On ANOVA Decompositions of Kernels and Gaussian Random Field Paths

The FANOVA (or “Sobol’-Hoeffding”) decomposition of multivariate functions has been used for high-dimensional model representation and global sensitivity analysis. When the objective function f has no simple analytic form and is costly to evaluate, computing FANOVA terms may be unaffordable due to numerical integration costs. Several approximate approaches relying on Gaussian random field (GRF) models have been proposed to alleviate these costs, where f is substituted by a (kriging) predictor or by conditional simulations. Here we focus on FANOVA decompositions of GRF sample paths, and we notably introduce an associated kernel decomposition into 4 d 4d terms called KANOVA. An interpretation in terms of tensor product projections is obtained, and it is shown that projected kernels control both the sparsity of GRF sample paths and the dependence structure between FANOVA effects. Applications on simulated data show the relevance of the approach for designing new classes of covariance kernels dedicated to high-dimensional kriging.

A generalisation of the fractional Brownian field based on non-Euclidean norms

We explore a generalisation of the L´evy fractional Brownian field on the Euclidean space based on replacing the Euclidean norm with another norm. A characterisation result for admissible norms yields a complete description of all self-similar Gaussian random fields with stationary increments. Several integral representations of the introduced random fields are derived. In a similar vein, several non-Euclidean variants of the fractional Poisson field are introduced and it is shown that they share the covariance structure with the fractional Brownian field and converge to it. The shape parameters of the Poisson and Brownian variants are related by convex geometry transforms, namely the radial pth mean body and the polar projection transforms.

Fast Update of Conditional Simulation Ensembles

Gaussian random field (GRF) conditional simulation is a key ingredient in many spatial statistics problems for computing Monte-Carlo estimators and quantifying uncertainties on non-linear functionals of GRFs conditional on data. Conditional simulations are known to often be computer intensive, especially when appealing to matrix decomposition approaches with a large number of simulation points. This work studies settings where conditioning observations are assimilated batch sequentially, with one point or a batch of points at each stage. Assuming that conditional simulations have been performed at a previous stage, the goal is to take advantage of already available sample paths and by-products to produce updated conditional simulations at mini- mal cost. Explicit formulae are provided, which allow updating an ensemble of sample paths conditioned on n ≥ 0 observations to an ensemble conditioned on n + q observations, for arbitrary q ≥ 1. Compared to direct approaches, the proposed formulae proveto substantially reduce computational complexity. Moreover, these formulae explicitly exhibit how the q new observations are updating the old sample paths. Detailed complexity calculations highlighting the benefits of this approach with respect to state-of-the-art algorithms are provided and are complemented by numerical experiments.

Continued fractions built from convex sets and convex functions

In a partially ordered semigroup with the duality (or polarity) transform, it is pos- sible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

Phosphocreatine interacts with phospholipids, affects membrane properties and exerts membrane-protective effects

A broad spectrum of beneficial effects has been ascribed to creatine (Cr), phosphocreatine (PCr) and their cyclic analogues cyclo-(cCr) and phospho-cyclocreatine (PcCr). Cr is widely used as nutritional supplement in sports and increasingly also as adjuvant treatment for pathologies such as myopathies and a plethora of neurodegenerative diseases. Additionally, Cr and its cyclic analogues have been proposed for anti-cancer treatment. The mechanisms involved in these pleiotropic effects are still controversial and far from being understood. The reversible conversion of Cr and ATP into PCr and ADP by creatine kinase, generating highly diffusible PCr energy reserves, is certainly an important element. However, some protective effects of Cr and analogues cannot be satisfactorily explained solely by effects on the cellular energy state. Here we used mainly liposome model systems to provide evidence for interaction of PCr and PcCr with different zwitterionic phospholipids by applying four independent, complementary biochemical and biophysical assays: (i) chemical binding assay, (ii) surface plasmon resonance spectroscopy (SPR), (iii) solid-state (31)P-NMR, and (iv) differential scanning calorimetry (DSC). SPR revealed low affinity PCr/phospholipid interaction that additionally induced changes in liposome shape as indicated by NMR and SPR. Additionally, DSC revealed evidence for membrane packing effects by PCr, as seen by altered lipid phase transition. Finally, PCr efficiently protected against membrane permeabilization in two different model systems: liposome-permeabilization by the membrane-active peptide melittin, and erythrocyte hemolysis by the oxidative drug doxorubicin, hypoosmotic stress or the mild detergent saponin. These findings suggest a new molecular basis for non-energy related functions of PCr and its cyclic analogue. PCr/phospholipid interaction and alteration of membrane structure may not only protect cellular membranes against various insults, but could have more general implications for many physiological membrane-related functions that are relevant for health and disease.

Comparative Analysis of Evolving Software Systems Using the Gini Coefficient

Software metrics offer us the promise of distilling useful information from vast amounts of software in order to track development progress, to gain insights into the nature of the software, and to identify potential problems. Unfortunately, however, many software metrics exhibit highly skewed, non-Gaussian distributions. As a consequence, usual ways of interpreting these metrics --- for example, in terms of "average" values --- can be highly misleading. Many metrics, it turns out, are distributed like wealth --- with high concentrations of values in selected locations. We propose to analyze software metrics using the Gini coefficient, a higher-order statistic widely used in economics to study the distribution of wealth. Our approach allows us not only to observe changes in software systems efficiently, but also to assess project risks and monitor the development process itself. We apply the Gini coefficient to numerous metrics over a range of software projects, and we show that many metrics not only display remarkably high Gini values, but that these values are remarkably consistent as a project evolves over time.

