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Unit Circle (unit + circle)
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] Estimation of an optimal mixed-phase inverse filterGEOPHYSICAL PROSPECTING, Issue 4 2000Bjorn Ursin Inverse filtering is applied to seismic data to remove the effect of the wavelet and to obtain an estimate of the reflectivity series. In many cases the wavelet is not known, and only an estimate of its autocorrelation function (ACF) can be computed. Solving the Yule-Walker equations gives the inverse filter which corresponds to a minimum-delay wavelet. When the wavelet is mixed delay, this inverse filter produces a poor result. By solving the extended Yule-Walker equations with the ACF of lag , on the main diagonal of the filter equations, it is possible to decompose the inverse filter into a finite-length filter convolved with an infinite-length filter. In a previous paper we proposed a mixed-delay inverse filter where the finite-length filter is maximum delay and the infinite-length filter is minimum delay. Here, we refine this technique by analysing the roots of the Z -transform polynomial of the finite-length filter. By varying the number of roots which are placed inside the unit circle of the mixed-delay inverse filter, at most 2, different filters are obtained. Applying each filter to a small data set (say a CMP gather), we choose the optimal filter to be the one for which the output has the largest Lp -norm, with p=5. This is done for increasing values of , to obtain a final optimal filter. From this optimal filter it is easy to construct the inverse wavelet which may be used as an estimate of the seismic wavelet. The new procedure has been applied to a synthetic wavelet and to an airgun wavelet to test its performance, and also to verify that the reconstructed wavelet is close to the original wavelet. The algorithm has also been applied to prestack marine seismic data, resulting in an improved stacked section compared with the one obtained by using a minimum-delay filter. [source] A hypersingular time-domain BEM for 2D dynamic crack analysis in anisotropic solidsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2009M. Wünsche Abstract A hypersingular time-domain boundary element method (BEM) for transient elastodynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack-faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the crack-opening-displacements (CODs). Special crack-tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time-domain BEM. Copyright © 2008 John Wiley & Sons, Ltd. [source] Convergence properties of bias-eliminating algorithms for errors-in-variables identificationINTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 9 2005Torsten Söderström Abstract This paper considers the problem of dynamic errors-in-variables identification. Convergence properties of the previously proposed bias-eliminating algorithms are investigated. An error dynamic equation for the bias-eliminating parameter estimates is derived. It is shown that the convergence of the bias-eliminating algorithms is basically determined by the eigenvalue of largest magnitude of a system matrix in the estimation error dynamic equation. When this system matrix has all its eigenvalues well inside the unit circle, the bias-eliminating algorithms can converge fast. In order to avoid possible divergence of the iteration-type bias-eliminating algorithms in the case of high noise, the bias-eliminating problem is re-formulated as a minimization problem associated with a concentrated loss function. A variable projection algorithm is proposed to efficiently solve the resulting minimization problem. A numerical simulation study is conducted to demonstrate the theoretical analysis. Copyright © 2005 John Wiley & Sons, Ltd. [source] Decentralized control of discrete-time linear time invariant systems with input saturationINTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 12 2010Ciprian Deliu Abstract We study decentralized stabilization of discrete-time linear time invariant (LTI) systems subject to actuator saturation using LTI controllers. The requirement of stabilization under both saturation constraints and decentralization imposes obvious necessary conditions on the open-loop plant, namely that its eigenvalues are in the closed unit disc and further that the eigenvalues on the unit circle are not decentralized fixed modes. The key contribution of this work is to provide a broad sufficient condition for decentralized stabilization under saturation. Specifically, we show through an iterative argument that the stabilization is possible: whenever (1) the open-loop eigenvalues are in the closed unit disc; (2) the eigenvalues on the unit circle are not decentralized fixed modes; and (3) these eigenvalues on the unit circle have algebraic multiplicity of 1. Copyright © 2009 John Wiley & Sons, Ltd. [source] Identification of Persistent Cycles in Non-Gaussian Long-Memory Time SeriesJOURNAL OF TIME SERIES ANALYSIS, Issue 4 2008Mohamed Boutahar Abstract., Asymptotic distribution is derived for the least squares estimates (LSE) in the unstable AR(p) process