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.

Statistical characterization and development of summary measures of non-normal distributions of nuclear morphometry data from prostate cancer cells for use as prognostic indicators

Nuclear morphometry (NM) uses image analysis to measure features of the cell nucleus which are classified as: bulk properties, shape or form, and DNA distribution. Studies have used these measurements as diagnostic and prognostic indicators of disease with inconclusive results. The distributional properties of these variables have not been systematically investigated although much of the medical data exhibit nonnormal distributions. Measurements are done on several hundred cells per patient so summary measurements reflecting the underlying distribution are needed.^ Distributional characteristics of 34 NM variables from prostate cancer cells were investigated using graphical and analytical techniques. Cells per sample ranged from 52 to 458. A small sample of patients with benign prostatic hyperplasia (BPH), representing non-cancer cells, was used for general comparison with the cancer cells.^ Data transformations such as log, square root and 1/x did not yield normality as measured by the Shapiro-Wilks test for normality. A modulus transformation, used for distributions having abnormal kurtosis values, also did not produce normality.^ Kernel density histograms of the 34 variables exhibited non-normality and 18 variables also exhibited bimodality. A bimodality coefficient was calculated and 3 variables: DNA concentration, shape and elongation, showed the strongest evidence of bimodality and were studied further.^ Two analytical approaches were used to obtain a summary measure for each variable for each patient: cluster analysis to determine significant clusters and a mixture model analysis using a two component model having a Gaussian distribution with equal variances. The mixture component parameters were used to bootstrap the log likelihood ratio to determine the significant number of components, 1 or 2. These summary measures were used as predictors of disease severity in several proportional odds logistic regression models. The disease severity scale had 5 levels and was constructed of 3 components: extracapsulary penetration (ECP), lymph node involvement (LN+) and seminal vesicle involvement (SV+) which represent surrogate measures of prognosis. The summary measures were not strong predictors of disease severity. There was some indication from the mixture model results that there were changes in mean levels and proportions of the components in the lower severity levels. ^

Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥0 is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.

Towards understanding thermal jet quenching via lattice simulations

After reviewing how simulations employing classical lattice gauge theory permit to test a conjectured Euclideanization property of a light-cone Wilson loop in a thermal non-Abelian plasma, we show how Euclidean data can in turn be used to estimate the transverse collision kernel, C(k⊥), characterizing the broadening of a high-energy jet. First results, based on data produced recently by Panero et al, suggest that C(k⊥) is enhanced over the known NLO result in a soft regime k⊥ < a few T. The shape of k3⊥ C(k⊥) is consistent with a Gaussian at small k⊥.

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.

Bounding Standard Gaussian Tail Probabilities

We review various inequalities for Mills' ratio (1 - Φ)= Ø, where Ø and Φ denote the standard Gaussian density and distribution function, respectively. Elementary considerations involving finite continued fractions lead to a general approximation scheme which implies and refines several known bounds.

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.

Global Optimization with Sparse and Local Gaussian Process Models

We present a novel surrogate model-based global optimization framework allowing a large number of function evaluations. The method, called SpLEGO, is based on a multi-scale expected improvement (EI) framework relying on both sparse and local Gaussian process (GP) models. First, a bi-objective approach relying on a global sparse GP model is used to determine potential next sampling regions. Local GP models are then constructed within each selected region. The method subsequently employs the standard expected improvement criterion to deal with the exploration-exploitation trade-off within selected local models, leading to a decision on where to perform the next function evaluation(s). The potential of our approach is demonstrated using the so-called Sparse Pseudo-input GP as a global model. The algorithm is tested on four benchmark problems, whose number of starting points ranges from 102 to 104. Our results show that SpLEGO is effective and capable of solving problems with large number of starting points, and it even provides significant advantages when compared with state-of-the-art EI algorithms.

Kernel bandwidth estimation for moving object detection in non-stabilized cameras

The evolution of the television market is led by 3DTV technology, and this tendency can accelerate during the next years according to expert forecasts. However, 3DTV delivery by broadcast networks is not currently developed enough, and acts as a bottleneck for the complete deployment of the technology. Thus, increasing interest is dedicated to ste-reo 3DTV formats compatible with current HDTV video equipment and infrastructure, as they may greatly encourage 3D acceptance. In this paper, different subsampling schemes for HDTV compatible transmission of both progressive and interlaced stereo 3DTV are studied and compared. The frequency characteristics and preserved frequency content of each scheme are analyzed, and a simple interpolation filter is specially designed. Finally, the advantages and disadvantages of the different schemes and filters are evaluated through quality testing on several progressive and interlaced video sequences.

