Graphical Diagnostics for Markov Models for Categorical Data

Markov models are widely used as a method for describing categorical data that exhibit stationary and nonstationary autocorrelation. However, diagnostic methods are a largely overlooked topic for Markov models. We introduce two types of residuals for this purpose: one for assessing the length of runs between state changes, and the other for assessing the frequency with which the process moves from any given state to the other states. Methods for calculating the sampling distribution of both types of residuals are presented, enabling objective interpretation through graphical summaries. The graphical summaries are formed using a modification of the probability integral transformation that is applicable for discrete data. Residuals from simulated datasets are presented to demonstrate when the model is, and is not, adequate for the data. The two types of residuals are used to highlight inadequacies of a model posed for real data on seabed fauna from the marine environment.

Supplemental materials, including an R-package RMC with functions to perform the diagnostic measures on the class of models considered in this article, are at the journal’s website. The R-package is also available at CRAN.     

Monotonicity criteria for difference schemes designed for hyperbolic equations

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K. Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations.     

Depth notions for orthogonal regression

Global depth, tangent depth and simplicial depths for classical and orthogonal regression are compared in examples, and properties that are useful for calculations are derived. The robustness of the maximum simplicial depth estimates is shown in examples. Algorithms for the calculation of depths for orthogonal regression are proposed, and tests for multiple regression are transferred to orthogonal regression. These tests are distribution free in the case of bivariate observations. For a particular test problem, the powers of tests that are based on simplicial depth and tangent depth are compared by simulations.     

Challenges for mathematical and statistical modelling for metrology

There are challenges in disseminating information concerning mathematical and statistical modelling for metrology. We specify these challenges and consider ways they are being met. In addition to journal papers, the value of supporting software and other aspects are indicated. Relevant work in the UK Software Support for Metrology programme is outlined. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Difference methods for solving boundary value problems for fractional differential equations

Difference schemes for second-order ordinary and partial differential equations with a fractional time derivative are considered. Stationary and nonstationary problems for the diffusion equation in one-and multidimensional domains are examined separately. The stability and convergence of the difference schemes for these equations are proved.     

Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media

Nonlinear dynamical stochastic models are ubiquitous in different areas. Their statistical properties are often of great interest, but are also very challenging to compute. Many excitable media models belong to such types of complex systems with large state dimensions and the associated covariance matrices have localized structures. In this article, a mathematical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed. Rigorous \linebreak mathematical analysis shows that the local effect from the diffusion results in an exponential decay of the components in the covariance matrix as a function of the distance while the global effect due to the mean field interaction synchronizes different components and contributes to a global covariance. The analysis is based on a comparison with an appropriate linear surrogate model, of which the covariance propagation can be computed explicitly. Two important applications of these theoretical results are discussed. They are the spatial averaging strategy for efficiently sampling the covariance matrix and the localization technique in data assimilation. Test examples of a linear model and a stochastically coupled FitzHugh-Nagumo model for excitable media are adopted to validate the theoretical results. The latter is also used for a systematical study of the spatial averaging strategy in efficiently sampling the covariance matrix in different dynamical regimes.     

Cascadic Multigrid for Finite Volume Methods for Elliptic Problems

In this paper, some effective cascadic multigrid methods are proposed for solving the large scale symmetric or nonsymmetric algebraic systems arising from the finite volume methods for second order elliptic problems. It is shown that these algorithms are optimal in both accuracy and computational complexity. Numerical experiments are reported to support our theory.     

A comparison of statistical methods for prenatal screening for Down syndrome

Currently, prenatal screening for Down Syndrome (DS) uses the mother's age as well as three biochemical markers for risk prediction. Risk calculations for the biochemical markers use a quadratic discriminant function. In this paper we compare several classification procedures to quadratic discrimination methods for biochemical-based DS risk prediction, based on data from a prospective multicentre prenatal screening study. We investigate alternative methods including linear discriminant methods, logistic regression methods, neural network methods, and classification and regression-tree methods. Several experiments are performed, and in each experiment resampling methods are used to create training and testing data sets. The procedures on the test data set are summarized by the area under their receiver operating characteristic curves. In each experiment this process is repeated 500 times and then the classification procedures are compared. We find that several methods are superior to the currently used quadratic discriminant method for risk estimation for these data. The implications of these results for prenatal screening programs are discussed.     