Internal and external effects on cognition: Evidence from memory training in very preterm-born children and cognition in patients with carotid artery stenosis before and after treatment

The present synopsis aims to integrate one study about memory training in very preterm-born children and two studies about cognition in patients with carotid artery stenosis before and after treatments. Preterm-born children are at increased risk of cognitive deficits and behavioural problems compared with peers born at term. This thesis determined whether memory training would improve cognitive functions in school-age very preterm-born children. Memory strategy training produced significant improvements in trained and non-trained cognitive functions; a core working memory training revealed significant effects on short-term memory and working memory tasks. Six months after training, children in both training groups showed better working memory performance than children in the waiting control group. This is evidence that memory training – an external influence on cognition – induces plastic changes in very preterm-born children. Patients with carotid artery stenosis are known to be at increased risk of cognitive impairment. We showed that patients with symptomatic or asymptomatic carotid artery stenosis were at higher risk for cognitive deficits than expected in a normative sample. This thesis seeks to link cognitive plasticity to internal factors like carotid stenosis. An external factor, which influences blood flow to the brain is the nature of the carotid artery stenosis treatment. Research on the effects of carotid artery stenosis treatment on cognition has produced inconsistent results. We found significant improvement in frontal lobe functions, visual memory and motor speed one year after treatment independent of the treatment type (best medical treatment, carotid artery stenting, carotid artery endarterectomy); providing evidence for ‘treatment-induced’ cognitive plasticity. Baseline performance was negatively associated with improvement in various cognitive functions after training in very preterm-born children and after treatment in patients with carotid artery stenosis. The present synopsis aims to integrate these findings into the current and relevant literature, and discuss consequences as well as methodological considerations resulting from the studies constituting the thesis at hand.

On degeneracy and invariances of random fields paths with applications in Gaussian process modelling

We study pathwise invariances and degeneracies of random fields with motivating applications in Gaussian process modelling. The key idea is that a number of structural properties one may wish to impose a priori on functions boil down to degeneracy properties under well-chosen linear operators. We first show in a second order set-up that almost sure degeneracy of random field paths under some class of linear operators defined in terms of signed measures can be controlled through the two first moments. A special focus is then put on the Gaussian case, where these results are revisited and extended to further linear operators thanks to state-of-the-art representations. Several degeneracy properties are tackled, including random fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promising results and perspectives in Gaussian process modelling. In a first numerical experiment, it is shown that dedicated kernels can be used to infer an axis of symmetry. Our second numerical experiment deals with conditional simulations of a solution to the heat equation, and it is found that adapted kernels notably enable improved predictions of non-linear functionals of the field such as its maximum.

Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

Multi-objective optimization algorithms aim at finding Pareto-optimal solutions. Recovering Pareto fronts or Pareto sets from a limited number of function evaluations are challenging problems. A popular approach in the case of expensive-to-evaluate functions is to appeal to metamodels. Kriging has been shown efficient as a base for sequential multi-objective optimization, notably through infill sampling criteria balancing exploitation and exploration such as the Expected Hypervolume Improvement. Here we consider Kriging metamodels not only for selecting new points, but as a tool for estimating the whole Pareto front and quantifying how much uncertainty remains on it at any stage of Kriging-based multi-objective optimization algorithms. Our approach relies on the Gaussian process interpretation of Kriging, and bases upon conditional simulations. Using concepts from random set theory, we propose to adapt the Vorob’ev expectation and deviation to capture the variability of the set of non-dominated points. Numerical experiments illustrate the potential of the proposed workflow, and it is shown on examples how Gaussian process simulations and the estimated Vorob’ev deviation can be used to monitor the ability of Kriging-based multi-objective optimization algorithms to accurately learn the Pareto front.

Atlas-Based Segmentation of Brain Tumor Images Using a Markov Random Field-Based Tumor Growth Model and Non-Rigid Registration

We propose a new and clinically oriented approach to perform atlas-based segmentation of brain tumor images. A mesh-free method is used to model tumor-induced soft tissue deformations in a healthy brain atlas image with subsequent registration of the modified atlas to a pathologic patient image. The atlas is seeded with a tumor position prior and tumor growth simulating the tumor mass effect is performed with the aim of improving the registration accuracy in case of patients with space-occupying lesions. We perform tests on 2D axial slices of five different patient data sets and show that the approach gives good results for the segmentation of white matter, grey matter, cerebrospinal fluid and the tumor.

Estimating and quantifying uncertainties on level sets using the Vorob'ev expectation and deviation with Gaussian process models

Several methods based on Kriging have recently been proposed for calculating a probability of failure involving costly-to-evaluate functions. A closely related problem is to estimate the set of inputs leading to a response exceeding a given threshold. Now, estimating such a level set—and not solely its volume—and quantifying uncertainties on it are not straightforward. Here we use notions from random set theory to obtain an estimate of the level set, together with a quantification of estimation uncertainty. We give explicit formulae in the Gaussian process set-up and provide a consistency result. We then illustrate how space-filling versus adaptive design strategies may sequentially reduce level set estimation uncertainty.