Locally Adaptive Wavelet Empirical Bayes Estimation of a Location Parameter

The traditional empirical Bayes (EB) model is considered with the parameter being a location parameter, in the situation when the Bayes estimator has a finite degree of smoothness and, possibly, jump discontinuities at several points. A nonlinear wavelet EB estimator based on wavelets with bounded supports is constructed, and it is shown that a finite number of jump discontinuities in the Bayes estimator do not affect the rate of convergence of the prior risk of the EB estimator to zero. It is also demonstrated that the estimator adjusts to the degree of smoothness of the Bayes estimator, locally, so that outside the neighborhoods of the points of discontinuities, the posterior risk has a high rate of convergence to zero. Hence, the technique suggested in the paper provides estimators which are significantly superior in several respects to those constructed earlier.     

Minimum-Variance Multitaper Spectral Estimation on the Sphere

We develop a method to estimate the power spectrum of a stochastic process on the sphere from data of limited geographical coverage. Our approach can be interpreted either as estimating the global power spectrum of a stationary process when only a portion of the data are available for analysis, or estimating the power spectrum from local data under the assumption that the data are locally stationary in a specified region. Restricting a global function to a spatial subdomain—whether by necessity or by design—is a windowing operation, and an equation like a convolution in the spectral domain relates the expected value of the windowed power spectrum to the underlying global power spectrum and the known power spectrum of the localization window. The best windows for the purpose of localized spectral analysis have their energy concentrated in the region of interest while possessing the smallest effective bandwidth as possible. Solving an optimization problem in the sense of Slepian (1960) yields a family of orthogonal windows of diminishing spatiospectral localization, the best concentrated of which we propose to use to form a weighted multitaper spectrum estimate in the sense of Thomson (1982). Such an estimate is both more representative of the target region and reduces the estimation variance when compared to estimates formed by any single bandlimited window. We describe how the weights applied to the individual spectral estimates in forming the multitaper estimate can be chosen such that the variance of the estimate is minimized.     

单位球面上的一极值问题

设Ω是R~n中的紧集,1相似文献   

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n ^–4/5 x ^–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.     

Weighted Approximation of Functions on the Unit Sphere

The direct and inverse theorems are established for the best approximation in the weighted L^p space on the unit sphere of R^d+1, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a K-functional defined using the power of the spherical h-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of R^d.     

Adaptive Estimation of Distributions Using Exponential Sub-Families

Abstract

An algorithm is presented that, for a large-dimensional exponential family G, finds a lower dimension exponential sub-family of G which contains distributions best fitting groups of identically distributed observations within a set of data. The data are therefore fitted to a family of distributions that has been adaptively chosen as representative of them. The algorithm is implemented in the special case in which G is a logspline family of distributions. An example data set is analyzed using the method.     

Biorthogonal Locally Supported Wavelets on the Sphere Based on Zonal Kernel Functions

This article presents a method for approximating spherical functions from discrete data of a block-wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotationinvariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from classical tensor product wavelet theory.     

C~n中单位球面上的不变g-函数及BMOA

设B是C~n中的单位球,S是C~n中的单位球面。■(f)是定义在S上的不变g-函数．设f∈ BMOA,若存在正测度集ES,使■(f)＜ ∞在E上成立,则■(f)＜ ∞在S上几乎处处成立,同时■(f)∈BMO(S)且存在常数C,使得‖■(f)‖≤C‖f‖.     

Cn中单位球面上的不变g-函数及BMOA

设B是Cn中的单位球,S是Cn中的单位球面,(g)(f)是定义在S上的不变g-函数.设f∈BMOA,若存在正测度集E(∪)S,使(g)(f)＜+∞在E上成立,则(g)(f)＜+∞在S上几乎处处成立,同时(g)(f)∈BMO(S)且存在常数C,使得‖(g)(f)‖*≤C‖f‖*.     

Adaptive Estimation of FCG Using Nonlinear State-Space Models

Abstract

In this paper, an efficient adaptive nonlinear algorithm for estimation and identification, the so-called adaptive Lainiotis filter (ALF), is applied to the problem of fatigue crack growth (FCG) estimation, identification, and prediction of the final crack (failure). A suitable nonlinear state-space FCG model is introduced for both ALF and extended Kalman filter (EKF). Both algorithms are tested in order to compare their efficiency. Through extensive analysis and simulation, it is demonstrated that the ALF has superior performance both in FCG estimation, as well as in predicting the remaining lifetime to failure. Furthermore, it is shown that the ALF is faster and easier to implement in a parallel/distributed processing mode, and much more robust than the classic EKF.     

