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Unit Sphere (unit + sphere)
Selected AbstractsDecision Theory Applied to an Instrumental Variables ModelECONOMETRICA, Issue 3 2007Gary Chamberlain This paper applies some general concepts in decision theory to a simple instrumental variables model. There are two endogenous variables linked by a single structural equation; k of the exogenous variables are excluded from this structural equation and provide the instrumental variables (IV). The reduced-form distribution of the endogenous variables conditional on the exogenous variables corresponds to independent draws from a bivariate normal distribution with linear regression functions and a known covariance matrix. A canonical form of the model has parameter vector (,, ,, ,), where ,is the parameter of interest and is normalized to be a point on the unit circle. The reduced-form coefficients on the instrumental variables are split into a scalar parameter ,and a parameter vector ,, which is normalized to be a point on the (k,1)-dimensional unit sphere; ,measures the strength of the association between the endogenous variables and the instrumental variables, and ,is a measure of direction. A prior distribution is introduced for the IV model. The parameters ,, ,, and ,are treated as independent random variables. The distribution for ,is uniform on the unit circle; the distribution for ,is uniform on the unit sphere with dimension k-1. These choices arise from the solution of a minimax problem. The prior for ,is left general. It turns out that given any positive value for ,, the Bayes estimator of ,does not depend on ,; it equals the maximum-likelihood estimator. This Bayes estimator has constant risk; because it minimizes average risk with respect to a proper prior, it is minimax. The same general concepts are applied to obtain confidence intervals. The prior distribution is used in two ways. The first way is to integrate out the nuisance parameter ,in the IV model. This gives an integrated likelihood function with two scalar parameters, ,and ,. Inverting a likelihood ratio test, based on the integrated likelihood function, provides a confidence interval for ,. This lacks finite sample optimality, but invariance arguments show that the risk function depends only on ,and not on ,or ,. The second approach to confidence sets aims for finite sample optimality by setting up a loss function that trades off coverage against the length of the interval. The automatic uniform priors are used for ,and ,, but a prior is also needed for the scalar ,, and no guidance is offered on this choice. The Bayes rule is a highest posterior density set. Invariance arguments show that the risk function depends only on ,and not on ,or ,. The optimality result combines average risk and maximum risk. The confidence set minimizes the average,with respect to the prior distribution for ,,of the maximum risk, where the maximization is with respect to ,and ,. [source] Spectral estimation on a sphere in geophysics and cosmologyGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2008F. A. Dahlen SUMMARY We address the problem of estimating the spherical-harmonic power spectrum of a statistically isotropic scalar signal from noise-contaminated data on a region of the unit sphere. Three different methods of spectral estimation are considered: (i) the spherical analogue of the one-dimensional (1-D) periodogram, (ii) the maximum-likelihood method and (iii) a spherical analogue of the 1-D multitaper method. The periodogram exhibits strong spectral leakage, especially for small regions of area A, 4,, and is generally unsuitable for spherical spectral analysis applications, just as it is in 1-D. The maximum-likelihood method is particularly useful in the case of nearly-whole-sphere coverage, A, 4,, and has been widely used in cosmology to estimate the spectrum of the cosmic microwave background radiation from spacecraft observations. The spherical multitaper method affords easy control over the fundamental trade-off between spectral resolution and variance, and is easily implemented regardless of the region size, requiring neither non-linear iteration nor large-scale matrix inversion. As a result, the method is ideally suited for most applications in geophysics, geodesy or planetary science, where the objective is to obtain a spatially localized estimate of the spectrum of a signal from noisy data within a pre-selected and typically small region. [source] Exploratory orientation data analysis with , sectionsJOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 5 2004K. Gerald Van Den Boogaart Since the domain of crystallographic orientations is three-dimensional and spherical, insightful visualization of them or visualization of related probability density functions requires (i) exploitation of the effect of a given orientation on the crystallographic axes, (ii) consideration of spherical means of the orientation probability density function, in particular with respect to one-dimensional totally geodesic submanifolds, and (iii) application of projections from the two-dimensional unit sphere onto the unit disk . The familiar crystallographic `pole figures' are actually mean values of the spherical Radon transform. The mathematical Radon transform associates a real-valued function f defined on a sphere with its mean values along one-dimensional circles with centre , the origin of the coordinate system, and spanned by two unit vectors. The family of views suggested here defines , sections in terms of simultaneous orientational relationships of two different crystal axes with two different specimen directions, such that their superposition yields a user-specified pole probability