Practical use of robust GCV and modified GCV for spline smoothing

Generalized cross-validation (GCV) is a popular parameter selection criterion for spline smoothing of noisy data, but it sometimes yields a severely undersmoothed estimate, especially if the sample size is small. Robust GCV (RGCV) and modified GCV are stable extensions of GCV, with the degree of stabilization depending on a parameter \(\gamma \in (0,1)\) for RGCV and on a parameter \(\rho >1\) for modified GCV. While there are favorable asymptotic results about the performance of RGCV and modified GCV, little is known for finite samples. In a large simulation study with cubic splines, we investigate the behavior of the optimal values of \(\gamma \) and \(\rho \), and identify simple practical rules to choose them that are close to optimal. With these rules, both RGCV and modified GCV perform significantly better than GCV. The performance is defined in terms of the Sobolev error, which is shown by example to be more consistent with a visual assessment of the fit than the prediction error (average squared error). The results are consistent with known asymptotic results.     

A hyperbolic surface with a square grid net

We prove the existence of a hyperbolic surface spread over the sphere for which the projection map has all its singular values on the extended real line and such that the preimage of the extended real line under the projection map is homeomorphic to the square grid in the plane. This answers a question raised by è. B. Vinberg. Supported by the Alexander von Humboldt Foundation. Supported by NSF grants DMS-0400636, DMS-0244421, and DMS-0244547.     

A canonical module characterization of Serre’s (R1)

In this short note, we give a characterization of domains satisfying Serre’s condition (R₁) in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author [9 Yanagawa, K. (2015). Dualizing complexes of seminormal a?ne semigroup rings and toric face rings. J. Algebra 425:367–391.[Crossref], [Web of Science ®] , [Google Scholar]] where the normality is described in terms of the “shape” of the canonical module.     

Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellis?s largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellis?s reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.     

On Configurational-Force-Driven Crack Propagation and Adaptive Mesh Refinement in Finite Inelasticity

We introduce a consistent variational framework for inelasticity at finite strains, yielding dual balances in physical and material space as the Euler equations. The formulation is employed for the simultaneous usage of configurational forces as both driving forces for crack propagation as well as h-adaptive mesh refinement. The theoretical basis builds upon a global balance of internal and external power, where the mechanical response is exclusively governed by two scalar functions, the free energy function and a dissipation potential. The resulting variational structure is exploited in the context of fracture mechanics and yields evolution equations for internal variables. In the discrete setting, we present a geometry model fully separated from the finite element mesh structure that represents structural changes of the material configuration due to crack propagation. Advanced meshing algorithms provide an optimal discretization at the crack tip. Local and global criteria are obtained via error estimators based on configurational forces being interpreted as indicators of an energetic misfit due to an insufficient discretization. The numerical handling is decomposed into a staggered algorithm scheme for the dual set of equilibrium equations in material and physical space and efficient mesh generation tools. Exemplary numerical examples are considered to illustrate the method and to underline the effects of inelastic material behaviour in the presented context. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High-resolution non-invasive 3D imaging of paint microstructure by synchrotron-based X-ray laminography

The characterisation of the microstructure and micromechanical behaviour of paint is key to a range of problems related to the conservation or technical art history of paintings. Synchrotron-based X-ray laminography is demonstrated in this paper to image the local sub-surface microstructure in paintings in a non-invasive and non-destructive way. Based on absorption and phase contrast, the method can provide high-resolution 3D maps of the paint stratigraphy, including the substrate, and visualise small features, such as pigment particles, voids, cracks, wood cells, canvas fibres etc. Reconstructions may be indicative of local density or chemical composition due to increased attenuation of X-rays by elements of higher atomic number. The paint layers and their interfaces can be distinguished via variations in morphology or composition. Results of feasibility tests on a painting mockup (oak panel, chalk ground, vermilion and lead white paint) are shown, where lateral and depth resolution of up to a few micrometres is demonstrated. The method is well adapted to study the temporal evolution of the stratigraphy in test specimens and offers an alternative to destructive sampling of original works of art.     

The use of multivariate MR imaging intensities versus metabolic data from MR spectroscopic imaging for brain tumour classification

This study investigated the value of information from both magnetic resonance imaging and magnetic resonance spectroscopic imaging (MRSI) to automated discrimination of brain tumours. The influence of imaging intensities and metabolic data was tested by comparing the use of MR spectra from MRSI, MR imaging intensities, peak integration values obtained from the MR spectra and a combination of the latter two. Three classification techniques were objectively compared: linear discriminant analysis, least squares support vector machines (LS-SVM) with a linear kernel as linear techniques and LS-SVM with radial basis function kernel as a nonlinear technique. Classifiers were evaluated over 100 stratified random splittings of the dataset into training and test sets. The area under the receiver operating characteristic (ROC) curve (AUC) was used as a global performance measure on test data. In general, all techniques obtained a high performance when using peak integration values with or without MR imaging intensities. For example for low- versus high-grade tumours, low- versus high-grade gliomas and gliomas versus meningiomas, the mean test AUC was higher than 0.91, 0.94, and 0.99, respectively, when both MR imaging intensities and peak integration values were used. The use of metabolic data from MRSI significantly improved automated classification of brain tumour types compared to the use of MR imaging intensities solely.     

Wolfram

Ohne Zusammenfassung