the role of difference equations in numerical analysis abstract the central role of difference equations and principally the role of their stability properties in numerical analysis is discussed by analyzing the special problem of approximating the solutions of ordinary differential equations we show how a deeper knowledge of the stability properties of difference equations may be useful in designing numerical methods

some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

an algebraic approximation for the matrix exponential in singularly perturbed boundary value problems a symmetric difference scheme for linear stiff or singularly perturbed boundary value problems of first order with constant coefficients is constructed being based on a stability function containing a matrix square root its essential feature is the unconditional stability in the absence of purely imaginary eigenvalues of the coefficient matrix local damping of errors uniform stability and uniform second order convergence are proved the computation of the specific matrix square root by a well known stable variant of newton s method is discussed a numerical example confirming the results is given

hypothesis testing in the polychotomous logistic model with an application to detecting gastrointestinal cancer we discuss the use of the trichotomous logistic model to discriminate between patients with gastrointestinal gi cancer patients with benign gi disease and normal subjects using symptoms and the concentrations of some serum proteins that are potentially indicative of malignancy as covariates a parsimonious model can be obtained by invoking an indistinguishability hypothesis which is appropriate when a covariate is considered to have no predictive value between categories it is shown that the polychotomous model can be re parameterised under the null hypothesis to give a reduced form which can be fitted by maximum likelihood the validity of the use of the same methods for retrospective sampling is discussed the approach is illustrated by the development of a logistic model to identify symptomatic and asymptomatic subjects with a high risk of gi cancer

an analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems stability and convergence results are derived for the box and trapezoidal schemes applied to boundary value problems for linear singularly perturbed first order systems of o d e s without turning points

automated allocation of mesh points for stiff two point boundary value problems abstract the numerical solution of a stiff two point boundary value problem is considered where a general procedure to automate the allocation of mesh points is obtained this procedure requires negligible prerequisites to the problem the automation is achieved by constructing a diffeomorphism acting on the domain of the independent variable this smooths out the sharp corners in the solution while it leaves the domain boundary and the boundary conditions unchanged the diffeomorphism is defined by differential equations which are coupled with the original equations to form an augmented system

posterior likelihood methods for multivariate survival data this article deals with the semiparametric analysis of multivariate survival data with random block group effects survival times within the same group are correlated as a consequence of a frailty random block effect the standard approaches assume either a parametric or a completely unknown baseline hazard function this paper considers an intermediate solution that is a nonparametric function that is reasonably smooth this is accomplished by a bayesian model in which the conditional proportional hazards model is used with a correlated prior process for the baseline hazard the posterior likelihood based on data as well as the prior process is similar to the discretized penalized likelihood for the frailty model the methodology is exemplified with the recurrent kidney infections data of mcgilchrist and aisbett 1991 biometrics 47 461 466 in which the times to infections within the same patients are expected to be correlated the reanalysis of the data has shown that the estimates of the parameters of interest and the associated standard errors depend on the prior knowledge about the smoothness of the baseline hazard

numerical methods for stiff two point boundary value problems we consider the two point boundary value problem for stiff systems of ordinary differential equations for systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions a priori estimates are obtained problems with turning points can be treated with this theory and we discuss this in detail we give robust difference approximations and present error estimates for these schemes in particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions numerical examples are presented for both linear and nonlinear problems

a riccati transformation method for solving linear bvps ii computational aspects in part i of this paper siamj numer anal 25 1988 pp 1055 1073 a riccati transformation method for solving linear boundary value problems for ordinary differential equations is presented here the implementation aspects of this method are discussed in particular a double sweep integration process which includes a strategy for solving a matrix riccati differential equation and forming a continuous representation of its solution is described the efficacy of the method is demonstrated on a number of singular perturbation problems particular attention is paid to the choice of scaling for the variables and its effect on the solution method the computational experience while not based upon a refined implementation indicates that the riccati method may well prove to be competitive with the other general methods

automatic smoothing of regression functions in generalized linear models abstract we consider the penalized likelihood method for estimating nonparametric regression functions in generalized linear models nelder and wedderburn 1972 and present a generalized cross validation procedure for empirically assessing an appropriate amount of smoothing in these estimates asymptotic arguments and numerical simulations are used to show that the generalized cross validatory procedure preforms well from the point of view of a weighted mean squared error criterion the methodology adds to the battery of graphical tools for model building and checking within the generalized linear model framework included are two examples motivated by medical and horticultural applications

