Infovis and Statistical Graphics: Different Goals,Different Looks

The importance of graphical displays in statistical practice has been recognized sporadically in the statistical literature over the past century, with wider awareness following Tukey's Exploratory Data Analysis and Tufte's books in the succeeding decades. But statistical graphics still occupy an awkward in-between position: within statistics, exploratory and graphical methods represent a minor subfield and are not well integrated with larger themes of modeling and inference. Outside of statistics, infographics (also called information visualization or Infovis) are huge, but their purveyors and enthusiasts appear largely to be uninterested in statistical principles.

We present here a set of goals for graphical displays discussed primarily from the statistical point of view and discuss some inherent contradictions in these goals that may be impeding communication between the fields of statistics and Infovis. One of our constructive suggestions, to Infovis practitioners and statisticians alike, is to try not to cram into a single graph what can be better displayed in two or more. We recognize that we offer only one perspective and intend this article to be a starting point for a wide-ranging discussion among graphic designers, statisticians, and users of statistical methods. The purpose of this article is not to criticize but to explore the different goals that lead researchers in different fields to value different aspects of data visualization.     

Guiding Data Analysts with Visual Statistical Strategies

Abstract

The concept of statistical strategy is introduced and used to develop a structured graphical user interface for guiding data analysis. The interface visually represents statistical strategies that are designed by expert data analysts to guide novices. The representation is an abstraction of the expert's concepts of the essence of a data analysis. We argue that an environment that visually guides and structures data analysis will improve data analysis productivity, accuracy, accessibility, and satisfaction in comparison to an environment without such aids, especially for novice data analysts. Our concepts are based on notions from cognitive science, and can be empirically evaluated. The interface consists of two interacting windows—the guidemap and the workmap. Each window contains a graph that has nodes and edges. The guidemap graph represents the statistical strategy for a specific statistical task (such as describing data). Nodes represent potential data analysis actions that can be taken by the system. Edges represent potential actions that can be taken by the analyst. The guidemap graph exists prior to the data analysis session, having been created by an expert. The workmap graph represents the complete history of all steps taken by the data analyst. It is constructed during the data analysis session as a result of the analyst's actions. Workmap nodes represent data sets, data models, or data analysis procedures that have been created or used by the analyst. Workmap edges represent the chronological sequence of the analyst's actions. One workmap node is highlighted to indicate which statistical object is the focus of the strategy. We illustrate our concepts with ViSta, the Visual Statistics system that we have developed.     

The Mode Tree: A Tool for Visualization of Nonparametric Density Features

Abstract

Recognition and extraction of features in a nonparametric density estimate are highly dependent on correct calibration. The data-driven choice of bandwidth h in kernel density estimation is a difficult one that is compounded by the fact that the globally optimal h is not generally optimal for all values of x. In recognition of this fact a new type of graphical tool, the mode tree, is proposed. The basic mode tree plot relates the locations of modes in density estimates with the bandwidths of those estimates. Additional information can be included on the plot indicating factors such as the size of modes, how modes split, and the locations of antimodes and bumps. The use of a mode tree in adaptive multimodality investigations is proposed, and an example is given to show the value in using a normal kernel, as opposed to the biweight or other kernels, in such investigations. Examples of such investigations are provided for Ahrens's chondrite data and van Winkle's Hidalgo stamp data. Finally, the bivariate mode tree is introduced, together with an example using Scott's lipid data.     

The model method: Students’ performance and its effectiveness

This study describes Singapore students’ (N = 607) performance on two tasks in a recently developed Mathematics Processing Instrument (MPI). The MPI comprised tasks sourced from Australia's NAPLAN and Singapore's PSLE. This study also examines students’ use of the model method to solve the two tasks. The model method is a visual problem-solving heuristic prevalently used in Singapore classrooms. The study found that students who solved the tasks using a visual method predominantly used the model method as a visual problem-solving strategy. Another interesting observation was the hindrance of successful problem solving caused by the persistence of prototypical images of model drawings. Implications include encouraging teachers to get their students to identify problem situations where the model method will both work and not work well, and making the role of the generator in the model method explicit in the mathematics textbooks.     

Higher order optimal approximation of Csiszar's f-divergence

Here are established various sharp and nearly optimal probabilistic inequalities giving the high order approximation of Csiszar's f-divergence between two probability measures, which is the most essential and general tool for their comparison. The above are done through Taylor's formula, generalized Taylor–Widder's formula, an alternative recent expansion formula. Based on these we give many representation formulae of Csiszar's distance, then we estimate in all directions their remainders by using either the norms approach or the modulus of continuity way. Most of the last probabilistic estimates are sharp or nearly sharp, attained by basic simple functions.     