On Error Bounds for Approximation Schemes for Non-Convex Degenerate Elliptic Equations

In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one space-dimension. The equations under consideration are the degenerate elliptic Isaacs equations with x-depending coefficients, and the results applies in particular to certain finite difference methods and control schemes based on the dynamic programming principle. Recently, Krylov, Barles, and Jakobsen obtained similar estimates for convex Hamilton–Jacobi–Bellman equations in arbitrary space-dimensions. Our results are only valid in one space-dimension, but they are the first results of this type for non-convex second-order equations.     

SOR-like Methods for Augmented Systems

Several SOR-like methods are proposed for solving augmented systems. These have many different applications in scientific computing, for example, constrained optimization and the finite element method for solving the Stokes equation. The convergence and the choice of optimal parameter for these algorithms are studied. The convergence and divergence regions for some algorithms are given, and the new algorithms are applied to solve the Stokes equations as well.     

A method for separating nearly multiple eigenvalues for Hermitian matrix

In this paper, we propose a numerical method to verify for nearly multiple eigenvalues of a Hermitian matrix not being strictly multiple eigenvalues. From approximate eigenvalues computed, it seems to be difficult to distinguish whether they are strictly multiple eigenvalues or simple ones, and if they are very close each other, the verification method for simple eigenvalues may fail to enclose them separately, because of singularity of the system in the verification. There are several methods for enclosing multiple and nearly multiple eigenvalues (e.g., [Rump, Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl. 324 (2001) 209–226]), For such cases, there is no result to decide the enclosed eigenvalues are nearly multiple or strictly multiple, up to now. So, for enclosed eigenvalues, we propose a numerical method to separate nearly multiple eigenvalues.     

Teacher time out as a site for studying mathematical knowledge for teaching

The special mathematical knowledge that is needed for teaching has been studied for decades but the methods for studying it have challenges. Some methods, such as measurement and cognitive interviews, are removed from the dynamics of teaching. Other methods, such as observation, are closer to practice but mostly involve an outsider perspective. Moreover, few methods tap into the tacit and often invisible demands that teachers encounter in teaching. This article develops an argument that teacher time outs in rehearsals and enactments might be a productive site for studying mathematical knowledge for teaching. Teacher time outs constitute a site for professional deliberation, which 1) preserves the complexity and gets inside the dynamics of teaching, where 2) tacit and implicit challenges and demands are made explicit, and where 3) insider and outsider perspectives are combined.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

Feedback methods for inverse simulation of dynamic models for engineering systems applications

Inverse simulation is a form of inverse modelling in which computer simulation methods are used to find the time histories of input variables that, for a given model, match a set of required output responses. Conventional inverse simulation methods for dynamic models are computationally intensive and can present difficulties for high-speed applications. This article includes a review of established methods of inverse simulation, giving some emphasis on iterative techniques that were first developed for aeronautical applications. It goes on to discuss the application of a different approach that is based on feedback principles. This feedback method is suitable for a wide range of linear and non-linear dynamic models and involves two distinct stages. The first stage involves design of a feedback loop around the given simulation model, and in the second stage, that closed-loop system is used for inversion of the model. Issues of robustness within closed-loop systems used in inverse simulation are not significant as there are no plant uncertainties or external disturbances. Thus, the process is simpler than that required for the development of a control system of equivalent complexity. Engineering applications of this feedback approach to inverse simulation are described through case studies that put particular emphasis on non-linear and multi-input multi-output models.     