a nonparametric multiple choice method within the random utility framework abstract many researchers use categorical data analysis to recover individual consumption preferences but the standard discrete choice models require restrictive assumptions to improve the flexibility of discrete choice data analysis we propose a nonparametric multiple choice model that applies the penalized likelihood method within the random utility framework we show that the deterministic component of the random utility function in the model is a cubic smoothing spline function the method subsumes the conventional conditional logit model mcfadden 1973 in zarembka p ed frontiers in econometrics as a special case in this paper we present the model describe the estimator provide the computational algorithm of the model and demonstrate the model by applying it to nonmarket valuation of recreation sites

stability of a three point scheme for linear second order singularly perturbed bvps with turning points in this paper we study stability properties of a numerical method applied to linear second order perturbed boundary value problems with boundary or interior layers the method consists of a three point difference scheme with variable stepsize the stability of which may be investigated by studying that of an lu factorization for the coefficient matrix of a suitable tridiagonal system we also propose a mesh selection strategy for treating boundary and interior layers which furnishes a good mesh on which the three point scheme is stable

tridiagonal matrices invertibility and conditioning abstract tridiagonal matrices arise in a large variety of applications most of the time they are diagonally dominant and this is indeed the case most extensively studied in this paper we study in a unified approach the invertibility and the conditioning of such matrices the results presented provide practical criteria for a tridiagonal and irreducible matrix to be both invertible and well conditioned an application to a singular perturbation boundary value problem is then presented

viewpoint based simplification using f divergences we propose a new viewpoint based simplification method for polygonal meshes driven by several f divergences such as kullback leibler hellinger and chi square these distances are a measure of discrimination between probability distributions the kullback leibler distance between the projected and the actual area distributions of the polygons in the scene already has been used as a measure of viewpoint quality in this paper we use the variation in those viewpoint distances to determine the error introduced by an edge collapse we apply the best half edge collapse as a decimation criterion the approximations produced by our method are close to the original model in terms of both visual and geometric criteria unlike many pure visibility driven methods our new approach does not completely remove hidden interiors in order to increase the visual quality of the simplified models this makes our approach more suitable for applications which require exact geometry tolerance but also require high visual quality

order barriers and characterizations for continuous mono implicit runge kutta schemes the mono implicit runge kutta mirk schemes a subset of the family of implicit runge kutta irk schemes were originally proposed for the numerical solution of initial value odes more than fifteen years ago dur ing the last decade a considerable amount of attention has been given to the use of these schemes in the numerical solution of boundary value ode prob lems where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes recent work in this area has seen the development of some software packages for bound ary value odes based on these schemes unfortunately these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval while the collocation schemes provide a natural continuous solution approximation the availability of a continuous solution is important not only to the user of the software but also within the code itself for example in estimation of errors defect control mesh selection and the provision of initial solution estimates for new meshes an approach for the construction of a continuous solution approximation based on the mirk schemes is suggested by recent work in the area of continuous extensions for explicit runge kutta schemes for initial value odes in this paper we describe our work in the investigation of continuous versions of the mirk schemes i we give some lower bounds relating the stage order to the minimal number of stages for general continuous irk schemes ii we establish lower bounds on the number of stages needed to derive continuous mirk schemes of orders 1 through 6 and iii we provide characterizations of these schemes having a minimal number of stages for each of these orders

stability of collocation at gaussian points symmetric runge kutta schemes are particularly useful for solving stiff two point boundary value problems such a stable schemes perform well in many cases but it is demonstrated that in some instances the stronger property of algebraic stability is required a characterization of symmetric algebraically stable runge kutta schemes is given the class of schemes thus defined turns out not to be very rich the only collocation schemes in it are those based on gauss points and other schemes in the class do not seem to offer any advantage over collocation at gaussian points

