Weighted Least-squares Sense Research Articles

The orbits of Triton and Nereid and the pole orientation of Neptune from Voyager,HubbleSpace Telescope, and Earth-based astrometry in 1847–2020

Context.New observations and new planetary and satellite ephemerides provide opportunities to improve the ephemerides for Triton and Nereid as well as relevant parameters. In particular, the observations include a lot of new accurate Earth-based positions reduced withGaiaastrometic catalogs and accurate positions obtained fromHubbleSpace Telescope.Aims.We aim to reliably improve the ephemerides for Triton and Nereid along with some parameters by using all the available astrometric data from 1847 to 2020 and by updating the dynamical model. We also aim to improve the geometrical descriptions based on the improved orbits of the two satellites and the pole orientation of Neptune.Methods.The orbits of Triton and Nereid are determined by fitting dynamical and observational model parameters to observations in a weighted least-squares sense. The dynamical model makes use of the new ephemerides from Jet Propulsion Laboratory for planets, DE440, and those for the inner satellites of Neptune, NEP090. For completeness, in addition to the gravitational effects considered by NEP081, the model also includes perturbations from inner satellites and a revised model for the motion of the pole orientation of Neptune. Moreover, model simplifications are investigated to speed up the motion equation integration. Since the pole orientation angles of Neptune at epoch are possibly improvable according to the preliminary post-fit sensitivity analysis, these angles are adjusted together with the satellite state vectors at epoch. Linear mapping of the covariance matrix is a measure of formal uncertainties of our orbit and pole solutions. However, to obtain more reliable accuracy estimations, it is necessary to consider the uncertainties in the observations and the unadjusted model parameters. To accomplish this, a method (BR-RS) that performs bootstrap resampling of observations (BR) and random sampling of unadjusted model parameters (RS) is used. Analytical representations are fitted to the orbit and pole solutions to provide their geometric descriptions.Results.The model we use can be fitted to the observations with their estimated accuracies. The new ephemerides,FORCES-8-MAIN-2020, covering years 1600–2650 are available online in SPICE format. The orbits are well determined with the orbital uncertainties expected to be within 200 km (about 10 mas as seen from the Earth) for Triton and 1000 km (50 mas) for Nereid for the next 100 yr as estimated by the BR-RS method. In particular, the correction in the Nereid mean orbit motion from the NEP081 solution is +4.′′9 yr−1, and has a BR-RS uncertainty of 0.′′24 yr−1. In the fitting process, we also determine the pole orientation of Neptune. At the initial epoch 1989 September 1 TDB, the right ascension and declination of the new pole orientation referred to the International Celestial Reference System areαp= 299.°339 ± 0.°012 (formal)∕ ± 0.°014 (BR-RS) andδp= 42.°985 ± 0.°016 (formal)∕ ± 0.°045 (BR-RS), respectively. From 1800 to 2200, the motion of the pole orientation is well constrained with a BR-RS uncertainty of about 0.°01–0.°05. We also provide geometrical descriptions for the new orbits and pole orientation.

Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation

We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. (IMA J Numer Anal 35:1698–1728, 2015), where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices), and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.

Analytical 3D rotation estimation using vector measurements with full variance-covariance matrix

Abstract 3D rotation estimation using vector measurements is investigated. A general stochastic model is employed in which no specific structure is assumed about the variance covariance matrix of the measurement errors, or in other words, different elements of the same vector, and/or different vectors can have different variances and can be arbitrarily correlated. The rotation matrix is estimated in two steps both of which are analytical. First, a free matrix, not necessarily special orthogonal, is estimated optimally in the weighted least-squares sense. In this step, the special case with only 2 non-collinear vector measurements is also treated. Second, a special orthogonal matrix is estimated which is closest to the previously estimated free matrix in the minimum Frobenius norm sense. Error analysis is performed, and the variance covariance matrix of the rotation estimation error is derived. Simulation is conducted and the statistical efficiency and consistency of the proposed method are validated.

Efficient WLS Design of IIR Digital Filters Using Partial Second-Order Factorization

In this brief, a novel algorithm is developed for the design of infinite-impulse-response digital filters in the weighted least-squares sense. To simplify the design problem, the Levy linearized function and the Sanathanan–Koerner iterative technique are utilized. In the proposed algorithm, a denominator polynomial is decomposed as a cascade of a few second-order factors (SOFs) and a higher order factor (HOF). They are sequentially updated in each iteration. When updating SOFs composed of poles that are tending to be outside the unit circle, necessary and sufficient stability conditions are taken into account, while the HOF is optimized without any stability constraint. The number of SOFs can be automatically determined in the proposed design procedure, which is convenient for practical designs. Simulation results demonstrate that the proposed algorithm can attain a balance between computational efficiency and design accuracy.

