on the numerical condition of bernstein bezier subdivision processes the linear map m that takes the bernstein coefficients of a polynomial p t on a given interval a b into those on any subinterval a b is specified by a stochastic matrix which depends only on the degree n of p t and the size and location of a t relative to a b we show that in the 11 110j norm the condition number of m has the simple form k m 2f max u7 vo where u5 m a b a and v b m b a are the barycentric coordinates of the subinterval midpoint m 2 a b and f denotes the zoom factor b a b a of the subdivision map this suggests a practical rule of thumb in assessing how far bezier curves and surfaces may be subdivided without exceeding prescribed worst case bounds on the typical errors in their control points the exponential growth of koo m with n also argues forcefully against the use of high degree forms in computer aided geometric design applications

partial non gaussian state space summary in this paper we suggest the use of simulation techniques to extend the applicability of the usual gaussian state space filtering and smoothing techniques to a class of nongaussian time series models this allows a fully bayesian or maximum likelihood analysis of some interesting models including outlier models discrete markov chain components multiplicative models and stochastic variance models finally we discuss at some length the use of a non gaussian model to seasonally adjust the published money supply figures

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

a monte carlo approach to nonnormal and nonlinear state space modeling abstract a solution to multivariate state space modeling forecasting and smoothing is discussed we allow for the possibilities of nonnormal errors and nonlinear functionals in the state equation the observational equation or both an adaptive monte carlo integration technique known as the gibbs sampler is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework the methodology is a general strategy for obtaining marginal posterior densities of coefficients in the model or of any of the unknown elements of the state space missing data problems including the k step ahead prediction problem also are easily incorporated into this framework we illustrate the broad applicability of our approach with two examples a problem involving nonnormal error distributions in a linear model setting and a one step ahead prediction problem in a situation where both the state and observational equations are nonlinear and involve unknown parameters

hierarchically refined and coarsened splines for moving interface problems with particular application to phase field models of prostate tumor growth abstract moving interface problems are ubiquitous in science and engineering to develop an accurate and efficient methodology for this class of problems we present algorithms for local h adaptivity of hierarchical b splines to be utilized in isogeometric analysis we extend bezier projection an efficient quadrature free local projection technique to the hierarchical setting in this case extraction operators may not be invertible to address this issue we develop a multi level reconstruction operator which maintains the locality properties of the projection we also introduce a balance parameter to control the overlap of hierarchical functions leading to improved numerical conditioning we apply our algorithms to the simulation of localized prostate cancer growth we model this disease using the phase field method and a set of diffusion reaction equations to account for the dynamics of nutrients and a key biomarker termed prostate specific antigen our results include examples on simple 2d and 3d domains and a more compelling tissue scale patient specific simulation which is run over a prostate anatomy extracted from medical images our methods for local h adaptivity efficiently capture the evolving interface between the tumor and the neighboring healthy tissue with remarkable accuracy in all cases

on gibbs sampling for state space models summary we show how to use the gibbs sampler to carry out bayesian inference on a linear state space model with errors that are a mixture of normals and coefficients that can switch over time our approach simultaneously generates the whole of the state vector given the mixture and coefficient indicator variables and simultaneously generates all the indicator variables conditional on the state vectors the states are generated efficiently using the kalman filter we illustrate our approach by several examples and empirically compare its performance to another gibbs sampler where the states are generated one at a time the empirical results suggest that our approach is both practical to implement and dominates the gibbs sampler that generates the states one at a time

bayes inference in regression models with arma p q errors we develop practical and exact methods of analyzing arma p 4 regression error models in a bayesian framework by using the gibbs sampling and metropolis hastings algorithms and we prove that the kernel of the proposed markov chain sampler converges to the true density the procedures can be applied to pure arma time series models and to determine features of the likelihood function by choosing appropriate diffuse priors our results are unconditional on the initial observations we also show how the algorithm can be further simplified for the important special cases of stationary ar p and invertible ma q models recursive transformations developed in this paper to diagonalize the covariance matrix of the errors should prove useful in frequentist estimation examples with simulated and actual economic data are presented kpj ourls gibbs sampling metropolis hastings algorithm data augmentation time series arma processes markov chain bayesian statistics

