Open-source machine learning: R meets Weka

Two of the prime open-source environments available for machine/statistical learning in data mining and knowledge discovery are the software packages Weka and R which have emerged from the machine learning and statistics communities, respectively. To make the different sets of tools from both environments available in a single unified system, an R package RWeka is suggested which interfaces Weka’s functionality to R. With only a thin layer of (mostly R) code, a set of general interface generators is provided which can set up interface functions with the usual “R look and feel”, re-using Weka’s standardized interface of learner classes (including classifiers, clusterers, associators, filters, loaders, savers, and stemmers) with associated methods.     

minPtest: a resampling based gene region-level testing procedure for genetic case-control studies

Current technologies generate a huge number of single nucleotide polymorphism (SNP) genotype measurements in case-control studies. The resulting multiple testing problem can be ameliorated by considering candidate gene regions. The minPtest R package provides the first widely accessible implementation of a gene region-level summary for each candidate gene using the min $P$ test. The latter is a permutation-based method that can be based on different univariate tests per SNP. The package brings together three different kinds of tests which were scattered over several R packages, and automatically selects the most appropriate one for the study design at hand. The implementation of the minPtest integrates two different parallel computing packages, thus optimally leveraging available resources for speedy results.     

Advances in integrative statistics for logic programming

We present recent developments on the syntax of Real, a library for interfacing two Prolog systems to the statistical language R. We focus on the changes in Prolog syntax within SWI-Prolog that accommodate greater syntactic integration, enhanced user experience and improved features for web-services. We recount the full syntax and functionality of Real as well as presenting a full application and sister packages which include Prolog code interfacing a number of common and useful tasks that can be delegated to R. We argue that Real is a powerful extension to logic programming, providing access to a popular statistical system that has complementary strengths in areas such as machine learning, statistical inference and visualisation. Furthermore, Real has a central role to play in the uptake of semantic web, computational biology and bioinformatics as application areas for research in logic programming.     

On directional convexity

Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R ^d → R is called D-convex, where D is a set of vectors in R ^d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R ^d, denoted by co^D(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R ² where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co ^D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R ³ (where D is the orthonormal basis of R ³), we construct arbitrarily large finite sets A with co ^D(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices. This research was supported by Charles University Grants No. 158/99 and 159/99.     

On Functional Separately Convex Hulls

Let D be a set of vectors in R ^d . A function f: R ^d → R is called D-convex if its restriction to each line parallel to a nonzero vector of D is a convex function. For a set A⊆ R ^d , the functional D-convex hull of A, denoted by co ^D (A) , is the intersection of the zero sets of all nonnegative D -convex functions that are 0 on A . We prove some results concerning the structure of functional D -convex hulls, e.g., a Krein—Milman-type theorem and a result on separation of connected components. We give a polynomial-time algorithm for computing co ^D (A) for a finite point set A (in any fixed dimension) in the case of D being a basis of R ^d (the case of separate convexity). This research is primarily motivated by questions concerning the so-called rank-one convexity, which is a particular case of D -convexity and is important in the theory of systems of nonlinear partial differential equations and in mathematical modeling of microstructures in solids. As a direct contribution to the study of rank-one convexity, we construct a configuration of 20 symmetric 2 x 2 matrices in a general (stable) position with a nontrivial functionally rank-one convex hull (answering a question of K. Zhang on the existence of higher-dimensional nontrivial configurations of points and matrices). Received October 3, 1995, and in revised form June 24, 1996.     

Convex surfaces which intersect each congruent copy of themselves in a connected set

LetK ⊂R ³ be a three-dimensional convex body such that, for every isometry ρ ofR ³, the boundaries ofK and ρK meet in a connected set. ThenK is a parallel set of some possibly degenerate linesegment.     

Characterization of Quasi-Coherent Modules that are Module Schemes

Let R be a commutative ring with unit, and let E be an R-module. We say the functor of R-modules E, defined by E(B) = E ? _R B, is a quasi-coherent R-module, and its dual E* is an R-module scheme. Both types of R-module functors are essential for the development of the theory of the linear representations of an affine R-group. We prove that a quasi-coherent R-module E is an R-module scheme if and only if E is a projective R-module of finite type, and, as a consequence, we also characterize finitely generated projective R-modules.     

Doob's Decomposition Theorem for Fuzzy (Super) Submartingales

Abstract

The notions of fuzzy random variables and fuzzy (super) submartingales are introduced. In this paper we provide the necessary and sufficient conditions of Doob's decomposition for fuzzy (super) submartingales. Finally, we discuss the decomposition of fuzzy (super) submartingales on R, and an example is given which explains that not every fuzzy (super) submartingale has Doob's decomposition.     

