2,999 research outputs found

    Scattered data fitting on surfaces using projected Powell-Sabin splines

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    We present C1 methods for either interpolating data or for fitting scattered data associated with a smooth function on a two-dimensional smooth manifold Ī© embedded into R3. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of local projections on the tangent planes. The data fitting method is a two-stage method. We illustrate the performance of the algorithms with some numerical examples, which, in particular, confirm the O(h3) order of convergence as the data becomes dens

    An introduction to regular splines and their application for initial value problems of ordinary differential equations

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    This report describes an application of the general method of integrating initial value problems by means of regular splines for equations with movable singularities. By defining the families of functions that make up the regular splines such that they closely resemble the behaviour of the solutions of the differential equation, it is possible to trace the location of the singularities very precisely. To demonstrate this we treat Riccati differential equations. These are known to possess solutions with poles, usually of the first order. This type of differential equation or system arises in describing chemical or biological processes or more general control processes. To make the report self contained it starts with an introduction to regular splines and develops the algebraic tools for the manipulation of rational splines. After the description of the integration procedure, the asymptotic behaviour of the systematic error is investigated. An example exhibits the results obtained from the program given in Appendix A. Then Riccati equations are introduced and methods for the determination of the singularities are developed. These methods are tested numerically with several examples. The results are given in Appendix B

    Interpolation and scattered data fitting on manifolds using projected Powellā€“Sabin splines

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    We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts {(UĪ¾ , Ī¾)}Ī¾āˆˆ satisfying certain conditions of smooth dependence on Ī¾. If is a C2-manifold embedded into R3, then projections into tangent planes can be employed. The data fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function

    Spline-based self-controlled case series method

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    The self-controlled case series (SCCS) method is an alternative to study designs such as cohort and case control methods and is used to investigate potential associations between the timing of vaccine or other drug exposures and adverse events. It requires information only on cases, individuals who have experienced the adverse event at least once, and automatically controls all fixed confounding variables that could modify the true association between exposure and adverse event. Time-varying confounders such as age, on the other hand, are not automatically controlled and must be allowed for explicitly. The original SCCS method used step functions to represent risk periods (windows of exposed time) and age effects. Hence, exposure risk periods and/or age groups have to be prespecified a priori, but a poor choice of group boundaries may lead to biased estimates. In this paper, we propose a nonparametric SCCS method in which both age and exposure effects are represented by spline functions at the same time. To avoid a numerical integration of the product of these two spline functions in the likelihood function of the SCCS method, we defined the first, second, and third integrals of I-splines based on the definition of integrals of M-splines. Simulation studies showed that the new method performs well. This new method is applied to data on pediatric vaccines
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