a fast algorithm for signal extraction influence and cross validation in state space models summary a fast algorithm is developed for computing the conditional mean and variance of the signal given the observations in a signal plus noise model the resulting recursions can be applied immediately to provide new and efficient formulae for smoothing part or all of the state vector the ideas of studentized residuals and leverage from regression analysis are generalized to state space models and the new algorithm is used to compute the various measures the results are also applied to obtain a new efficient algorithm for polynomial spline smoothing suppose observations are generated by a gaussian signal plus noise process with the signal described by a state space model such a model for the signal occurs often in practice for example when the signal is the output of a stochastic difference or differential equation this paper presents a new algorithm for signal extraction that is the computa tion of the conditional mean and variance of the signal given the observations the usual method for signal extraction is to use the kalman filter see for example anderson moore 1979 p 105 followed by a smoothing step using an algorithm such as the fixed interval smoothing algorithm anderson moore 1979 p 187 our approach also uses the kalman filter but for the smoothing step introduces a new set of recursions which are more efficient than those for the fixed interval smoothing algorithm as for the fixed interval smoothing algorithm recursions for estimating the signal and its variance are carried out separately so that considerable additional savings can be made if the signal estimate alone is required the development uses ideas introduced by ansley kohn 1987a a remarkable property of the new recursions is that they can be applied immediately

bezier projection a unified approach for local projection and quadrature free refinement and coarsening of nurbs and t splines with particular application to isogeometric design and analysis abstract we introduce bezier projection as an element based local projection methodology for b splines nurbs and t splines this new approach relies on the concept of bezier extraction and an associated operation introduced here spline reconstruction enabling the use of bezier projection in standard finite element codes bezier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global l 2 projection bezier projection is used to develop a unified framework for spline operations including cell subdivision and merging degree elevation and reduction basis roughening and smoothing and spline reparameterization in fact bezier projection provides a quadrature free approach to refinement and coarsening of splines in this sense bezier projection provides the fundamental building block for h p k r adaptivity in isogeometric analysis

smoothing and interpolation with the state space model abstract a new result dealing with smoothing and interpolation in the state space model is developed and explored the result simplifies the derivation of existing smoothing algorithms and provides alternate forms that have analytic and practical computing advantages the connection to signal extraction and interpolation is explored and diffuse specifications are considered

sector delayed hopf type mixed mode oscillations in a prototypical three time scale model we consider a three dimensional three time scale system that was first proposed by krupa et al 2008 under the additional assumption that two singular perturbation parameters are present in the equations while the presence of three scales was shown to give rise to canard induced periodic mixed mode oscillations mmos desroches et al 2012 in the parameter regime studied by krupa et al 2008 we additionally observe mixed mode patterns that display delayed hopf type behaviour ne shtadt 1987 we present analytical and numerical evidence for the occurrence of stable periodic dynamics that realises both mechanisms and we discuss the transition between them to the best of our knowledge the resulting mixed sector delayed hopf type mmo trajectories represent a novel class of mixed mode dynamics in singularly perturbed systems of ordinary differential equations

the role of the bezier extraction operator for t splines of arbitrary degree linear dependencies partition of unity property nesting behaviour and local refinement we determine linear dependencies and the partition of unity property of t spline meshes of arbitrary degree using the bezier extraction operator local refinement strategies for standard semi standard and nonstandard t splines also by making use of the bezier extraction operator are presented for meshes of even and odd polynomial degrees a technique is presented to determine the nesting between two t spline meshes again exploiting the bezier extraction operator finally the hierarchical refinement of standard semi standard and non standard t spline meshes is discussed this technique utilises the reconstruction operator which is the inverse of the bezier extraction operator

bayesian model discrimination and bayes factors for linear gaussian state space models summary it is shown how to discriminate between different linear gaussian state space models for a given time series by means of a bayesian approach which chooses the model that minimizes the expected loss a practical implementation of this procedure requires a fully bayesian analysis for both the state vector and the unknown hyperparameters and is carried out by markov chain monte carlo methods an application to some non standard situations such as testing hypotheses on the boundary of the parameter space discriminating non nested models and discrimination of more than two models is discussed in detail

dynamic linear models with switching abstract the problem of modeling change in a vector time series is studied using a dynamic linear model with measurement matrices that switch according to a time varying independent random process we derive filtered estimators for the usual state vectors and also for the state occupancy probabilities of the underlying nonstationary measurement process a maximum likelihood estimation procedure is given that uses a pseudo expectation maximization algorithm in the initial stages and nonlinear optimization we relate the models to those considered previously in the literature and give an application involving the tracking of multiple targets