Invariant smoothness measures for surfaces

Two novel smoothness measures for surfaces are presented in this paper. The second and third order smoothness are defined as the squared normal curvature and the squared variation in normal curvature integrated over all directions in the tangent plane. Both quantities are truly geometric in the sense that they are invariant with respect to the actual parametrization of the surface. All the same, all formulae are derived in terms of an arbitrary parametrization. In addition to providing a basis for variational surface construction, the second and third order smoothness can also be used for evaluation and assessment of the quality of an existing surface. This revised version was published online in August 2006 with corrections to the Cover Date.     

Cluster robust error estimates for the Rayleigh-Ritz approximation II: Estimates for eigenvalues

This is the second part of a paper that deals with error estimates for the Rayleigh-Ritz approximations of the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses the approximation of eigenvalues. Two kinds of estimates are considered: (i) estimates for the eigenvalue errors via the best approximation errors for the corresponding invariant subspaces, and (ii) estimates for the same via the corresponding residuals. Estimates of these two kinds are needed for, respectively, the a priori and a posteriory error analysis of numerical methods for computing eigenvalues. The paper’s major concern is to ensure that the estimates in question are accurate and ‘cluster robust’, i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues among those of interest. The paper’s main new results introduce estimates for clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.     

Algorithms for range restricted iterative methods for linear discrete ill-posed problems

Range restricted iterative methods based on the Arnoldi process are attractive for the solution of large nonsymmetric linear discrete ill-posed problems with error-contaminated data (right-hand side). Several derivations of this type of iterative methods are compared in Neuman et al. (Linear Algebra Appl. in press). We describe MATLAB codes for the best of these implementations. MATLAB codes for range restricted iterative methods for symmetric linear discrete ill-posed problems are also presented.     

A simple formula for mean multiplexing delay for independent regenerative sources

Simple expressions are given for the mean delay, mean waiting time, and mean busy period length in a multiplexer. Data streams with active periods having a general distribution are permitted, and the data rate during the active periods can be random. Data can also arrive in batches. The key restrictions of the model are that the sources are independent, idle periods are exponentially distributed, and a source generates at least enough data during an active period to keep the server busy throughout the period. The exact formulas allow evaluation of the error in approximations such as a heavy traffic diffusion approximation.Both continuous and discrete time models are considered. The discrete-time model includes that studied by Viterbi and subsequently generalized by Neuts. The Pollaczek-Khinchine formula for the mean amount of work in anM/GI/1 queue is retrieved as a limiting case.Preliminary version presented at IEEE INFOCOM, San Francisco, April 1993.     

A new approach for the derivation of bounds for the Jensen difference

There are many useful applications of Jensen's inequality in several fields of science, and due to this reason, a lot of results are devoted to this inequality in the literature. The main theme of this article is to present a new method of finding estimates of the Jensen difference for differentiable functions. By applying definition of convex function, and integral Jensen's inequality for concave function in the identity pertaining the Jensen difference, we derive bounds for the Jensen difference. We present integral version of the bounds in Riemann sense as well. The sharpness of the proposed bounds through examples are discussed, and we conclude that the proposed bounds are better than some existing bounds even with weaker conditions. Also, we present some new variants of the Hermite–Hadamard and Hölder inequalities and some new inequalities for geometric, quasi-arithmetic, and power means. Finally, we give some applications in information theory.     

Appropriate problems for learning and for performing—an issue for teacher training

Selecting, modifying or creating appropriate problems for mathematics class has become an activity of increaing importance in the professional development of German mathematics teachers. But rather than asking in general: “What is a good problem?” there should be a stronger emphasis on considering the specific goal of a problem, e.g.: “What are the ingredients that make a problem appropriate for initiating a learning process” or “What are the characteristics that make a problem appropriate for its use in a central test?” We propose a guiding scheme for teachers that turns out to be especially helpful, since the newly introduced orientation on outcome standards a) leads to a critical predominance of test items and b) expects teachers to design adequate problems for specific learning processes (e.g. problem solving, reasoning and modelling activities).