N‐person Prisoner's Dilemma†

Multi‐person versions of Prisoner's Dilemma are widely applicable in the social sciences. Examination of two important classes of real‐world situations reveals that although both can appropriately be called Prisoner's Dilemma, they have incompatible payoff structures. Thus Prisoner's Dilemma games constitute an important but apparently ambiguous set of models.

We therefore undertake a taxonomy of multi‐person Prisoner's Dilemma. Some aspects of the well‐studied two‐person case provide a useful beginning for the task. In the general multi‐person form, however, some properties of the two‐person game are found incompatible with others and so are dropped. Additional properties are suggested by strategic considerations and the associated social phenomena. We demonstrate interdependencies among the various properties and relate some of them to a simple graphical representation.     

Growth Curve Model with Hierarchical Within-Individuals Design Matrices

This paper deals with some inferential problems under an extended growth curve model with several hierarchical within-individuals design matrices. The model includes the one whose mean structure consists of polynomial growth curves with different degrees. First we consider the case when the covariance matrix is unknown positive definite. We derive a LR test for examining the hierarchical structure for within individuals design matrices and a model selection criterion. Next we consider the case when a random coefficients covariance structure is assumed, under certain assumption of between-individual design matrices. Similar inferential problems are also considered. The dental measurement data (see, e.g., Potthoff and Roy (1964, Biometrika, 51, 313-326)) is reexamined, based on extended growth curve models.     

Divide-and-Conquer With Sequential Monte Carlo

We propose a novel class of Sequential Monte Carlo (SMC) algorithms, appropriate for inference in probabilistic graphical models. This class of algorithms adopts a divide-and-conquer approach based upon an auxiliary tree-structured decomposition of the model of interest, turning the overall inferential task into a collection of recursively solved subproblems. The proposed method is applicable to a broad class of probabilistic graphical models, including models with loops. Unlike a standard SMC sampler, the proposed divide-and-conquer SMC employs multiple independent populations of weighted particles, which are resampled, merged, and propagated as the method progresses. We illustrate empirically that this approach can outperform standard methods in terms of the accuracy of the posterior expectation and marginal likelihood approximations. Divide-and-conquer SMC also opens up novel parallel implementation options and the possibility of concentrating the computational effort on the most challenging subproblems. We demonstrate its performance on a Markov random field and on a hierarchical logistic regression problem. Supplementary materials including proofs and additional numerical results are available online.     

Predicting stock returns and assessing prediction performance

Email: r.d.baker{at}salford.ac.uk Received on 1 November 2006. Accepted on 15 March 2007. We use regression methods to predict the expected monthly returnon stocks and the covariance matrix of returns, the predictorvariables being a company's ‘fundamentals’, suchas dividend yield and the history of previous returns. Predictionsare evaluated out of sample for shares traded on the LondonStock Exchange from 1976 to 2005. We explore and evaluate manymodelling and inferential approaches, including the use of weightedregression, discounted regression, shrinkage of regression coefficientsand the transformation to normality of predictor variables.We also investigate alternative covariance matrix models, suchas a two-index model and a shrinkage model. Using suitable statisticsto enable the out-of-sample performance of competing methodologiesto be compared is crucial, and we develop some new statisticsand a graphical aid for this purpose. What is original in thispaper is an evaluation of many modelling and inferential proceduresfor which conflicting claims have been made in the literatureand the development of new measures of portfolio performance.     

Subitising activity relative to units construction: a case study

Subitising, a quick apprehension of the numerosity of a small set of items, has been found to change from an individual's reliance on perceptual to conceptual processes. In this study, we utilised a constructivist teaching experiment methodology to investigate how the subitising activity of one preschool student, Amy, related to her construction of prenumerical units. Subitising and counting tasks were designed to assess and perturb Amy's thinking relative to her construction of units, and to observe changes in Amy's activity associated with the different tasks. Findings indicate that as Amy's subitising activity changed from perceptual to conceptual, she constructed subitised motor units and subitised figurative units. Implications of this study suggest that the construction of subitised units may support young children's later development of arithmetic units.     

Blow-up analysis for a nonlocal reaction-diffusion system with time-dependent coefficients

ABSTRACT

A blow-up analysis for a nonlocal reaction-diffusion system with time-dependent coefficients is investigated under null Dirichlet boundary conditions. Based on the Kaplan's method, comparison principle and modified differential inequality technique, we establish a blow-up criteria and derive the bounds for the blow-up time under the appropriate measures in whole-dimensional space.     

The Graphical Horseshoe Estimator for Inverse Covariance Matrices

We develop a new estimator of the inverse covariance matrix for high-dimensional multivariate normal data using the horseshoe prior. The proposed graphical horseshoe estimator has attractive properties compared to other popular estimators, such as the graphical lasso and the graphical smoothly clipped absolute deviation. The most prominent benefit is that when the true inverse covariance matrix is sparse, the graphical horseshoe provides estimates with small information divergence from the sampling model. The posterior mean under the graphical horseshoe prior can also be almost unbiased under certain conditions. In addition to these theoretical results, we also provide a full Gibbs sampler for implementing our estimator. MATLAB code is available for download from github at http://github.com/liyf1988/GHS. The graphical horseshoe estimator compares favorably to existing techniques in simulations and in a human gene network data analysis. Supplementary materials for this article are available online.     