high order finite difference schemes for the solution of second order bvps we introduce new methods in the class of boundary value methods bvms to solve boundary value problems bvps for a second order ode these formulae correspond to the high order generalizations of classical finite difference schemes for the first and second derivatives in this research we carry out the analysis of the conditioning and of the time reversal symmetry of the discrete solution for a linear convection diffusion ode problem we present numerical examples emphasizing the good convergence behavior of the new schemes finally we show how these methods can be applied in several space dimensions on a uniform mesh

regression and ordered categorical variables read before the royal statistical society by professor r l plackett on behalf of the late professor anderson at a meeting organized by the research section on wednesday october 5th 1983 professor j b copas in the chair summary a general approach to regression modelling for ordered categorical response variables y is given which is equally applicable to ordered and unordered y the regressor variables xt xi x may be continuous or categorical the method is based on the logistic family which contains a hierarchy of regression models ranging from ordered to unordered models ordered properties of the former the stereotype model are established the choice between models is made empirically on the basis of model fit this is particularly important for assessed ordered categorical response variables where it is not obvious a priori whether or not the ordering is relevant to the regression relationship model simplification is investigated in terms of whether or not the response categories are distinguishable with respect to x the models are fitted iteratively using the method of maximum likelihood examples are given

krylov approximations for matrix square roots in stiff boundary value problems recently we proposed an algebraic difference scheme with extended stability properties for linear boundary value problems involving stiff differential equations of first order here an efficient approximation scheme is presented for matrix square roots which provides the stabilization of that scheme in case of stiffness it combines the use of low rank matrix approximations from projections onto krylov subspaces with an accelerated sign iteration for the matrix square root the krylov approximation being accurate in eigenspaces with large eigenvalues preserves the stability of the scheme and the 0 n3 square root computation need be performed only in lower dimension operation counts and numerical results show that the effort for the numerical scheme is essentially proportional to the number of stiff components but not to the norm of the coefficient matrix approximation properties of low rank krylov matrices which may be of independent interest are analyzed 1 the sqrt one step difference scheme a standard approximation for the differential equation in the linear boundary value problem bvp u x a x u x g x x e 0 1 u x er bou o b u l 8 is the trapezoidal rule on a suitable grid 0 xo 1 this restriction leads to unacceptable small stepsizes in regions where the solution is smooth this situation is usually referred to as stiffness in certain cases however a relaxation received march 27 1990 revised december 5 1990 1991 mathematics subject classification primary 65l10 65l20 65f30

penalized likelihood estimation of hazard and intensity functions summary a non parametric estimate is proposed for the hazard function using penalized likelihood methods the estimate turns out to be a hyperbolic spline function which is continuous with its first derivative discontinuous only at a finite number of points it always assumes positive values unlike anderson and senthilselvan s 1980 quadratic spline estimate this method is modified for the estimation of the intensity function of a nonstationary poisson process an example is given for comparison of the new estimator with the estimator of anderson and senthilselvan 1980

boundary value techniques for initial value problems in ordinary differential equations the numerical solution of initial value problems in ordinary differential equations by means of boundary value techniques is considered we discuss a finite difference method which was already investigated by fox in 1954 and fox and mitchell in 1957 hereby we concentrate on explaining the fundamentals of the method because for initial value problems the boundary value method seems to be fairly unknown we further propose and discuss new jalerkin methods for initial value problems along the lines of the boundary value approach

the accurate numerical solution of highly oscillatory ordinary differential equations we consider systems of ordinary differential equations with rapidly oscillating solutions conventional numerical methods require an excessively small time step t 0 eh where h is the step size necessary for the resolution of a smooth function of t and 1 e measures the size of the large eigenvalues of the system s jacobian for the linear problem with well separated large eigenvalues we introduce smooth transformations which lead to the separation of the time scales and computation with a large time step t 0 h for more general problems including systems with weak polynomial nonlinearities we develop an asymptotic theory which leads to expansions whose terms are suitable for numerical approximation resonances can be detected and resolved often with a large time step t 0 h passage through resonance in nonautonomous systems can be resolved by a moderate time step t 0 eh