n维单位球面上的Toeplitz符号演算

探讨了C^n中单位球面S上Berezin变换和Toeplitz算子的性质,证明了由{Tφ,φ∈L^∞ （S）}所生成的C^＊-代数中算子T的符号恰好为单位球B上函数T（称为T的Berezin变换）的非切向边界值．此外,本文还得到了经典Toeplitz符号演算的有趣推广．     

A Note on Certain Block Spaces on the Unit Sphere

In this note, we clarify a relation between block spaces and the Hardy space. We obtain Bq^0.v belong to H^1（S^n-1）＋L（ln＋L）^1＋v（s^n-1）,v〉-1,q〉1,Furthermore,if v≥ 0, q 〉 1. we verify that block spaces Rq^0.v（S^n-1）are proper subspaces of H1 （S^n- 1）,     

Density Estimation on the Stiefel Manifold

This paper develops the theory of density estimation on the Stiefel manifoldV_{k, m}, whereV_{k, m}is represented by the set ofm×kmatricesXsuch thatX′X=I_k, thek×kidentity matrix. The density estimation by the method of kernels is considered, proposing two classes of kernel density estimators with small smoothing parameter matrices and for kernel functions of matrix argument. Asymptotic behavior of various statistical measures of the kernel density estimators is investigated for small smoothing parameter matrix and/or for large sample size. Some decompositions of the Stiefel manifoldV_{k, m}play useful roles in the investigation, and the general discussion is applied and examined for a special kernel function. Alternative methods of density estimation are suggested, using decompositions ofV_{k, m}.     

球面均匀分布的拟合优度检验

证明了基于惯量矩的d维单位球面上样本服从均匀分布的基本特征,得到球面均匀分布协差阵特征根估计的强相合性及渐近多元正态性.提出了检验球面上样本均匀性的渐近卡方统计量,证明了拟合优度检验的相合性并做检验功效的随机模拟.     

Visual Servoing on the Sphere Using Conformal Geometric Algebra

This paper deals with a visual servoing scheme, which uses a paracatadioptric sensor as visual input. The paracatadioptric sensor provides a wide field of view while maintaining the single center of projection, which is a desirable property of these sensors. The projection induced by this sensor is nonlinear. In this paper a linear model of this projection is presented, the model is developed using the conformal geometric algebra framework which allows to represent nonlinear conformal transformations using a special type of multivectors called versors. With this model we relate the feature time variation with the camera velocity to design a velocity controller used in a visual servoing task.     

Weak Type Inequalities for Moduli of Smoothness on Unit Sphere

For a periodic function L_p(T),Ul'yanov proved the now classical inequalities in 1968 w(f,t)_q≤C(f_O~t(u~(-θ)w(f,u)_p)~(q_1)(du)/u)~(1/(q_1),(1) where 1≤p相似文献   

g-Natural Contact Metrics on Unit Tangent Sphere Bundles

We construct a three-parameter family of contact metric structures on the unit tangent sphere bundle T ₁ M of a Riemannian manifold M and we study some of their special properties related to the Levi-Civita connection. More precisely, we give the necessary and sufficient conditions for a constructed contact metric structure to be K-contact, Sasakian, to satisfy some variational conditions or to define a strongly pseudo-convex CR-structure. The obtained results generalize classical theorems on the standard contact metric structure of T ₁ M.     

Neural Networks for Optimal Approximation of Continuous Functions on the Unit Sphere

We prove that, to approximate continuous target functions defined on the unit sphere of ${\mathbb{R}^d}$ , neural networks are at least as good as algebraic polynomials.     

模糊仿生间接自适应控制用于逼近误差的估计

针对一类不确定非线性系统,提出将生物个体对外界环境的适应对策引入到间接模糊自适应控制方法.方法的特点是将生物个体的主动自适应性与模糊控制相结合,将反应生物特性能力的结构单位生态位作为模糊规则后件.在仿生模糊系统的基础上,利用Lyapunov综合方法形成状态反馈仿生自适应控制算法,关键问题是重构误差有界,得到的系统具有良好逼近性.仿生控制律有两部分构成,一部分是等效控制,另一部分是边界控制,边界参数自适应律被用于这个边界控制,根据稳定性理论,证明闭环系统的全局稳定性.最后对倒立摆系统进行了仿真,结果证明了该方法的有效性和可行性.     

On Banach Spaces Whose Unit Sphere Determines Polynomials

In this paper we study the problem of characterizing the real Banach spaces whose unit sphere determines polynomials, i.e., if two polynomials coincide in the unit sphere, is this sufficient to guarantee that they are identical？ We show that, in the frame of spaces with unconditional basis, non- reflexivity is a sufficient, although not necessary, condition for the above question to have an affirmative answer. We prove that the only lp^n spaces having this property are those with p irrational, while the only lp spaces which do not enjoy it are those with p an even integer. We also introduce a class of polynomial determining sets in any real Banach space.