Learning an L1-regularized Gaussian Bayesian Network in the Equivalence Class Space

Learning the structure of a graphical model from data is a common task in a wide range of practical applications. In this paper, we focus on Gaussian Bayesian networks, i.e., on continuous data and directed acyclic graphs with a joint probability density of all variables given by a Gaussian. We propose to work in an equivalence class search space, specifically using the k-greedy equivalence search algorithm. This, combined with regularization techniques to guide the structure search, can learn sparse networks close to the one that generated the data. We provide results on some synthetic networks and on modeling the gene network of the two biological pathways regulating the biosynthesis of isoprenoids for the Arabidopsis thaliana plant

Nutritive value of palm kernel meal in diets for growing rabbits

The aim of this work was to determine the nutritive value of palm kernel meal (PKM) in diets for growing rabbits. In Experiment 1, 20 New Zealand × Californian growing rabbits 50 d-old were used to determine energy, crude protein, fibre and fat digestibility of PKM. The nutritive value was estimated by the difference method using a basal diet and another diet made by substituting 200 g/kg of basal diet with PKM. Energy, crude protein, ether extract and neutral detergent fibre of PKM digestibilities were, respectively, 0.549 (±0.056, SE), 0.541 (±0.069), 0.850 (±0.048) and 0.430 (±0.101), and the digestible energy concentration was 10.9 MJ/kg (±1.03) DM. In Experiment 2, 412 rabbits were allocated at random to the two experimental diets to measure growing performance. Inclusion of 200 g PKM/kg in the diet did not affect feed or digestible energy intake but decreased slightly (by around 5%) average daily gain (P = 0.003) and feed efficiency (P < 0.001). Neither mortality nor Clostridium perfringens counts in soft faeces were affected by type of diet. Palm kernel meal can be considered a palatable source of fibre, protein and fat for rabbits and can substitute significant amounts of other fibrous ingredients in the diet without adverse effects on growth performance.

Hand Image Segmentation by means of Gaussian Multiscale Aggregation for biometric applications

Applying biometrics to daily scenarios involves demanding requirements in terms of software and hardware. On the contrary, current biometric techniques are also being adapted to present-day devices, like mobile phones, laptops and the like, which are far from meeting the previous stated requirements. In fact, achieving a combination of both necessities is one of the most difficult problems at present in biometrics. Therefore, this paper presents a segmentation algorithm able to provide suitable solutions in terms of precision for hand biometric recognition, considering a wide range of backgrounds like carpets, glass, grass, mud, pavement, plastic, tiles or wood. Results highlight that segmentation accuracy is carried out with high rates of precision (F-measure 88%)), presenting competitive time results when compared to state-of-the-art segmentation algorithms time performance

Gaussian Multiscale Aggregation oriented to Hand Biometric Segmentation in Mobile Devices

New trends in biometrics are oriented to mobile devices in order to increase the overall security in daily actions like bank account access, e-commerce or even document protection within the mobile. However, applying biometrics to mobile devices imply challenging aspects in biometric data acquisition, feature extraction or private data storage. Concretely, this paper attempts to deal with the problem of hand segmentation given a picture of the hand in an unknown background, requiring an accurate result in terms of hand isolation. For the sake of user acceptability, no restrictions are done on background, and therefore, hand images can be taken without any constraint, resulting segmentation in an exigent task. Multiscale aggregation strategies are proposed in order to solve this problem due to their accurate results in unconstrained and complicated scenarios, together with their properties in time performance. This method is evaluated with a public synthetic database with 480000 images considering different backgrounds and illumination environments. The results obtained in terms of accuracy and time performance highlight their capability of being a suitable solution for the problem of hand segmentation in contact-less environments, outperforming competitive methods in literature like Lossy Data Compression image segmentation (LDC).

Gaussian Multiscale Aggregation Applied to Segmentation in Hand Biometrics

This paper presents an image segmentation algorithm based on Gaussian multiscale aggregation oriented to hand biometric applications. The method is able to isolate the hand from a wide variety of background textures such as carpets, fabric, glass, grass, soil or stones. The evaluation was carried out by using a publicly available synthetic database with 408,000 hand images in different backgrounds, comparing the performance in terms of accuracy and computational cost to two competitive segmentation methods existing in literature, namely Lossy Data Compression (LDC) and Normalized Cuts (NCuts). The results highlight that the proposed method outperforms current competitive segmentation methods with regard to computational cost, time performance, accuracy and memory usage.