纵向数据边际模型非参数光滑方法的比较

对于纵向数据边际模型的均值函数, 有很多非参数估计方法, 其中回归样条, 光滑样条, 似乎不相关(SUR)核估计等方法在工作协方差阵正确指定时具有最小的渐近方差. 回归样条的渐近偏差与工作协方差阵无关, 而SUR核估计和光滑样条估计的渐近偏差却依赖于工作协方差阵. 本文主要研究了回归样条, 光滑样条和SUR核估计的效率问题. 通过模拟比较发现回归样条估计的表现比较稳定, 在大多数情况下比光滑样条估计和SUR核估计的效率高.     

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves, the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed. The quartic Hermite spline curves not only have the same interpolation and conti-nuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C2 continuity by the shape parameters when the interpolation conditions are fixed.     

广义非参数回归的B样本贝叶斯估计

本文主要研究广义非参数模型B样条Bayes估计 .将回归函数按照B样条基展开 ,我们不具体选择节点的个数 ,而是节点个数取均匀的无信息先验 ,样条函数系数取正态先验 ,用B样条函数的后验均值估计回归函数 .并给出了回归函数B样条Bayes估计的MCMC的模拟计算方法 .通过对Logistic非参数回归的模拟研究 ,表明B样条Bayes估计得到了很好的估计效果     

Forcing under Anti‐Foundation Axiom: An expression of the stalks

We introduce a new simple way of defining the forcing method that works well in the usual setting under FA, the Foundation Axiom, and moreover works even under Aczel's AFA, the Anti‐Foundation Axiom. This new way allows us to have an intuition about what happens in defining the forcing relation. The main tool is H. Friedman's method of defining the extensional membership relation ∈ by means of the intensional membership relation ε . Analogously to the usual forcing and the usual generic extension for FA‐models, we can justify the existence of generic filters and can obtain the Forcing Theorem and the Minimal Model Theorem with some modifications. These results are on the line of works to investigate whether model theory for AFA‐set theory can be developed in a similar way to that for FA‐set theory. Aczel pointed out that the quotient of transition systems by the largest bisimulation and transition relations have the essentially same theory as the set theory with AFA. Therefore, we could hope that, by using our new method, some open problems about transition systems turn out to be consistent or independent. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Airflow patterns for various approximations to velocity—pressure-gradient data

The effect of various ways of approximating velocity—pressure-gradient data on the computation of pressure fields in a grain bin is studied. The experimental data are also approximated by a cubic spline. The usual approximating formulas produce differing pressure patterns whenever the plenum pressure is sufficiently high to introduce velocities beyond the measured range.     

A cubic spline approximation and application of TAGE iterative method for the solution of two point boundary value problems with forcing function in integral form

In this article, we report an efficient high order numerical method based on cubic spline approximation and application of alternating group explicit method for the solution of two point non-linear boundary value problems, whose forcing functions are in integral form, on a non-uniform mesh. The proposed method is applicable when the internal grid points of solution interval are odd in number. The proposed cubic spline method is also applicable to integro-differential equations having singularities. Computational results are given to demonstrate the utility of the method.     

A Method for allowing Local Editing of Globally Optimized Splines

The present work describes an algorithm for modifying spline curves in the neighborhood of an editing point while preserving global smoothness properties. Current spline algorithms have either a graphical editing mode in which the user edits local properties of the spline, or a globally optimizing mode, in which the spline coefficients are determined such that overall properties e. g. smoothness, distance to support points, or physical behavior are optimized. With globally optimized splines, editing parameters at one point causes their transmission through the whole spline. Hence, the user has the impression that it is not possible to change the shape of the spline without disturbing the overall behavior. This can be circumvented by forcing the trajectory of the globally optimized curve to lie in the close vicinity of the original curve far away from the edited point. The present work describes an algorithm for local editing of spline curves that are produced by a global optimizer. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multinomial spline approximation scheme using spline quasi-interpolants

This paper presents a multinomial spline approximation scheme based on spline quasi-interpolants. The scheme can be considered as an extension of the usual Bernstein approximation for complex exponentials. Error estimates and numerical examples are given to show that this new scheme could produce highly accurate results.     

Structure of weak Descartes systems

A weak Descartes system is a basis of functions such that every ordered subset is a weak Tchebycheff system, the canonical example being the usual spline basis involving truncated power functions. By examining the intervals of degeneracy for a WD-system, we show that it is possible to produce a new basis that has a simple and convenient structure similar to the spline basis.     

AnO(h 6) quintic spline collocation method for fourth order two-point boundary value problems

AnO(h ⁶) collocation method based on quintic splines is developed and analyzed for general fourth-order linear two-point boundary value problems. The method determines a quintic spline approximation to the solution by forcing it to satisfy a high order perturbation of the original boundary value problem at the nodal points of the spline. A variation of this method is formulated as a deferred correction method. The error analysis of the new method and its numerical behavior is presented.This research was supported by AFOSR grant 84-0385.     

Numerical approaches to the functional distribution of anomalous diffusion with both traps and flights

The functional distributions of particle trajectories have wide applications. This paper focuses on providing effective computation methods for the models, which characterize the distribution of the functionals of the paths of anomalous diffusion with both traps and flights. Two kinds of discretization schemes are proposed for the time fractional substantial derivatives. The Galerkin method with interval spline scaling bases is used for the space approximation; compared with the usual finite element or spectral polynomial bases, the spline scaling bases have the advantages of keeping the Toeplitz structure of the stiffness matrix, and being easy to generate the matrix elements and to perform preconditioning. The rigorous stability analyses for both the semi and the full discrete schemes are skillfully developed. Under the assumptions of the regularity of the exact solution, the convergence of the provided schemes is also theoretically proved and numerically verified. Moreover, the theoretical background of the selected basis function and the implementation details of the algorithms involved are described in detail.     