on the degree elevation of bernstein polynomial representation the polynomials determined in the bernstein bezier basis enjoy considerable popularity in computer aided design cad applications the common situation in these applications is that polynomials given in the basis of degree n have to be represented in the basis of higher degree the corresponding transformation algorithms are called algorithms for degree elevation of bernstein polynomial representations these algorithms are only then of practical importance if they do not require the ill conditioned conversion between the bernstein and the power basis we discuss all the algorithms of this kind known in the literature and compare them to the new ones we establish some among the latter are better conditioned and not more expensive than the currently used ones all these algorithms can be applied componentwise to vector valued polynomial bezier representations of curves or surfaces

disturbance smoother for state space models summary this paper develops a method to evaluate the smoothed estimator of the disturbance vector in a state space model together with its mean squared error matrix this disturbance smoother also leads to an efficient smoother for the state vector applications include a method to calculate auxiliary residuals for unobserved components time series models and an em algorithm for estimating covariance parameters in a state space model

legendre bernstein basis transformations abstract the bernstein form of a polynomial offers valuable insight into its geometrical behavior and has thus won widespread acceptance as the basis for bezier curves and surfaces for least squares approximation problems on the other hand the use of orthogonal bases such as the legendre polynomials permits simple and efficient constructions for convergent sequences of approximants we derive the transformation matrices that map the bernstein and legendre forms of a degree n polynomial on 0 1 into each other and examine the stability of this linear map in the p 1 and norms the condition number of the legendre bernstein transformation matrix grows at a significantly slower rate with n than in the well studied power bernstein case and at a dramatically slower rate than for other common e g bernstein hermite or power hermite basis conversions the utility of legendre representations in approximation problems and their relatively stable transformation to bernstein bezier form argue for more widespread applications of legendre methods in cagd algorithms

bayes regression with autoregressive errors a gibbs sampling approach abstract this paper develops a practical framework for the bayesian analysis of gaussian and student t regression models with autocorrelated errors as is customary in classical estimation procedures the posteriors are conditioned on the initial observations recourse is taken to the method of gibbs sampling an iterative markovian sampling method and it is shown that the proposed approach can readily deal with high order autoregressive processes without requiring an importance sampling function or other tuning constants several examples including one with ar 4 errors are used to illustrate the ideas

covariance structure of the gibbs sampler with applications to the comparisons of estimators and augmentation schemes summary we study the covariance structure of a markov chain generated by the gibbs sampler with emphasis on data augmentation when applied to a bayesian missing data problem the gibbs sampler produces two natural approximations for the posterior distribution of the parameter vector the empirical distribution based on the sampled values of the parameter vector and a mixture of complete data posteriors we prove that rao blackwellization causes a one lag delay for the autocovariances among dependent samples obtained from data augmentation and consequently the mixture approximation produces estimates with smaller variances than the empirical approximation the covariance structure results are used to compare different augmentation schemes it is shown that collapsing and grouping random components in a gibbs sampler with two or three components usually result in more efficient sampling schemes

the simulation smoother for time series models summary recently suggested procedures for simulating from the posterior density of states given a gaussian state space time series are refined and extended we introduce and study the simulation smoother which draws from the multivariate posterior distribution of the disturbances of the model so avoiding the degeneracies inherent in state samplers the technique is important in gibbs sampling with non gaussian time series models and for performing bayesian analysis of gaussian time series

reduced difference polynomials and self intersection computations abstract a reduced difference polynomial f u v p u p v u v may be associated with any given univariate polynomial p t t 0 1 such that the locus f u v 0 identifies the pairs of distinct values u and v satisfying p u p v the bernstein coefficients of f u v on 0 1 2 can be determined from those of p t through a simple algorithm and can be restricted to any subdomain of 0 1 2 by standard subdivision methods by constructing the reduced difference polynomials f u v and g u v associated with the coordinate components of a polynomial curve r t x t y t a quadtree decomposition of 0 1 2 guided by the variation diminishing property yields a numerically stable scheme for isolating real solutions of the system f u v g u v 0 which identify self intersections of the curve r t through the kantorovich theorem for guaranteed convergence of newton raphson iterations to a unique solution the self intersections can be efficiently computed to machine precision by generalizing the reduced difference polynomial to encompass products of univariate polynomials the method can be readily extended to compute the self intersections of rational curves and of the rational offsets to pythagorean hodograph curves