a model to simplify 2d triangle meshes with irregular shapes we present a 2d triangle mesh simplification model which is able to produce high quality approximations of any original planar mesh regardless of the shape of the original mesh this method consists of two phases a self organizing algorithm and a triangulation algorithm the self organizing algorithm is an unsupervised incremental clustering algorithm which provides us a set of nodes representing the best approximation of the original mesh the triangulation algorithm reconstructs the simplified mesh from the planar points obtained by the self organizing training process some examples are detailed with the purpose of demonstrating the ability of the model to perform the task of simplifying an original mesh with irregular shape

numerical methods for second order singular pertubation problems abstract by using sufficient conditions which ensure the well conditioning of tridiagonal matrices we derive numerical methods for second order singular pertubations boundary values problems under hypotheses similar to the ones usually used in the continuous theory methods are derived which use constant step sizes h o 1 2 in the more general case methods with variable step sizes are defined which solve the problem with n 1 steps examples on standard test problems are shown

inferences for case control and semiparametric two sample density ratio models summary we consider inference in general binary response regression models under retrospective sampling plans prentice pyke 1979 discovered that inference for the odds ratio parameter in a logistic model can be based on a prospective likelihood even though the sampling scheme is retrospective we show that the estimating function obtained from the prospective likelihood is optimal in a class of unbiased estimating functions also we link casecontrol sampling with a two sample biased sampling problem where the ratio of two densities is assumed to take a known parametric form connections between this model and the cox proportional hazards model are pointed out large and small sample size behaviour of the proposed estimators is studied

on the existence of maximum likelihood estimates in logistic regression models summary the problems of existence uniqueness and location of maximum likelihood estimates in log linear models have received special attention in the literature haberman 1974 chapter 2 wedderburn 1976 silvapulle 1981 for multinomial logistic regression models we prove existence theorems by considering the possible patterns of data points which fall into three mutually exclusive and exhaustive categories complete separation quasicomplete separation and overlap our results suggest general rules for identifying infinite parameter estimates in log linear models for frequency tables

diagnostic and sampling models in screening summary screening methods are developed which raise the proportion of successes in the retained population to a predetermined level specification regions for the screening variables are derived within both the diagnostic and sampling paradigms dawid 1976 logistic and normal models are used for the two frameworks and the results are illustrated by an example of screening for malnourishment

robust logistic discrimination summary logistic discrimination is a well established method for allocating observations to one of two or more populations in this paper we show how the logistic model can be adapted to make it robust against outlying observations from the populations the new model is successfully tested using simulated data against ordinary logistic discrimination and also against logistic discrimination where parameters are fitted using resistant methods the model is then used on data collected from the teeth of rats logistic discrimination is a partially parametric method for discrimination applicable in many situations and under several sampling schemes notable references to the method are day kerridge 1967 anderson 1972 1979 1982 and anderson blair 1982 the appeal of the logistic method for discrimination is that only the ratio of likelihoods has to be modelled and not the full distributional form of the underlying variables press wilson 1978 and byth mclachlan 1980 show that in many circumstances logistic discrimination is preferable to the usual linear discrimination based on either normal populations with equal covariance matrices or fisher s linear discriminant for the two group case suppose x is a vector of observations h1 and h2 are the two groups and a a parameter vector then the log likelihood ratio is modelled as

fitting logistic models under case control or choice based sampling on compare deux approches dans le cas important de l ajustement des modeles de regression logistique

boundary value problems and dichotomic stability since the conditioning of a boundary value problem bvp is closely related to the existence of a dichotomic fundamental solution i e where one set of modes is increasing and a complementary set is decreasing it is important to have discretization methods that conserve this dichotomy property the conditions this imposes on such a method are investigated in this paper they are worked out in more detail for scalar second order equations the central difference scheme and for linear first order systems as well for the latter type both one step methods including collocation and multistep methods those that may be used in multiple shooting are examined

isotonic logistic discrimination summary we propose an isotonic logistic discrimination procedure which generalises linear logistic discrimination by allowing linear boundaries to be more flexibly shaped as monotone functions of the discriminant variables under each of three familiar sampling schemes for obtaining a training dataset namely prospective mixture and retrospective we provide the corresponding likelihood based inference an application to a cancer study is given in addition we consider theoretical comparisons of our method with two recent algorithmic monotone discrimination procedures

numerical methods for singular perturbation problems consider the two point boundary value problem for a stiff system of ordinary differential equations an adaptive method to solve these problems even when turning points are present is discussed