ON THE TITCHMARSH-WEYL THEORY OF ORDINARY SYMMETRIC ODD-ORDER DIFFERENTIAL EXPRESSIONS AND A DIRECT CONVERGENCE THEOREM

Abstract

In this paper the odd-order differential equation M[y] λ wy on the interval (O,∞), associated with the symmetric differential expression M of (2k-1)st order (k ≥ 2) with w a positive weight function and λ a complex number, is shown to possess k-Titchmarsh-Weyl solutions for every non-real λ in the underlying Hilbert space L² _w(O, ∞) having identical representation for every non-real λ. In terms of these solutions the Green's function associated with the singular boundary value problem is shown to possess identical representation for all non-real λ which has been further made use of in the third-order case to establish a direct convergence eigenfunction expansion theorem. The symmetric spectral matrix appearing in the expansion theorem has been characterized in terms of the Titchmarsh-Weyl m-coefficients.     

ON THE STRUCTURE OF MINIMAL R n ACTIONS

Abstract

In this paper we study several concepts and models which are relevant in describing both the topological and dynamical structure of a typical R ⁿ flow. Some of these ideas originated in our earlier papers, and those of other authors, and we here attempt to synthesise these concepts. We start with shear—a notion which describes how little equicontinuity the flow contains. We move to R ⁿ suspensions which depend on particular R ⁿ cocycles and easily obtain a crude representation of the flow as a tower—a partial suspension over a base flow which contains the shear. Rudolph's deep theory of suspension models is modified to provide a new suspension model which incorporates the shear as the base of the tower. Finally we investigate towers in the context of a special class of automorphisms to see when these objects are themselves suspensions.     

A probabilistic representation of solutions of the incompressible Navier-Stokes equations in R3

A new probabilistic representation is presented for solutions of the incompressible Navier-Stokes equations in R³ with given forcing and initial velocity. This representation expresses solutions as scaled conditional expectations of functionals of a Markov process indexed by the nodes of a binary tree. It gives existence and uniqueness of weak solutions for all time under relatively simple conditions on the forcing and initial data. These conditions involve comparison of the forcing and initial data with majorizing kernels.     

The taxicab locus of Apollonius: promoting exploratory routes to the punchline using rich undergraduate tasks

ABSTRACT

A key motivational tactic in undergraduate mathematics teaching is to launch topics with fundamental questions that originate from surprising or remarkable phenomena. Nonetheless, constructing a sequence of tasks that promotes students' own routes to resolving such questions is challenging. This note aims to address this challenge in two ways. First, to illustrate the motivational tactic, the taxicab manifestation of a locus attributed to Apollonius is introduced and a natural question arising from comparison with the analogous Euclidean locus is considered, namely, does the taxicab locus of Apollonius ever coincide with a taxicab circle? Second, a companion sequence of rich undergraduate tasks is elaborated using theoretical design principles, with the tasks culminating in this fundamental geometric question. This note therefore provides a design approach that can be replicated in undergraduate teaching contexts based around similarly motivating mathematical phenomena.     

Markovian representation of a bilinear time series model and maximum likelihood estimation of the parameters

A bilinear time series (BLTS) model is expressed in the form of Akaike's Markovian representation in order to use the Kalman recursive estimation approach. It is shown that Akaike's Markovian representation of autoregressive moving average models of orderp and q (ARMA(p,q)) and that of the bilinear model are equivalent. This equivalence facilitates the maximum likelihood estimation of the parameters involved in the bilinear model, which otherwise is an unwieldy problem. The present approach can easily be extended to take into account missing observations     

A Note on Weyl Modules for ∞ and ∞

In this article we extend Jantzen's formula for the determinant of the contravariant form on highest weight   _n -modules to a formula for highest weight   _∞-modules. This, in turn, has applications to the representation theory of (degenerate and nondegenerate) cyclotomic Hecke algebras.     

A Nonstandard Finite Difference Scheme for Nonlinear Heat Transfer in a Thin Finite Rod

A nonstandard finite difference scheme is constructed to solve an initial-boundary value problem involving a quartic nonlinearity that arises in heat transfer involving conduction with thermal radiation. It is noted that the positivity condition is equivalent to the usual linear stability criteria and it is shown that the representation of the nonlinear term in the finite difference scheme, in addition to the magnitudes of the equation parameters, has a direct bearing on the scheme's stability. Finally, solution profiles are plotted and avenues of further inquiry are discussed.