An analytic comparison of regularization methods for Gaussian Processes

Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. They have many applications in the field of Computer Experiments, in particular to perform sensitivity analysis, adaptive design of experiments and global optimization. Nearly all of the applications of GPs require the inversion of a covariance matrix that, in practice, is often ill-conditioned. Regularization methodologies are then employed with consequences on the GPs that need to be better understood.The two principal methods to deal with ill-conditioned covariance matrices are i) pseudoinverse and ii) adding a positive constant to the diagonal (the so-called nugget regularization).The first part of this paper provides an algebraic comparison of PI and nugget regularizations. Redundant points, responsible for covariance matrix singularity, are defined. It is proven that pseudoinverse regularization, contrarily to nugget regularization, averages the output values and makes the variance zero at redundant points. However, pseudoinverse and nugget regularizations become equivalent as the nugget value vanishes. A measure for data-model discrepancy is proposed which serves for choosing a regularization technique.In the second part of the paper, a distribution-wise GP is introduced that interpolates Gaussian distributions instead of data points. Distribution-wise GP can be seen as an improved regularization method for GPs

Mathematical modeling for the activation of t-lymphocytes: population balance modeling with non conventional growth law

International audienceThis article focuses on the mathematical modeling techniques applied to the particulate processes in biological sciences particularly Immunology. T-cell activation is studied with respect to their surface protein (CD3) concentration that is described in-terms of two particulate processes, growth and birth processes. The production of new proteins and change in their concentrations (on cells) has created the idea of balance law in the form of hyperbolic problems. In order to add a new dimension to the problem, CPBM (Cell Population Balance Model) is used with respect to multi-proteins concentrations. A mathematical model is used to find analytical solution while Finite volume scheme is used to approximate numerically the solution of PBE (Population Balance Equation) in order to anticipate and analyze the density of activated T-cells according to the given initial protein concentration

Bimodal distribution of the solid products in a magmatic chamber: Modelling by fractional crystallization and coupling of the chemical exchanges with the differential melt/solid transport ================================================ Répartition bimodale des produits solides d'une chambre magmatique : modélisation par cristallisation fractionnée avec couplage des échanges chimiques et du transport différentiel magma/solide

International audienceOur aim is to explain the possible bimodality of the compositions of the magmatic rocks of the same province. In order to do so, we present a model for the crystallization of a magmatic chamber, coupling the three phenomena: solidification, sedimentation, chemical reactions between the solid and the liquid. These three phenomena make two independent dimensionless parameters appear: the ratios of the solidification rate to the transport velocity, and of the chemical kinetics to the transport velocity. The model is written for one independent chemical component. It is shown that, for certain values of the dimensionless parameters, the chemical composition of the chamber can present a bimodal distribution, starting from uniform initial conditions. This model shows that the coupling between three elementary phenomena is enough to explain the bimodality, or more generally the appearance of discontinuities of chemical compositions, without making any additional assumption. ==================================================== Nous cherchons à rendre compte de la bimodalité possible des compositions des roches magmatiques d'une même province. Pour cela, nous présentons un modèle de cristallisation d'une chambre magmatique couplant les trois phénomènes : solidification, sédimentation, réactions chimiques entre le solide et le liquide. Ceux-ci font apparaître deux paramètres sans dimension indépendants, exprimant les rapports respectifs de la vitesse de solidification sur la vitesse du déplacement solide/liquide, et de la cinétique d'échange sur la vitesse de déplacement. Le modèle est écrit pour un constituant chimique indépendant. On montre que, pour certaines valeurs des paramètres sans dimension, la composition chimique de la chambre peut présenter une répartition bimodale, alors que les conditions initiales sont uniformes. Ce modèle montre que le couplage entre trois phénomènes élémentaires suffit à rendre compte de la bimodalité, ou plus généralement, de l'apparition de discontinuités de compositions, sans faire intervenir d'hypothèse additionnelle

Carotid Atherosclerotic Markers in CADASIL

Purpose: Cerebral autosomal dominant arteriopathy with subcortical infarcts and leukoencephalopathy (CADASIL) is a cerebral small vessel disease caused by mutations of the NOTCH3 gene. Marked variations in disease severity have raised the hypothesis that non-genetic factors may modulate the expressivity of the phenotype. The aim of the current study was to evaluate whether atherosclerosis, assessed by carotid duplex ultrasonography, is associated with variations in the clinical and MRI phenotype of CADASIL. Methods: Data from 144 consecutive patients enrolled in an ongoing prospective cohort study were collected. Degree of disability was assessed by the modified Rankin Scale, that of cognitive impairment by the Mattis Dementia Rating Scale (MDRS). The total volume of the brain, of lacunar lesions and of white matter hyperintensities, the number of cerebral microhemorrhages, and parameters derived from histograms of apparent diffusion coefficient were measured on cerebral MRI. Atherosclerosis was evaluated by B-mode ultrasonography of carotid arteries. Both the carotid intima-media thickness cIMT) and the presence of carotid plaques or stenosis were recorded. Results: Higher cIMT was found to be independently associated with lower MDRS scores when this score was less than the quartile limit (p = 0.02). Only a trend for a positive association was detected between cIMT and the Rankin score (p = 0.06). There was no significant association between carotid markers and the occurrence of stroke or MRI parameters except for diffusion data. The mean and peak values of MRI diffusion histograms were found positively associated with the presence of plaques (p < 0.01). Conclusion: The results suggest that the severity of atherosclerosis may relate to cognitive decline in CADASIL and that this effect is possibly related to the degree of microstructural cerebral tissue lesions. Longitudinal studies are needed to confirm these results. Copyright (C) 2010 S. Karger AG, Base