density function. Thus, the spherical averaging and the spherical projection onto the unit disk determine the distortion of the display. Commonly, spherical projections preserving either volume or angle are favoured. This rich family displays f completely, i.e. if f is given or can be determined unambiguously, then it is uniquely represented by several subsets of these views. A computer code enables the user to specify and control interactively the display of linked views, which is comprehensible as the user is in control of the display. [source] Regularized, fast, and robust analytical Q-ball imagingMAGNETIC RESONANCE IN MEDICINE, Issue 3 2007Maxime Descoteaux Abstract We propose a regularized, fast, and robust analytical solution for the Q-ball imaging (QBI) reconstruction of the orientation distribution function (ODF) together with its detailed validation and a discussion on its benefits over the state-of-the-art. Our analytical solution is achieved by modeling the raw high angular resolution diffusion imaging signal with a spherical harmonic basis that incorporates a regularization term based on the Laplace,Beltrami operator defined on the unit sphere. This leads to an elegant mathematical simplification of the Funk,Radon transform which approximates the ODF. We prove a new corollary of the Funk,Hecke theorem to obtain this simplification. Then, we show that the Laplace,Beltrami regularization is theoretically and practically better than Tikhonov regularization. At the cost of slightly reducing angular resolution, the Laplace,Beltrami regularization reduces ODF estimation errors and improves fiber detection while reducing angular error in the ODF maxima detected. Finally, a careful quantitative validation is performed against ground truth from synthetic data and against real data from a biological phantom and a human brain dataset. We show that our technique is also able to recover known fiber crossings in the human brain and provides the practical advantage of being up to 15 times faster than original numerical QBI method. Magn Reson Med 58:497,510, 2007. © 2007 Wiley-Liss, Inc. [source] On the boundedness of some potential-type operators with oscillating kernelsMATHEMATISCHE NACHRICHTEN, Issue 5 2005Denis N. Karasev Abstract We consider a class of multidimensional potential-type operators with kernels that have singularities at the origin and on the unit sphere and that are oscillating at infinity. We describe some convex sets in the (1/p, 1/q)-plane for which these operators are bounded from Lp into Lq and indicate domains where they are not bounded. We also reveal some effects which show that oscillation and singularities of the kernels may strongly influence on the picture of boundedness of the operators under consideration. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] An anisotropic viscoelastic model for collagenous soft tissues at large strains , Computational aspectsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2009Alexander E. Ehret This paper focusses on computational aspects related to a recently proposed anisotropic viscoelastic model for soft biological tissues at large strains [1]. A key aspect of this model is the generalisation of micromechanically motivated one-dimensional constitutive equations to three dimensions by numerical integration over the unit sphere. A strong effect of this procedure on the accuracy and in particular on the material symmetry of the model is observed. Finally a finite element example of an artery subject to normotensive blood pressure is presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] PLUG-IN ESTIMATION OF GENERAL LEVEL SETSAUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 1 2006Antonio Cuevas Summary Given an unknown function (e.g. a probability density, a regression function, ,) f and a constant c, the problem of estimating the level set L(c) ={f,c} is considered. This problem is tackled in a very general framework, which allows f to be defined on a metric space different from . Such a degree of generality is motivated by practical considerations and, in fact, an example with astronomical data is analyzed where the domain of f is the unit sphere. A plug-in approach is followed; that is, L(c) is estimated by Ln(c) ={fn,c}, where fn is an estimator of f. Two results are obtained concerning consistency and convergence rates, with respect to the Hausdorff metric, of the boundaries ,Ln(c) towards ,L(c). Also, the consistency of Ln(c) to L(c) is shown, under mild conditions, with respect to the L1 distance. Special attention is paid to the particular case of spherical data. [source] A practical method for computing the largest M -eigenvalue of a fourth-order partially symmetric tensorNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2009Yiju Wang Abstract In this paper, we consider a bi-quadratic homogeneous polynomial optimization problem over two unit spheres arising in nonlinear elastic material analysis and in entanglement studies in quantum physics. The problem is equivalent to computing the largest M -eigenvalue of a fourth-order tensor. To solve the problem, we propose a practical method whose validity is guaranteed theoretically. To make the sequence generated by the method converge to a good solution of the problem, we also develop an initialization scheme. The given numerical experiments show the effectiveness of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd. [source] Mean curvature flows and isotopy of maps between spheresCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 8 2004Mao-Pei Tsui Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is isotopic to a constant map. © 2004 Wiley Periodicals, Inc. [source] | ||||||||||