Set Of Support Points Research Articles

Optimal breeding-value prediction using a sparse selection index.

Genomic prediction uses DNA sequences and phenotypes to predict genetic values. In homogeneous populations, theory indicates that the accuracy of genomic prediction increases with sample size. However, differences in allele frequencies and linkage disequilibrium patterns can lead to heterogeneity in SNP effects. In this context, calibrating genomic predictions using a large, potentially heterogeneous, training data set may not lead to optimal prediction accuracy. Some studies tried to address this sample size/homogeneity trade-off using training set optimization algorithms; however, this approach assumes that a single training data set is optimum for all individuals in the prediction set. Here, we propose an approach that identifies, for each individual in the prediction set, a subset from the training data (i.e., a set of support points) from which predictions are derived. The methodology that we propose is a sparse selection index (SSI) that integrates selection index methodology with sparsity-inducing techniques commonly used for high-dimensional regression. The sparsity of the resulting index is controlled by a regularization parameter (λ); the G-Best Linear Unbiased Predictor (G-BLUP) (the prediction method most commonly used in plant and animal breeding) appears as a special case which happens when λ = 0. In this study, we present the methodology and demonstrate (using two wheat data sets with phenotypes collected in 10 different environments) that the SSI can achieve significant (anywhere between 5 and 10%) gains in prediction accuracy relative to the G-BLUP.

Adaptive independent sticky MCMC algorithms

Monte Carlo methods have become essential tools to solve complex Bayesian inference problems in different fields, such as computational statistics, machine learning, and statistical signal processing. In this work, we introduce a novel class of adaptive Monte Carlo methods, called adaptive independent sticky Markov Chain Monte Carlo (MCMC) algorithms, to sample efficiently from any bounded target probability density function (pdf). The new class of algorithms employs adaptive non-parametric proposal densities, which become closer and closer to the target as the number of iterations increases. The proposal pdf is built using interpolation procedures based on a set of support points which is constructed iteratively from previously drawn samples. The algorithm’s efficiency is ensured by a test that supervises the evolution of the set of support points. This extra stage controls the computational cost and the convergence of the proposal density to the target. Each part of the novel family of algorithms is discussed and several examples of specific methods are provided. Although the novel algorithms are presented for univariate target densities, we show how they can be easily extended to the multivariate context by embedding them within a Gibbs-type sampler or the hit and run algorithm. The ergodicity is ensured and discussed. An overview of the related works in the literature is also provided, emphasizing that several well-known existing methods (like the adaptive rejection Metropolis sampling (ARMS) scheme) are encompassed by the new class of algorithms proposed here. Eight numerical examples (including the inference of the hyper-parameters of Gaussian processes, widely used in machine learning for signal processing applications) illustrate the efficiency of sticky schemes, both as stand-alone methods to sample from complicated one-dimensional pdfs and within Gibbs samplers in order to draw from multi-dimensional target distributions.

Two French Oil Terminals in the WWII

Often, literature suggests that the Atlantic Wall was a set of support points (Stutzpunkte) and fortresses (Festungen) extended from Norway up to Spain. This concept is not completely wrong but it masks the logistics involved in its defence activity. In this article the parallel destiny of the Talards and Bouchemaine oil terminals active in France during the German occupation and both having failed a complete destruction is considered. Original air recognition images, archive documents, information from experts and visits on the terrain helped a lot in clarifying their history, organization and preservation states. From previous publications and the present article, it appears now clear that the Atlantic Wall study must be addressed as an interdisciplinary matter with the contributions of experts in different fields.

Block-based Thiele-like blending rational interpolation

Thiele-type continued fraction interpolation may be the favoured nonlinear interpolation in the sense that it is constructed by means of the inverse differences which can be calculated recursively and produce useful intermediate results. However, Thiele's interpolation is in fact a point-based interpolation by which we mean that a new interpolating continued fraction with one more degree of its numerator or denominator polynomial is obtained by adding a tail to the current one, or in other words, this can be reshaped simply by adding a new support point to the current set of support points once at a time. In this paper we introduce so-called block-based inverse differences to extend the point-based Thiele-type interpolation to the block-based Thiele-like blending rational interpolation. The construction process may be outlined as follows: first of all, divide the original set of support points into some subsets (blocks), then construct each block by using whatever interpolation means, linear or rational, and finally assemble these blocks by Thiele's method to shape the whole interpolation scheme. Clearly, many flexible rational interpolation schemes can be obtained in this case including the classical Thiele's continued fraction interpolation as its special case. As an extension, a bivariate analogy is also discussed and numerical examples are given to show the effectiveness of our method.

A Neville-like method via continued fractions

As we know, the classical Neville's algorithm is an effective method used to solve the interpolation problem by polynomials. In this paper, we adopt the idea of the Neville's algorithm to construct a kind of blending rational interpolants via continued fractions. For a given set of support points, there are many ways to build up the interpolation schemes, by which we mean that there are many choices to make to determine the initial interpolants on subsets of support points and then update them step by step to form a solution to the full interpolation problem. Numerical examples are given to show the advantage of our method and a multivariate analogy is also discussed.

A graphical approach for the construction of constrained a and l-optimal designs using efficiency plots

In this paper, a graphical approach for computing constrained optimal designs in regression experiments is presented. The method is simple to implement and can be used in design problems where there are two optimality criteria, and these criteria are expressible as convex functionals of information matrices. The method is applied to seek constrained L and D-optimal designs in quadratic and cubic polynomial models. Several interesting features are noted, including (i) ifξ0 and ξ1 are optimal designs for two optimality criteria φ0 and φ1 and the two designs have the same support, the optimal design for a convex combination of φ0 and φ1 can have a different set of support points, and (ii) D-restricted extrapolation designs. are generally more efficient than A-restricted extrapolation designs for estimating the model parameters.

Subordination, Extreme points and support points

Let s(F) denote the set of functions subordinate to a function F analytic in the open unit disk Δ. Let be the set of functions f analytic in Δ such that where ∞s(f)denotes the closed convex hull of s(f). Let denote those functions analytic in Δ such that the set of support points of s( f) is . In this paper we investigate the problems of describing functions in and Recently, Abu-Muhanna showed that univalent functions, with convex and satisfying a smoothness condition at ∞ must be in . We show that : is closed in the space of analytic functions. Consequently any F univalent with convex is in Moreover if such an F is not a half-plane mapping it belongs to

Some topological properties of support points of convex sets

It is shown, as a corollary to Trojanski’s renorming theorem, that the set of support points (of certain closed convex sets) is connected. Some connectedness properties are established for support functionals, and it is also shown that the support points form an Fσ.