Spline analysis of Hydrographic data

The basic problem involved in determining where the ship can not go is an attempt to reconstruct the sea bed. The interpolation of points necessary to reconstruct the sea bed was done using a bicubic spline. This method was chosen because of the similarities between the boundary conditions believed to be characteristic of the modeling problem and those of the natural spline. These include the continuity of the first and second derivatives, and the minimum curvature exhibited by the spline method which is characteristic of the sea bottom. The major problem faced in modeling the sea bed was selecting the extra data points needed in order to find a meaningful solution. This selection was done both by intuition and by constructing splines to model the possible behavior along a straight line. The results were two different models: a ridge model, characterized by a single shallow ridge in the center of the region; and a hill model, characterized by two smaller ridges. By varying one of these extra data points (called critical points), several models of both these extremes as well as intermediate models were generated. However, it was found that the number of given points did not permit a definitive model. Data was needed inside the region, especially at the critical points and at the exterior points in order to better define the boundary. The boundary could not be reliably determined since our spline model does not allow for accurate extrapolation. Thus, the model, although close to what is believed to be the correct model, is not good enough to allow for navigation because of the limited number of given data points.     

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??Inspired by intuitive meanings of truncated power basis's coefficients, the local penalization based on range's linear decreasing function is given in penalized spline regression model. This method gives less penalization to fitting curve where data is with more volatility, which makes fitted curve controls tradeoff between goodness-of-fit and smoothness better. Simulations show that regression models with local penalized spline obtain lower information rules' scores than global penalized spline when the data is with heteroskedasticity.     

Comparative Analysis of Cubic Spline and Kernel Estimation of a Probit Function

The least-squares cubic spline and the kernel estimators produce comparable mean squared errors, although the kernel produces smaller mean squared errors when the variable increases away from 0. Mean squared error increases with an increase in the number of knots (for the cubic spline) or reduced band width (for the kernel estimator). The cubic spline produces smaller mean squared errors when all observations are made at knots than when they are spaced out between knots. Irrespective of the exact form of the probit function g(x), the cubic spline estimator is asymptotically unbiased, while the kernel estimator only converges to g(x) under certain conditions. Moreover, the cubic spline is a smooth function, which is twice differentiable on the interval [0,1].     

Interpolation by splines satisfying mixed boundary conditions

We consider interpolation of Hermite data by splines of degreen withk given knots, satisfying boundary conditions which may involve derivatives at both end points (e.g., a periodicity condition). It is shown that, for a certain class of boundary conditions, a necessary and sufficient condition for the existence of a unique solution is that the data points and knots interlace properly and that there does not exist a polynomial solution of degreen?k. The method of proof is to show that any spline interpolating zero data vanishes identically, rather than the usual determinantal approach.     

Ergodicity of Truncated Stochastic Navier Stokes with Deterministic Forcing and Dispersion

Turbulence in idealized geophysical flows is a very rich and important topic. The anisotropic effects of explicit deterministic forcing, dispersive effects from rotation due to the \(\beta \)-plane and F-plane, and topography together with random forcing all combine to produce a remarkable number of realistic phenomena. These effects have been studied through careful numerical experiments in the truncated geophysical models. These important results include transitions between coherent jets and vortices, and direct and inverse turbulence cascades as parameters are varied, and it is a contemporary challenge to explain these diverse statistical predictions. Here we contribute to these issues by proving with full mathematical rigor that for any values of the deterministic forcing, the \(\beta \)- and F-plane effects and topography, with minimal stochastic forcing, there is geometric ergodicity for any finite Galerkin truncation. This means that there is a unique smooth invariant measure which attracts all statistical initial data at an exponential rate. In particular, this rigorous statistical theory guarantees that there are no bifurcations to multiple stable and unstable statistical steady states as geophysical parameters are varied in contrast to claims in the applied literature. The proof utilizes a new statistical Lyapunov function to account for enstrophy exchanges between the statistical mean and the variance fluctuations due to the deterministic forcing. It also requires careful proofs of hypoellipticity with geophysical effects and uses geometric control theory to establish reachability. To illustrate the necessity of these conditions, a two-dimensional example is developed which has the square of the Euclidean norm as the Lyapunov function and is hypoelliptic with nonzero noise forcing, yet fails to be reachable or ergodic.     

Exponential Box-Like Splines on Nonuniform Grids

We generalize the exponential box spline by allowing it to have arbitrarily spaced knots in any of its directions and derive the corresponding recurrence and differentiation rules. The corresponding spline space is spanned by the shifts of finitely many such splines and contains the usual family of exponential polynomials. The (local) linear independence of the spanning set is equivalent to a geometric condition closely related to unimodularity. January 10, 1996. Date revised: December 9, 1997. Date accepted: March 18, 1998.     

Some extremal properties of multivariate polynomial splines in the metric Lp (Rd )

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the Bd instead of the usual multivariate cardinal interpolation oper-ators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asymptoti-cally optimal for the Kolmogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on Bd in the metric Lp(Bd).     

Restricted Systems

Restricted systems are introduced and characterized. They are related with usual properties of spline spaces with relevant consequences in geometric modelling. It is a weaker property than local linear independence but it is preserved under a wide range of transformations.     

Finite difference approximations to one-dimensional parabolic equations using a cubic spline technique

This paper uses a cubic spline approximation to produce finite difference representations of the homogeneous heat equation in one spatial variable; it is shown that the usual explicit and implicit formulae are particular cases of the formulations given here. Formulae for truncation error and conditions for stability are derived. Numerical results are given for a simple example.