Symplectic algebras and poisson algebras

Abstract

A ring R is called left P-injective if for every a ∈ R, aR = r(l(a)) where l( ? ) and r( ? ) denote left and right annihilators respectively. The ring R is called left GP-injective if for any 0 ≠ a ∈ R, there exists n > 0 such that a ⁿ  ≠ 0 and a ⁿ R = r(l(a ⁿ )). As a response to an open question on GP -injective rings, an example of a left GP-injective ring which is not left P-injective is given. It is also proved here that a ring R is left FP -injective if and only if every matrix ring 𝕄 _n (R) is left GP-injective.     

Sparsity Inducing Prior Distributions for Correlation Matrices of Longitudinal Data

For longitudinal data, the modeling of a correlation matrix ?R can be a difficult statistical task due to both the positive definite and the unit diagonal constraints. Because the number of parameters increases quadratically in the dimension, it is often useful to consider a sparse parameterization. We introduce a pair of prior distributions on the set of correlation matrices for longitudinal data through the partial autocorrelations (PACs), which vary independently over (?1,1). The first prior shrinks each of the PACs toward zero with increasingly aggressive shrinkage in lag. The second prior (a selection prior) is a mixture of a zero point mass and a continuous component for each PAC, allowing for a sparse representation. The structure implied under our priors is readily interpretable for time-ordered responses because each zero PAC implies a conditional independence relationship in the distribution of the data. Selection priors on the PACs provide a computationally attractive alternative to selection on the elements of ?R or ?R^{? 1} for ordered data. These priors allow for data-dependent shrinkage/selection under an intuitive parameterization in an unconstrained setting. The proposed priors are compared to standard methods through a simulation study and illustrated using a multivariate probit data example. Supplemental materials for this article (appendix, data, and R code) are available online.     

Hopf Algebra Approaches to Nonlinear Algebraic Equations

Abstract

Let M be a k-vector space and R ∈ Hom(M ^?p , M ^?q ), we present a general version of the FRT-construction, we provide a method for examining whether an FRT-bialgebra A(R) has a pre-braided structure and whether M can be regarded as an A(R)-dimodule. We show that the FRT-relation plays a fundamental role in determining the algebra structure on the FRT-bialgebra and the compatibility condition of relevant dimodule. As an example, we give a Hopf algebra approach for solving both homogeneous and non-homogeneous nonlinear (algebraic) equations.     

Prospects and challenges in R package development

R, a software package for statistical computing and graphics, has evolved into the lingua franca of (computational) statistics. One of the cornerstones of R’s success is the decentralized and modularized way of creating software using a multi-tiered development model: The R Development Core Team provides the “base system”, which delivers basic statistical functionality, and many other developers contribute code in the form of extensions in a standardized format via so-called packages. In order to be accessible by a broader audience, packages are made available via standardized source code repositories. To support such a loosely coupled development model, repositories should be able to verify that the provided packages meet certain formal quality criteria and “work”: both relative to the development of the base R system as well as with other packages (interoperability). However, established quality assurance systems and collaborative infrastructures typically face several challenges, some of which we will discuss in this paper.     

Intelligent Support for Exploratory Data Analysis

Abstract

Exploratory data analysis (EDA) is as much a matter of strategy as of selecting specific statistical operations. We have developed a knowledge-based planning system, called AIDE, to help users with EDA. AIDE strikes a balance between conventional statistical packages, which need guidance for every step in the exploration, and autonomous systems, which leave the user entirely out of the decision-making process. AIDE's processing is based on artificial intelligence planning techniques, which give us a useful means of representing some types of statistical strategy. In this article we describe the design of AIDE and its behavior in exploring a small, complex data set.     

Adaptive Isotopic Approximation of Nonsingular Curves: the Parameterizability and Nonlocal Isotopy Approach

We consider domain subdivision algorithms for computing isotopic approximations of a nonsingular algebraic curve. The curve is given by a polynomial equation f(X,Y)=0. Two algorithms in this area are from Snyder (1992) SIGGRAPH Comput. Graphics, 26(2), 121 and Plantinga and Vegter (2004) In Proc. Eurographics Symposium on Geometry Processing, pp. 245–254. We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parameterizability criterion for subdivision, and like Plantinga and Vegter, we exploit nonlocal isotopy. We further extend our algorithm in two important and practical directions: first, we allow subdivision cells to be rectangles with arbitrary but bounded aspect ratios. Second, we extend the input domains to be regions R ₀ with arbitrary geometry and which might not be simply connected. Our algorithm halts as long as the curve has no singularities in the region, and intersects the boundary of R ₀ transversally. Our algorithm is practical and easy to implement exactly. We report some very encouraging experimental results, showing that our algorithms can be much more efficient than the algorithms of Plantinga–Vegter and Snyder.     

