Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter

Golub, Gene H.; Heath, Michael T.; Wahba, Grace

doi:10.1080/00401706.1979.10489751

Cited by 3,050 publications

(1,606 citation statements)

References 34 publications

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“…For instance, the Generalized Cross-Validation method (GCV) (Golub et al, 1979) could be extended to estimate several regularization parameters. However, contrary to the ReML approach, the GCV method does not provide any estimate of (multiple) noise components.…”

Section: Discussionmentioning

confidence: 99%

MEG Source Localization under Multiple Constraints: An Extended Bayesian Framework

Mattout¹,

Phillips²,

Henson³

et al.

Complex Medical Engineering

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To use Electroencephalography (EEG) and Magnetoencephalography (MEG) as functional brain 3D imaging techniques, identifiable distributed source models are required. The reconstruction of EEG/ MEG sources rests on inverting these models and is ill-posed because the solution does not depend continuously on the data and there is no unique solution in the absence of prior information or constraints. We have described a general framework that can account for several priors in a common inverse solution. An empirical Bayesian framework based on hierarchical linear models was proposed for the analysis of functional neuroimaging data [Friston, K., Penny, W., Phillips, C., Kiebel, S., Hinton, G., Ashburner, J., 2002. Classical and Bayesian inference in neuroimaging: theory. NeuroImage 16, 465 -483] and was evaluated recently in the context of EEG [Phillips, C., Mattout, J., Rugg, M.D., Maquet, P., Friston, K., 2005. An empirical Bayesian solution to the source reconstruction problem in EEG. NeuroImage 24, 997 -1011]. The approach consists of estimating the expected source distribution and its conditional variance that is constrained by an empirically determined mixture of prior variance components. Estimation uses Expectation-Maximization (EM) to give the Restricted Maximum Likelihood (ReML) estimate of the variance components (in terms of hyperparameters) and the Maximum A Posteriori (MAP) estimate of the source parameters. In this paper, we extend the framework to compare different combinations of priors, using a second level of inference based on Bayesian model selection. Using MonteCarlo simulations, ReML is first compared to a classic Weighted Minimum Norm (WMN) solution under a single constraint. Then, the ReML estimates are evaluated using various combinations of priors. Both standard criterion and ROC-based measures were used to assess localization and detection performance. The empirical Bayes approach proved useful as: (1) ReML was significantly better than WMN for single priors; (2) valid location priors improved ReML source localization; (3) invalid location priors did not significantly impair performance. Finally, we show how model selection, using the logevidence, can be used to select the best combination of priors. This enables a global strategy for multiple prior-based regularization of the MEG/EEG source reconstruction. D

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Section: Discussionmentioning

confidence: 99%

MEG Source Localization under Multiple Constraints: An Extended Bayesian Framework

Mattout¹,

Phillips²,

Henson³

et al.

Complex Medical Engineering

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show abstract

“…The common approach is for example L-curve method (LCM) proposed by Hansen [13] and applied for instance in [14]. Another well-known scheme can be general cross validation method (GCV) introduced in [15]. However, for nonlinear inverse problems rigorous procedures to determine the optimal value of the regularization parameter are rarely available [16].…”

Section: Iteratively Regularized Gauss-newton Methodsmentioning

confidence: 99%

Comparison of two methods for magnetic field synthesis on a solenoid’s axis

Ziolkowski

2011

Electr Eng

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Magnetic field synthesis problems have been discussed at length in scientific literature but they still remain as a topic of research in electrical engineering, physics and medical applications. In these disciplines, there is often a need to design an electromagnetic arrangement which can generate a magnetic field of required distribution. Such an arrangement can also work as an active shield. The aim of an active shield is to generate a specified magnetic field which counteracts the external magnetic fields in a protected region. The idea of active shielding is to construct a suitable arrangement of coils, excited with currents that generate an opposite magnetic field sufficient to cancel out the unwanted external fields. The opposite field must have the same frequency and amplitude as the external field. If the incident field presents a wide bandwidth, the final aim is to generate an opposite field in the same frequency range or at least in a range as large as possible. Two independent methods of magnetic fields synthesis, i.e. iteratively regularized Gauss-Newton method and Genetic Algorithm coupled with Bezier curves-based method, are discussed and compared in this paper.

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“…The second point is the appropriate choice of parameters for reconstruction and the regularization, i.e., K and a. Some methods were reported concerning the choice of a for the case that data include noise with an unknown norm [Golub et al, 1979], and the consideration will be important in the future work.…”

Section: Application To Measured Datamentioning

confidence: 99%

Finite difference analyses of Schumann resonance and reconstruction of lightning distribution

Ando

Hayakawa

Швец³

et al. 2005

Radio Science

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[1] This paper deals with the computation of extremely low frequency propagation associated with the Schumann resonance phenomena and the reconstruction algorithm for source lightning location on the basis of measured Schumann resonance data. The finite difference equations are derived in terms of discretized magnetic fields in the spherical coordinates, introducing the azimuthal symmetry for simplicity. The most reliable electron and neutral density models in the atmosphere and the ionosphere can be used to describe extremely low frequency wave propagation. A linear inverse problem for the reconstruction is formulated using the computed spectra as a set of basis functions to identify lightning distributions with respect to the distances from any observatories to the global thunderstorm centers. Numerical experiments allow us to evaluate properties and precision of the solution in the absence or in the presence of noise in the initial spectral data. The inverse problem is applied to the experimental data collected at a field site in Japan, and distribution of global lightning activity is reconstructed for the data covering the period from March to December 1999. The reconstructed data show a reasonable set of distances from the observatory to well-known global thunderstorm centers, and they indicate the seasonal drift of lightning activity. The problems that were involved in solving the inverse problem are also discussed.

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Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter

Cited by 3,050 publications

References 34 publications

MEG Source Localization under Multiple Constraints: An Extended Bayesian Framework

MEG Source Localization under Multiple Constraints: An Extended Bayesian Framework

Comparison of two methods for magnetic field synthesis on a solenoid’s axis

Finite difference analyses of Schumann resonance and reconstruction of lightning distribution

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