Between moving least-squares and moving least- $$\ell _1$$ ℓ 1

Given function values at scattered points in $${\mathbb {R}}^d$$ , possibly with noise, one of the ways of generating approximation to the function is by the method of moving least-squares (MLS). The method consists of computing local polynomials which approximate the data in a locally weighted least-squares sense. The resulting approximation is smooth, and is well approximating if the underlying function is smooth. Yet, as any least-squares based method it is quite sensitive to outliers in the data. It is well known that least- $$\ell _1$$ approximations are not sensitive to outliers. However, due to the nature of the $$\ell _1$$ norm, using it in the framework of a “moving” approximation will not give a smooth, or even a continuous approximation. This paper suggests an error measure which is between the $$\ell _1$$ and the $$\ell _2$$ norms, with the advantages of both. Namely, yielding smooth approximations which are not too sensitive to outliers. A fast iterative method for computing the approximation is demonstrated and analyzed. It is shown that for a scattered data taken from a smooth function, with few outliers, the new approximation gives an $$O(h)$$ approximation error to the function.

A Recursive Method for the Approximation of LTI Systems Using Subband Processing

Using the subband technique, an LTI system can be implemented by the composition of an analysis filterbank, followed by a transfer matrix (subband model) and a synthesis filterbank. The advantage of this approach is that it offers a good tradeoff between latency and computational complexity. In this paper we propose an optimization method for approximating an LTI system using the subband technique. The proposed method includes optimal allocation of parameters from different FIR entries of the subband model, while keeping constant the total number of parameters, for a better utilization of the available coefficients. The optimization is done in a weighted least-squares sense considering either linear or logarithmic amplitude scale. Simulation results demonstrate the advantages of the proposed method when compared with classical implementation approaches using pole-zero transfer functions or segmented FFT algorithms.

Automatic Range Image Registration in the Markov Chain

In this paper, a novel entropy that can describe both long and short-tailed probability distributions of constituents of a thermodynamic system out of its thermodynamic limit is first derived from the Lyapunov function for a Markov chain. We then maximize this entropy for the estimation of the probabilities of possible correspondences established using the traditional closest point criterion between two overlapping range images. When we change our viewpoint to look carefully at the minimum solution to the probability estimate of the correspondences, the iterative range image registration process can also be modeled as a Markov chain in which lessons from past experience in estimating those probabilities are learned. To impose the two-way constraint, outliers are explicitly modeled due to the almost ubiquitous occurrence of occlusion, appearance, and disappearance of points in either image. The estimated probabilities of the correspondences are finally embedded into the powerful mean field annealing scheme for global optimization, leading the camera motion parameters to be estimated in the weighted least-squares sense. A comparative study using real images shows that the proposed algorithm usually outperforms the state-of-the-art ICP variants and the latest genetic algorithm for automatic overlapping range image registration.

Least-squares gradient calculation from multi-point observations of scalar and vector fields: methodology and applications with Cluster in the plasmasphere

Abstract. This paper describes a general-purpose algorithm for computing the gradients in space and time of a scalar field, a vector field, or a divergence-free vector field, from in situ measurements by one or more spacecraft. The algorithm provides total error estimates on the computed gradient, including the effects of measurement errors, the errors due to a lack of spatio-temporal homogeneity, and errors due to small-scale fluctuations. It also has the ability to diagnose the conditioning of the problem. Optimal use is made of the data, in terms of exploiting the maximum amount of information relative to the uncertainty on the data, by solving the problem in a weighted least-squares sense. The method is illustrated using Cluster magnetic field and electron density data to compute various gradients during a traversal of the inner magnetosphere. In particular, Cluster is shown to cross azimuthal density structure, and the existence of field-aligned currents in the plasmasphere is demonstrated.

Iterative greedy algorithm for solving the FIR paraunitary approximation problem

In this paper, a method for approximating a multi-input multi-output (MIMO) transfer function by a causal finite-impulse response (FIR) paraunitary (PU) system in a weighted least-squares sense is presented. Using a complete parameterization of FIR PU systems in terms of Householder-like building blocks, an iterative algorithm is proposed that is greedy in the sense that the observed mean-squared error at each iteration is guaranteed to not increase. For certain design problems in which there is a phase-type ambiguity in the desired response, which is formally defined in the paper, a phase feedback modification is proposed in which the phase of the FIR approximant is fed back to the desired response. With this modification in effect, it is shown that the resulting iterative algorithm not only still remains greedy, but also offers a better magnitude-type fit to the desired response. Simulation results show the usefulness and versatility of the proposed algorithm with respect to the design of principal component filter bank (PCFB)-like filter banks and the FIR PU interpolation problem. Concerning the PCFB design problem, it is shown that as the McMillan degree of the FIR PU approximant increases, the resulting filter bank behaves more and more like the infinite-order PCFB, consistent with intuition. In particular, this PCFB-like behavior is shown in terms of filter response shape, multiresolution, coding gain, noise reduction with zeroth-order Wiener filtering in the subbands, and power minimization for discrete multitone (DMT)-type transmultiplexers.