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments

backward error analysis of cyclic reduction for the solution of tridiagonal systems tridiagonal systems play a fundamental role in matrix computation in particular in recent years parallel algorithms for the solution of tridiagonal systems have been developed among these the cyclic reduction algorithm is particularly interesting here the stability of the cyclic reduction method is studied under the assumption of diagonal dominance a backward error analysis is made yielding a representation of the error matrix for the factorization and for the solution of the linear system the results are compared with those for lu factorization

penalized maximum likelihood estimation in logistic regression and discrimination summary maximum likelihood estimation of the parameters of the binary logistic regression model for pr h i x is discussed with separate discussion of sampling from i the conditional distribution of h given x ii the joint distribution of h and x and iii the conditional distribution of x given h difficulties associated with continuous x in the latter sampling scheme are discussed to avoid these penalized maximum likelihood estimates are introduced which give estimates of the logistic parameters and a nonparametric spline estimate of the marginal distribution of x extensions to multinomial logistic regression are outlined

decoupling and stability of algorithms for boundary value problems the ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals it is shown that most methods like multiple shooting collocation invariant imbedding and difference methods derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later for instance when solving a linear system this decoupling is related to the dichotomy of the ordinary differential equations in fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed how this stability is related to the use of the boundary conditions is also investigated

multiple group logistic discrimination among the many possible approaches suggested for statistical discrimination the logistic method can be classified midway between fully distributional solutions of which the assumption of multivariate normality is a classical example 16l and the distribution free techniques using for instance kernel or nearest neighbor methods l 32 therefore it is often called a partially parametric or partially distributional method l l this central position of the logistic model makes it one of the most attractive and widely used tools for solving regression and discrimination problems indeed since there are fewer distributional assumptions than for fully parametric models the logistic method is applicable to a larger family of multivariate distributions involving both discrete and continuous variables moreover in spite of its wide applicability and generality the method remains feasible and easy to use in contrast with nonparametric methods this paper is intended to review the basic ideas and principles of logistic discrimination 415 22 25 and also to bring additional results to some of the queries raised by professor anderson before his death we restrict our attention to discrimination between qualitatively distinct groups and do not envisage the case where groups are quantitatively distinct or ordered 3 12 14 we dedicate this paper to our friend and mentor the late professor j a anderson for his fundamental contribution to discriminant analysis and for his continuous support of our research efforts

a fast adaptive numerical method for stiff two point boundary value problems we describe a robust adaptive algorithm for the solution of singularly perturbed two point boundary value problems many different phenomena can arise in such problems including boundary layers dense oscillations and complicated or ill conditioned internal transition regions working with an integral equation reformulation of the original differential equation we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved based on this method we have constructed a black box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution the solver is direct and of arbitrarily high order accuracy and requires an amount of time proportional to the number of grid points

boundedness and stability of solutions to difference equations this paper is concerned with the qualitative behaviour of solutions to difference equations we focus on boundedness and stability of solutions and we present a unified theory that applies both to autonomous and nonautonomous equations and to nonlinear equations as well as linear equations our presentation brings together new established and hard to find results from the literature and provides a theory that is both memorable and easy to apply we show how the theoretical results given here relate to some of those in the established literature and by means of simple examples we indicate how the use of lipschitz constants in this way can provide useful insights into the qualitative behaviour of solutions to some nonlinear problems including those arising in numerical analysis

some aspects of using iv software to solve bvp via a riccati transformation we consider a numerical method for the solution of linear boundary value problems bvps with separated boundary conditions bcs for ordinary differential equations odes by using a particular linear transformation the problem is reformulated as three initial value problems ivps one of them being a riccati differential equation the nonlinearity of the riccati de makes it necessary to deal with the eventuality of modifying the structure of integration so called multiple embedding equations must thus be derived and a marching algorithm is incorporated a stability analysis of the three ivps indicates clearly for what class of problems the method is suited and why stiff iv methods should then frequently be used this leads us to consider the undesirable effect of superstability of implicit schemes which can occur when integrating the riccati de we interpret this occurrence in terms of the original bvp and the resulting analysis suggests some final considerations including the possibility of corrective measures

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough