53 resultados para Logarithmic Least Squares Method


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In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

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A new formulation of a pose refinement technique using ``active'' models is described. An error term derived from the detection of image derivatives close to an initial object hypothesis is linearised and solved by least squares. The method is particularly well suited to problems involving external geometrical constraints (such as the ground-plane constraint). We show that the method is able to recover both the pose of a rigid model, and the structure of a deformable model. We report an initial assessment of the performance and cost of pose and structure recovery using the active model in comparison with our previously reported ``passive'' model-based techniques in the context of traffic surveillance. The new method is more stable, and requires fewer iterations, especially when the number of free parameters increases, but shows somewhat poorer convergence.

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The influence matrix is used in ordinary least-squares applications for monitoring statistical multiple-regression analyses. Concepts related to the influence matrix provide diagnostics on the influence of individual data on the analysis - the analysis change that would occur by leaving one observation out, and the effective information content (degrees of freedom for signal) in any sub-set of the analysed data. In this paper, the corresponding concepts have been derived in the context of linear statistical data assimilation in numerical weather prediction. An approximate method to compute the diagonal elements of the influence matrix (the self-sensitivities) has been developed for a large-dimension variational data assimilation system (the four-dimensional variational system of the European Centre for Medium-Range Weather Forecasts). Results show that, in the boreal spring 2003 operational system, 15% of the global influence is due to the assimilated observations in any one analysis, and the complementary 85% is the influence of the prior (background) information, a short-range forecast containing information from earlier assimilated observations. About 25% of the observational information is currently provided by surface-based observing systems, and 75% by satellite systems. Low-influence data points usually occur in data-rich areas, while high-influence data points are in data-sparse areas or in dynamically active regions. Background-error correlations also play an important role: high correlation diminishes the observation influence and amplifies the importance of the surrounding real and pseudo observations (prior information in observation space). Incorrect specifications of background and observation-error covariance matrices can be identified, interpreted and better understood by the use of influence-matrix diagnostics for the variety of observation types and observed variables used in the data assimilation system. Copyright © 2004 Royal Meteorological Society

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A unified approach is proposed for sparse kernel data modelling that includes regression and classification as well as probability density function estimation. The orthogonal-least-squares forward selection method based on the leave-one-out test criteria is presented within this unified data-modelling framework to construct sparse kernel models that generalise well. Examples from regression, classification and density estimation applications are used to illustrate the effectiveness of this generic sparse kernel data modelling approach.

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This correspondence introduces a new orthogonal forward regression (OFR) model identification algorithm using D-optimality for model structure selection and is based on an M-estimators of parameter estimates. M-estimator is a classical robust parameter estimation technique to tackle bad data conditions such as outliers. Computationally, The M-estimator can be derived using an iterative reweighted least squares (IRLS) algorithm. D-optimality is a model structure robustness criterion in experimental design to tackle ill-conditioning in model Structure. The orthogonal forward regression (OFR), often based on the modified Gram-Schmidt procedure, is an efficient method incorporating structure selection and parameter estimation simultaneously. The basic idea of the proposed approach is to incorporate an IRLS inner loop into the modified Gram-Schmidt procedure. In this manner, the OFR algorithm for parsimonious model structure determination is extended to bad data conditions with improved performance via the derivation of parameter M-estimators with inherent robustness to outliers. Numerical examples are included to demonstrate the effectiveness of the proposed algorithm.

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New construction algorithms for radial basis function (RBF) network modelling are introduced based on the A-optimality and D-optimality experimental design criteria respectively. We utilize new cost functions, based on experimental design criteria, for model selection that simultaneously optimizes model approximation, parameter variance (A-optimality) or model robustness (D-optimality). The proposed approaches are based on the forward orthogonal least-squares (OLS) algorithm, such that the new A-optimality- and D-optimality-based cost functions are constructed on the basis of an orthogonalization process that gains computational advantages and hence maintains the inherent computational efficiency associated with the conventional forward OLS approach. The proposed approach enhances the very popular forward OLS-algorithm-based RBF model construction method since the resultant RBF models are constructed in a manner that the system dynamics approximation capability, model adequacy and robustness are optimized simultaneously. The numerical examples provided show significant improvement based on the D-optimality design criterion, demonstrating that there is significant room for improvement in modelling via the popular RBF neural network.

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A new robust neurofuzzy model construction algorithm has been introduced for the modeling of a priori unknown dynamical systems from observed finite data sets in the form of a set of fuzzy rules. Based on a Takagi-Sugeno (T-S) inference mechanism a one to one mapping between a fuzzy rule base and a model matrix feature subspace is established. This link enables rule based knowledge to be extracted from matrix subspace to enhance model transparency. In order to achieve maximized model robustness and sparsity, a new robust extended Gram-Schmidt (G-S) method has been introduced via two effective and complementary approaches of regularization and D-optimality experimental design. Model rule bases are decomposed into orthogonal subspaces, so as to enhance model transparency with the capability of interpreting the derived rule base energy level. A locally regularized orthogonal least squares algorithm, combined with a D-optimality used for subspace based rule selection, has been extended for fuzzy rule regularization and subspace based information extraction. By using a weighting for the D-optimality cost function, the entire model construction procedure becomes automatic. Numerical examples are included to demonstrate the effectiveness of the proposed new algorithm.

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A new identification algorithm is introduced for the Hammerstein model consisting of a nonlinear static function followed by a linear dynamical model. The nonlinear static function is characterised by using the Bezier-Bernstein approximation. The identification method is based on a hybrid scheme including the applications of the inverse of de Casteljau's algorithm, the least squares algorithm and the Gauss-Newton algorithm subject to constraints. The related work and the extension of the proposed algorithm to multi-input multi-output systems are discussed. Numerical examples including systems with some hard nonlinearities are used to illustrate the efficacy of the proposed approach through comparisons with other approaches.

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Most active-contour methods are based either on maximizing the image contrast under the contour or on minimizing the sum of squared distances between contour and image 'features'. The Marginalized Likelihood Ratio (MLR) contour model uses a contrast-based measure of goodness-of-fit for the contour and thus falls into the first class. The point of departure from previous models consists in marginalizing this contrast measure over unmodelled shape variations. The MLR model naturally leads to the EM Contour algorithm, in which pose optimization is carried out by iterated least-squares, as in feature-based contour methods. The difference with respect to other feature-based algorithms is that the EM Contour algorithm minimizes squared distances from Bayes least-squares (marginalized) estimates of contour locations, rather than from 'strongest features' in the neighborhood of the contour. Within the framework of the MLR model, alternatives to the EM algorithm can also be derived: one of these alternatives is the empirical-information method. Tracking experiments demonstrate the robustness of pose estimates given by the MLR model, and support the theoretical expectation that the EM Contour algorithm is more robust than either feature-based methods or the empirical-information method. (c) 2005 Elsevier B.V. All rights reserved.

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This paper employs a state space system description to provide a pole placement scheme via state feedback. It is shown that when a recursive least squares estimation scheme is used, the feedback employed can be expressed simply in terms of the estimated system parameters. To complement the state feedback approach, a method employing both state feedback and linear output feedback is discussed. Both methods arc then compared with the previous output polynomial type feedback schemes.

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A common problem in many data based modelling algorithms such as associative memory networks is the problem of the curse of dimensionality. In this paper, a new two-stage neurofuzzy system design and construction algorithm (NeuDeC) for nonlinear dynamical processes is introduced to effectively tackle this problem. A new simple preprocessing method is initially derived and applied to reduce the rule base, followed by a fine model detection process based on the reduced rule set by using forward orthogonal least squares model structure detection. In both stages, new A-optimality experimental design-based criteria we used. In the preprocessing stage, a lower bound of the A-optimality design criterion is derived and applied as a subset selection metric, but in the later stage, the A-optimality design criterion is incorporated into a new composite cost function that minimises model prediction error as well as penalises the model parameter variance. The utilisation of NeuDeC leads to unbiased model parameters with low parameter variance and the additional benefit of a parsimonious model structure. Numerical examples are included to demonstrate the effectiveness of this new modelling approach for high dimensional inputs.

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In this paper, we propose a new on-line learning algorithm for the non-linear system identification: the swarm intelligence aided multi-innovation recursive least squares (SI-MRLS) algorithm. The SI-MRLS algorithm applies the particle swarm optimization (PSO) to construct a flexible radial basis function (RBF) model so that both the model structure and output weights can be adapted. By replacing an insignificant RBF node with a new one based on the increment of error variance criterion at every iteration, the model remains at a limited size. The multi-innovation RLS algorithm is used to update the RBF output weights which are known to have better accuracy than the classic RLS. The proposed method can produces a parsimonious model with good performance. Simulation result are also shown to verify the SI-MRLS algorithm.

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Studies of the 1H n.m.r. and electronic spectra of a series of alkenylferrocenes including (E) and (Z) stereoisomers of various styrylferrocenes, have provided methods of structure elucidation. Crystals of the title compound are monoclinic, space group P21/c with Z= 4 in a unit cell of dimensions a= 17.603(2), b= 10.218(2), c= 10.072 Å, β= 103.27(2)°. The structure has been determined by the heavy-atom method from diffractometer data and refind by full-matrix least-squares techniques to R= 0.043 for 2 219 unique reflections.

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The molecular structure of trans-[PtCl(CCPh)(PEt2Ph)2] has been determined by X-ray diffraction methods. The crystals are monoclinic, space group P21, with a= 12.359(3), b= 13.015(3), c= 9.031(2)Å, β= 101.65(2)°, and Z= 2. The structure has been solved by the heavy-atom method and refined by full-matrix least squares to R 0.046 for 1 877 diffractometric intensity data. The crystals contain discrete molecules in which the platinum coordination is square planar. The phenylethynyl group is non-linear, with a Pt–CC angle of 163(2)°. Selected bond lengths are Pt–Cl 2.407(5) and Pt–C 1.98(2)Å. The structural trans influences of CCPh, CHCH2, and CH2SiMe3 ligands in platinum(II) complexes are compared; there is only a small dependence on hybridization at the ligating carbon atom.

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The molecular structure of trans-[PtCl(CHCH2)(PEt2Ph)2] has been determined by X-ray diffraction methods. The crystals are orthorhombic, space group Pbcn, with a= 10.686(2), b= 13.832(4), c= 16.129(4)Å, and Z= 4. The structure has been solved by the heavy-atom method and refined by full-matrix least squares to R 0.044 for 1 420 diffractometric intensity data. The crystals contain discrete molecules in which the platinum co-ordination is square planar. The Pt–Cl bond vector coincides with a crystallographic diad axis about which the atoms of the vinyl group are disordered. Selected bond lengths (Å) are Pt–Cl 2.398(4), Pt–P 2.295(3), and Pt–C 2.03(2). The Pt–CC angle is 127(2)°. From a survey of the available structural data it is concluded that there is little, if any, back donation from platinum to carbon in platinum–alkenyl linkages.