Automatic 3d free form shape matching using the graduated assignment algorithm

Three-dimensional free form shape matching is a fundamental problem in both the machine vision and pattern recognition literatures. However, the automatic approach to 3D free form shape matching still remains open. In this paper, we propose using k closest points in the second view for the automatic 3D free form shape matching. For the sake of computational efficiency, the optimised k-D tree is employed for the search of the k closest points. Since occlusion and appearance and disappearance of points almost always occur, slack variables have to be employed, explicitly modelling outliers in the process of matching. Then the relative quality of each possible point match is estimated using the graduated assignment algorithm, leading the camera motion parameters to be estimated by the quaternion method in the weighted least-squares sense. The experimental results based on both synthetic data and real images without any pre-processing show the effectiveness and efficiency of the proposed algorithm for the automatic matching of overlapping 3D free form shapes with either sparse or dense points.

An automated system for phase identification and quantitative composition determination using the electron microprobe: Theory and applications

Previously developed methods of phase classification using the electron microprobe have been based on setting threshold values or relative ranks for wt% oxide values. These methods perform poorly for minerals that have a significant solid solution, and provide no mechanism for checking the accuracy of the analysis or the accuracy of the classification. Because of these limitations, previous algorithms for phase classification cannot be completely automated to yield high-quality phase classifications for a broad range of assemblages. A new algorithm for phase classification using a transformation from wt% oxide to cation and finally to additive/exchange components is introduced. This algorithm can be completely automated, evaluates the accuracy of the mineral analysis and phase classification, and provides quantitative information about the composition of the phase. The algorithm works by exploiting the least-squares nature of the transformation from wt% oxide to additive/exchange components. This transformation determines the stoichiometric model-mineral composition nearest to the actual mineral composition (in a weighted least-squares sense). The difference between the stoichiometric model composition and the actual composition is determined for all possible phases. The stoichiometric model that has the smallest difference between the model and actual composition and has permissible values for the exchange components identifies the phase. If no stoichiometric mineral model fulfills these requirements (non-permissible exchange components or large difference between actual and model compositions), then the analysis is either of poor quality or there is no stoichiometric model available for the phase being analyzed. The ability of this algorithm to provide robust, automated phase classification and compositional analysis allows the electron microprobe to be used for modal analysis, as well as analysis of compositional variation with a phase and many other applications. The MATLAB package APHID (Additive/exchange PHase IDentification) implements this algorithm.

Fitting Dynamic Models to the Geosat Sea Level Observations in the Tropical Pacific Ocean. Part II: A Linear, Wind-driven Model

Abstract The Geosat altimeter sea level observations in the tropical Pacific Ocean are used to evaluate the Performance of a linear wind-driven equatorial wave model. The question posed is the extent to which such a model can describe the observed sea level variations. The Kalman filter and optimal smoother are used to obtain a solution that is an optimal fit to the observation in a weighted least-squares sense. The total mean variance of the Geosat sea level observation is 98.1 cm2, of which 36.6 cm2 is due to measurement errors, leaving 61.5 cm2 for the oceanographic signal to be explained. The model is found to account for about 68% of this signal Variance and the remainder is ascribed to the effects of physical mechanisms missing from the model. This result suggests that the Geosat data contains sufficient information for testing yet more sophisticated models. Utility of an approximate filter and smoother based on the asymptotic time limit of the estimation error covariance is also examined and compar...

Adaptive optimal load balancing in a nonhomogeneous multiserver system with a central job scheduler

A model comprising several servers, each equipped with its own queue and with possibly different service speeds, is considered. Each server receives a dedicated arrival stream of jobs; there is also a stream of generic jobs that arrive to a job scheduler and can be individually allocated to any of the servers. It is shown that if the arrival streams are all Poisson and all jobs have the same exponentially distributed service requirements, the probabilistic splitting of the generic stream that minimizes the average job response time is such that it balances the server idle times in a weighted least-squares sense, where the weighting coefficients are related to the service speeds of the servers. The corresponding result holds for nonexponentially distributed service times if the service speeds are all equal. This result is used to develop adaptive quasi-static algorithms for allocating jobs in the generic arrival stream when the load parameters are unknown. The algorithms utilize server idle-time measurements which are sent periodically to the central job scheduler. A model is developed for these measurements, and the result mentioned is used to cast the problem into one of finding a projection of the root of an affine function, when only noisy values of the function can be observed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Aberration-balancing technique for radially symmetric amplitude distributions: a generalization of the Maréchal approach

The Maréchal balancing theory for constant-amplitude circular pupils is extended to radially symmetric nonnegative amplitude distributions. A convenient technique based on the solution of a set of linear equations is proposed. It permits the determination of optimum balanced wave fronts in the weighted least-squares sense by using corresponding moments of the relevant apodizing function. Relations with the fundamental theory of orthogonal polynomials are discussed. Formulas for fast evaluation of peak intensity degradation in the far-field pattern and corresponding Maréchal tolerances are derived. Numerical results presented may be important in the design and analysis of optical systems for use with laser beams.