Graphs and their parallel groups

Given an immersion of a manifoldf: M→R ^n+k , dimensionM=n, the parallel groupP(f) off is formed by the diffeomorphisms ofM such that the normalk-planes at points of each orbit are parallel. In [3] we studied the parallel group of a plane closed curve. Here we concentrate on immersionsf: R ⁿ →R ⁿ⁺¹, special attention being paid to graphs of smooth maps fromR toR. Graphs of smooth mapsf: S ⁿ →R ^m are also dealt with and we characterise those maps of which the graph has nontrivial parallel group. To end up we find a sufficient condition for the triviality of the tangent group.     

Computing denumerants in numerical 3-semigroups

Abstract

As far as we know, usual computer algebra packages can not compute denumerants for almost medium (about a hundred digits) or almost medium-large (about a thousand digits) input data in a reasonably time cost on an ordinary computer. Implemented algorithms can manage numerical n-semigroups for small input data.

Basically, the denumerant of a non-negative element s ∈ ? is the number of non-negative integer solutions of certain linear non-negative Diophantine equation which constant term is equal to s.

Here we are interested in denumerants of numerical 3-semigroups which have almost medium input data. A new algorithm for computing denumerants is given for this task. It can manage almost medium input data in the worst case and medium-large or even large input data in some cases.     

Root Sublattices and Torus-Restriction of Prehomogeneous Vector Spaces Associated with Dynkin Quivers

For a Dynkin quiver Γ with r vertices, a subset S of the vertices of Γ, and an r-tuple d = (d⁽¹⁾, d⁽²⁾,…, d^(r)) of positive integers, we define a “torus-restricted” representation (G_S, R _d (Γ)) in natural way. Here we put G_S = G₁ × G₂ × … ×G_r, where each G_i is either SL(d⁽ⁱ⁾) or GL(d⁽ⁱ⁾) according to S containing i or not. In this paper, for a prescribed torus-restriction S, we give a necessary and sufficient condition on d that R _d (Γ) has only finitely many G_S-orbits. This can be paraphrased as a condition whether or not d is contained in a certain lattice spanned by positive roots of Γ. We also discuss the prehomogeneity of (G_S, R _d (Γ)).     

Comparison of Methods for Image Analysis on cDNA Microarray Data

Microarrays are part of a new class of biotechnologies which allow the monitoring of expression levels for thousands of genes simultaneously. Image analysis is an important aspect of microarray experiments, one that can have a potentially large impact on subsequent analyses such as clustering or the identification of differentially expressed genes. This article reviews a number of existing image analysis approaches for cDNA microarray experiments and proposes new addressing, segmentation, and background correction methods for extracting information from microarray scanned images. The segmentation component uses a seeded region growing algorithm which makes provision for spots of different shapes and sizes. The background estimation approach is based on an image analysis technique known as morphological opening. These new image analysis procedures are implemented in a software package named Spot, built on the R environment for statistical computing. The statistical properties of the different segmentation and background adjustment methods are examined using microarray data from a study of lipid metabolism in mice. It is shown that in some cases background adjustment can substantially reduce the precision—that is, increase the variability—of low-intensity spot values. In contrast, the choice of segmentation procedure has a smaller impact. The comparison further suggests that seeded region growing segmentation with morphological background correction provides precise and accurate estimates of foreground and background intensities.     

Orders in regular rings with minimal condition for principal right ideals

ABSTRACT

A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ?_(p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M _m (R) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M _m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a ₁ R + ˙˙˙ a _m R = dR, with m ≥ 2, a ₁,…, a _m, d ∈ R, there exist u ₁ ∈ U(R) and u ₂,…, u _m ∈ W(R) such that a ₁ u ₁ + ? a _m u _m = Rd.     

A domain decomposition method for conformal mapping onto a rectangle

Letg be the function which maps conformally a rectangleR onto a simply connected domainG so that the four vertices ofR are mapped respectively onto four specified pointsz ₁,z ₂,z ₃,z ₄ onG. This paper is concerned with the study of a domain decomposition method for computing approximations tog and to an associated domain functional in cases where: (i)G is bounded by two parallel straight lines and two Jordan arcs. (ii) The four pointsz ₁,z ₂,z ₃,Z ₄, are the corners where the two straight lines meet the two arcs.Communicated by Dieter Gaier.