooth and accurate satellite orbit on the basis of a kinematic orbit and a force model. Such a smoothed orbit is needed e.g. in the gravity field modeling procedure that is based on satellite accelerations. The procedure makes use of residual satellite accelerations defined as the difference between the observed accelerations and the reference ones. Computation of the reference accelerations requires an accurately determined set of satellite positions, which can be taken over from the smoothed orbit. The algorithm for computing the smoothed orbit is based on B‐splines. A component of the orbit (i.e. a set of X‐, Y ‐, or Z‐ coordinates in an inertial frame) is parameterized as a linear combination of B‐spline functions with corresponding coefficients. These coefficients are determined by means of a least‐square adjustment scheme. A force model is exploited to compute satellite accelerations, which are used as additional constraints. In order to find the balance between errors in the positions and unaccuracies of the computed accelerations, the generalized cross‐validation technique is implemented. Thanks to the property of a local support of B‐splines, the procedure is very efficient numerically.A numerical example demonstrates a performance of the proposed procedure. Furthermore, it shows that the solution with acceleration constraints is better than the solution based on the kinematic orbit only. The more precise the force model is, the more accurate smoothed orbit is obtained. Finally, a set of real CHAMP data is considered in the context of gravity field modeling. Smoothed orbits, which are computed both with and without acceleration constraints, as well as the original kinematic orbit are employed for calculating reference accelerations. The “Delft approach” is applied to recover gravity field models, which are compared with the EIGEN‐CG01C model. The results show that usage of the smoothed orbit obviously improves the gravity field model, especially if the smoothed orbit is computed with acceleration constraints.

Multilevel regularization of wavelet based fitting of scattered data ? some experiments

In [6], an adaptive method to approximate unorganized clouds of points by smooth surfaces based on wavelets has been described. The general fitting algorithm operates on a coarse-to-fine basis. It selects on each refinement level in a first step a reduced number of wavelets which are appropriate to represent the features of the data set. In a second step, the fitting surface is constructed as the linear combination of the wavelets which minimizes the distance to the data in a least squares sense. This is followed by a thresholding procedure on the wavelet coefficients to discard those which are too small to contribute much to the surface representation. In this paper, we firstly generalize this strategy to a classically regularized least squares functional by adding a Sobolev norm, taking advantage of the capability of wavelets to characterize Sobolev spaces of even fractional order. After recalling the usual cross-validation technique to determine the involved smoothing parameters, some examples of fitting severely irregularly distributed data, synthetically produced and of geophysical origin, are presented. In order to reduce computational costs, we then introduce a multilevel generalized cross-validation technique which goes beyond the Sobolev formulation and exploits the hierarchical setting based on wavelets. We illustrate the performance of the new strategy on some geophysical data.

Unbiased iterative reconstruction of polarization and space-charge profiles from thermal-wave experiments

The thermal-wave technique or laser-intensity modulation method is an important tool for the non-destructive probing of space-charge and polarization profiles in electrets. Analysing the experimental data requires solving a Fredholm integral equation which is known to be an ill-conditioned problem. This paper presents an iterative approach that is capable of reconstructing inherently unsmooth distributions. The deviations from the true profiles are slightly smaller than those obtained with Tikhonov regularization, while the computational burden is not a limiting factor on modern personal computers. The optimum number of iterations is estimated using the randomized generalized cross-validation technique. Results are shown for a number of model distributions, as well as for experimental data from a layered polyvinylidene fluoride film sandwich.

CGLS-GCV: a hybrid algorithm for low-rank-deficient problems

Given A=A+E∈ C m×n , where rank( A)⪡min( m, n), and b ̃ =b+ε , we investigate the following problems: (a) the construction of approximate minimum norm solutions of the least squares problem min‖ Ax− b‖, and (b) the computation of approximations of the column (row) subspace of A. We propose an algorithm for solving these problems based on conjugate gradient iterations followed by regularization in the generated Krylov subspace. Regularization is introduced for estimating rank( A) and implemented using the generalized cross-validation technique. We report the outcome of numerical experiments, showing that the new algorithm yields results with accuracy comparable to that of the SVD, but at a lower computational cost.

Extending the smoothing spline estimate

An extension of the one-parameter class of functionals considered in the Smoothing Spline Estimate framework is proposed to better deal with scaling problems or anisotropies of data to be approximated. In particular, a matrix, L, of regularizing parameters is introduced in place of the typical parameter λ. By means of an appropriate coordinates transformation, the resulting variational problem can be rephrased in terms of the “traditional” Smoothing Spline Estimate method, and the Generalized Cross Validation technique can consequently be applied to find optimal values for L. Following this approach, numerical experiments have been performed on synthetic 2D data, and the results compared with those obtained with Smoothing Spline Estimate and Hyper Basis Function methods.

Analysis of the Rate of Dissociation of Ligand Complexes

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