An analysis of covariance parameters in Gaussian process-based optimization

The need for globally optimizing expensive-to-evaluate functions frequently occurs in many real-world applications. Among the methods developed for solving such problems, the Efficient Global Optimization (EGO) is regarded as one of the state-of-the-art unconstrained continuous optimization algorithms. The surrogate model used in EGO is a Gaussian process (GP) conditional on data points. The most important control on the efficiency of the EGO algorithm is the GP covariance function (or kernel), which is taken as a parameterized function. In this paper, we theoretically and empirically analyze the effect of the covariance parameters, the so-called "characteristic length scale" and "nugget", on EGO performance. More precisely, we analyze the EGO algorithm with fixed covariance parameters and compare them to the standard setting where they are statistically estimated. The limit behavior of EGO with very small or very large characteristic length scales is identified. Experiments show that a "small" nugget should be preferred to its maximum likelihood estimate. Overall, this study contributes to a better theoretical and practical understanding of a key optimization algorithm

Parameter Optimization in Groundwater using Proper Orthogonal Decomposition as a Reduced Modeling Technique

http://www.epsmso.gr/all_conf_index/abstracts/ic-scce_2012_abs054.pdfInternational audienceThis paper deals with different approaches of applying Proper Orthogonal Decomposition in the field of groundwater flow, specifically the Richards equation, which is a convection-diffusion partial differential equation governing the behaviour of unsaturated fluid flow through a porous medium. The motivation for this research is the need to reduce computational complexity in inverse modelling studies, where a significant number of simulations are needed to determine suitable model parameters. Three different methods of implementing Proper Orthogonal Decomposition are explored. The first method is the Petrov-Galerkin method, a method well suited to speeding up linear problems. The second method is a "Hybrid" method, and proposes a linearization of all non-linear functions, building upon the Petrov-Galerkin approach. As such, it is suitable for use in the non-saturated groundwater zone. The third method combines the use of kriging and Proper Orthogonal to create a non-intrusive model for comparison purposes. With these three methods, the suitability of Proper Orthogonal as a reduced modelling method for unsaturated groundwater flow is shown

Gated blood-pool SPECT evaluation of changes after radiofrequency catheter ablation of accessory pathways Evidence for persistent ventricular preexcitation despite successful therapy

AbstractOBJECTIVESThis study was designed to prospectively evaluate the effects of radiofrequency ablation in Wolff-Parkinson-White (WPW) syndrome by scintigraphic analysis.BACKGROUNDThe functional changes triggered by radiofrequency current ablation of atrioventricular accessory pathways are not fully known.METHODSForty-four patients with WPW syndrome were consecutively investigated before and 48 h after radiofrequency therapy. Fourteen patients had right sided atrioventricular pathways and 30 patients had left sided bypass-tracts. Planar gated imaging and gated blood pool tomography were performed in all of these patients.RESULTSA significant increase in the left ventricular ejection fraction (LVEF) was demonstrated in patients with left preexcitation (62.2 ± 7.9% before ablation against 64.4 ± 6.3% after ablation, p = 0.02) but not for those with right sided anomalous pathway. Phase analysis only gave significant differences following ablation of right sided pathways (left-to-right phase difference = 14.4 ± 13.8° before ablation versus 7.5 ± 7.2° after ablation, p < 0.05). Early abnormal ventricular contraction persisted in 12 patients with right accessory pathways and in 8 patients with left accessory pathways despite the complete disappearance of any abnormal conduction as proven electrophysiologically.CONCLUSIONSFollowing catheter ablation of atrioventricular accessory pathways: 1) an improvement of left ventricular function may be seen, particularly in patients with left sided accessory pathways, and 2) unexpected persistence of local ventricular preexcitation at the site of successful ablation may be detected

Shotgun lipidomics and mass spectrometry imaging unveil diversity and dynamics in Gammarus fossarum lipid composition

Sentinel species are playing an indispensable role in monitoring environmental pollution in aquatic ecosystems. Many pollutants found in water prove to be endocrine disrupting chemicals that could cause disruptions in lipid homeostasis in aquatic species. A comprehensive profiling of the lipidome of these species is thus an essential step toward understanding the mechanism of toxicity induced by pollutants. Both the composition and spatial distribution of lipids in freshwater crustacean Gammarus fossarum were extensively examined herein. The baseline lipidome of gammarids of different sex and reproductive stages was established by high throughput shotgun lipidomics. Spatial lipid mapping by high resolution mass spectrometry imaging led to the discovery of sulfate-based lipids in hepato-pancreas and their accumulation in mature oocytes. A diverse and dynamic lipid composition in G. fossarum was uncovered, which deepens our understanding of the biochemical changes during development and which could serve as a reference for future ecotoxicological studies.Approches Protéomique et Lipidomique pour la compréhension des mécanismes moléculaires de toxicité en lien avec l'altération du métabolisme lipidique chez l'espèce sentinelle Gammarus fossarum durant le cycle de reproductio