driven by a non-Gaussian long-memory disturbance. The characteristic polynomial of the autoregressive process is assumed to have pairs of complex roots on the unit circle. In order to describe the limiting distribution of the LSE, two limit theorems involving long-memory processes are established in this article. The first theorem gives the limiting distribution of the weighted sum, is a non-Gaussian long-memory moving-average process and (cn,k,1 , k , n) is a given sequence of weights; the second theorem is a functional central limit theorem for the sine and cosine Fourier transforms [source] Nonlinear Riemann,Hilbert problems with circular target curvesMATHEMATISCHE NACHRICHTEN, Issue 9 2008Christer Glader Abstract The paper gives a systematic and self-contained treatment of the nonlinear Riemann,Hilbert problem with circular target curves |w , c | = r, sometimes also called the generalized modulus problem. We assume that c and r are Hölder continuous functions on the unit circle and describe the complete set of solutions w in the disk algebra H, , C and in the Hardy space H, of bounded holomorphic functions. The approach is based on the interplay with the Nehari problem of best approximation by bounded holomorphic functions. It is shown that the considered problems fall into three classes (regular, singular, and void) and we give criteria which allow to classify a given problem. For regular problems the target manifold is covered by the traces of solutions with winding number zero in a schlicht manner. Counterexamples demonstrate that this need not be so if the boundary condition is merely continuous. Paying special attention to constructive aspects of the matter we show how the Nevanlinna parametrization of the full solution set can be obtained from one particular solution of arbitrary winding number. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circleMATHEMATISCHE NACHRICHTEN, Issue 1-2 2007Lev Aizenberg Abstract Let D be a unit disk andM be an open arc of the unit circle whose Lebesgue measure satisfies 0 < l (M) < 2,. Our first result characterizes the restriction of the holomorphic functions f , ,(D), which are in the Hardy class ,1 near the arcM and are denoted by ,, ,1M(,,), to the open arcM. This result is a direct consequence of the complete description of the space of holomorphic functions in the unit disk which are represented by the Carleman formulas on the open arc M. As an application of the above characterization, we present an extension theorem for a function f , L1(M) from any symmetric sub-arc L , M of the unit circle, such that , M, to a function f , ,, ,1L(,,). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Limits of zeros of orthogonal polynomials on the circleMATHEMATISCHE NACHRICHTEN, Issue 12-13 2005Barry Simon Abstract We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a result of Alfaro and Vigil (which answered a question of P. Turán): namely, for n < N , one can freely prescribe the n -th polynomial and N , n zeros of the N -th one. We shall also describe all possible limit sets of zeros within the unit disk. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Numerical methods for palindromic eigenvalue problems: Computing the anti-triangular Schur formNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2009D. Steven Mackey Abstract We present structure-preserving numerical methods for the eigenvalue problem of complex palindromic pencils. Such problems arise in control theory, as well as from palindromic linearizations of higher degree palindromic matrix polynomials. A key ingredient of these methods is the development of an appropriate condensed form,the anti-triangular Schur form. Ill-conditioned problems with eigenvalues near the unit circle, in particular near ±1, are discussed. We show how a combination of unstructured methods followed by a structured refinement can be used to solve such problems accurately. Copyright © 2008 John Wiley & Sons, Ltd. [source] Asymptotics in Knuth's parking problem for caravans,RANDOM STRUCTURES AND ALGORITHMS, Issue 1 2006Jean Bertoin Abstract We consider a generalized version of Knuth's parking problem, in which caravans consisting of a random number of cars arrive at random on the unit circle. Then each car turns clockwise until it finds a free space to park. Extending a recent work by Chassaing and Louchard Random Struct Algor 21(1) (2002), 76,119, we relate the asymptotics for the sizes of blocks formed by occupied spots with the dynamics of the additive coalescent. According to the behavior of the caravans' size tail distribution, several qualitatively different versions of the eternal additive coalescent are involved. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006 [source] Fine structure of the zeros of orthogonal polynomials III: Periodic recursion coefficientsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2006Barry Simon We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings inverse to the density of zeros. Zeros away from the a.c. spectrum have limit points mod p and only finitely many of them. © 2005 Wiley Periodicals, Inc. [source] | |||||||||||||