General Least-squares Solution Research Articles

Optimized motion estimation for MRE data with reduced motion encodes

Motion estimation is an essential step common to all magnetic resonance elastography (MRE) methods. For dynamic techniques, the motion is obtained from a sinusoidal fit of the image phase at multiple, uniformly spaced relative phase offsets, φ, between the motion and the motion encoding gradients (MEGs). Generally, eight values of φ sampled at the Nyquist interval π/4 over [0, 2π). We introduce a method, termed reduced motion encoding (RME), that reduces the number of φ required, thereby reducing the imaging time for an MRE acquisition. A frequency-domain algorithm was implemented using the discrete Fourier transform (DFT) to derive the general least-squares solution for the motion amplitude and phase given an arbitrary number of φ. A closed form representation of the condition number of the transformation matrix which is used for estimating motion was introduced to determine the sensitivity to noise for different sampling patterns of φ. Simulation results confirmed the minimum error sampling patterns suggested from the condition number maps. The minimum noise in the motion estimate is obtained when the sampled φ are essentially evenly distributed over the range [0, π) with an interval π/n, where n is the number of φ sampled, or alternatively with an interval 2π/n over the range [0, 2π) which represents the Nyquist interval. Simulations also show that the noise level decreases as n increases as expected. The decrease in noise is the largest when n is small and it becomes less significant as n increases. The algorithm also makes it possible to estimate the motion from only two values of φ, which cannot be accomplished with traditional methods because sampling at the Nyquist interval is indeterminate. Finally, noise levels in motion estimated from phantom studies and in vivo results taken with different n agreed with that predicted by simulation and condition number calculations.

Implementation of a generalized least-squares method for determining calibration curves from data with general uncertainty structures

The determination of a best-fit calibration curve that describes the response of a measuring system to the value of a standard is one of the most widely used procedures in metrology. The mathematical basis for a generalized least-squares solution to this problem is reviewed. Examples of the application of a software implementation of the method are presented to illustrate the treatment of calibration problems with different uncertainty structures for the calibration data, including correlated data. The examples concern the calibration of analysers to measure the composition of natural gas and the calibration of a gas flow dilutor.

Measuring change in controlled longitudinal studies.

This paper examines the implications of the correlational structure of repeated measurements for three indices of change that can be used to evaluate treatment effects in longitudinal studies with scheduled assessment times and fixed total duration. The generalized least squares (GLS) regression of repeated measurements on time, which is usually reserved for complex mixed model solutions, takes the correlational structure of the repeated measurements into account, whereas simple gain scores and ordinary least squares (OLS) regression calculations do not. Nevertheless, the GLS solution is equivalent to OLS under conditions of compound symmetry and is equivalent to the analysis of simple gain scores in the presence of an autoregressive (order 1) correlational structure. The understanding of these relationships is important with regard to the frequently heard criticisms of the simpler definitions of treatment response in repeated measurement designs.

Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index

Krylov subspace methods have been recently considered to solve singular linear systems Ax = b . In this paper, we derive the necessary and sufficient conditions guaranteeing that a Krylov subspace method converges to a vector A D b+ Px 0, where A D is the Drazin inverse of A and P is the projection P = I−A D A . Let k be the index of A. We further show that A D b+Px 0, x 0 ∈ R (A k−1)+ N (A) , is a generalized least-squares solution of Ax = b in R (A k)+ N (A) . Finally, we present the convergence bounds for the quasi-minimal residual algorithm (QMR) and transpose-free quasi-minimal residual algorithm (TFQMR). The index k of A in this paper can be arbitrary, which extends to the main results of Freund and Hochbruck (Numer. Linear Algebra Appl. 1 (1994) 403–420) that only considers the case k = 1 .

Generalized thermobarometry: Solution of the inverse chemical equilibrium problem using data for individual species

The inverse chemical equilibrium problem is the determination of unknown equilibrium pressure, temperature, and chemical potentials of s species, given measurements of their thermochemical constants and the compositions of phases in which they occur. Of the s species for which free energy approximations can be made, c will be compositionally independent, i.e., form a basis of composition space and hence of chemical potential space. This means that if the equilibrium model is correct, it should be possible to express the chemical potentials of all s species as linear combinations of the chemical potentials of any c basis species. The inverse chemical equilibrium problem can then be stated: g i(p,T) − μ i = 0 i= 1…c g j(p,T)− ∑ i=1 c v ijμ i=0 j=c+1…s where g i ( p, T) is a measured approximation to the apparent free energy of formation of species i, determined from thermochemical constants and observed compositions; μ i is the unknown equilibrium chemical potential of species i; and v ij are stoichiometric coefficients relating the compositions of the s − c compositionally dependent species to the c basis species. The problem has s equations and c + 2 unknowns, hence is underdetermined, exact, or overdetermined, depending on the relative magnitudes of s and c. Because of errors in measurement, or failure to preserve equilibrium compositions, overdetermined systems will usually be inconsistent. In such cases, the ordinary least-squares solution to the problem may be found by finding p, T, and μ i that minimize ƒ Tƒ where ƒ is the s component vector obtained by evaluating (1). If desired, an error covariance matrix V can be incorporated to obtain a generalized least-squares solution at the minimum of ƒ TV −1ƒ . Confidence regions can be approximated by contouring the sum of squares and using Monte Carlo techniques. The formulation is readily extended to include data from directly calibrated equilibria as equations of form: ΔG j ( p, T) − ∑ ν ij μ i = 0. Solutions to underdetermined problems can be constrained by replacing some of the equations with inequalities such as: g i 0( p, T) − μ i ≥ 0.

A note on adjustment of free networks

The present paper deals with the least-squares adjustment where the design matrix (A) is rank-deficient. The adjusted parameters\(\hat x\) as well as their variance-covariance matrix (\(\sum _{\hat x} \)) can be obtained as in the “standard” adjustment whereA has the full column rank, supplemented with constraints,\(C\hat x = w\), whereC is the constraint matrix andw is sometimes called the “constant vector”. In this analysis only the inner adjustment constraints are considered, whereC has the full row rank equal to the rank deficiency ofA, andAC T =0. Perhaps the most important outcome points to the three kinds of results 1) A general least-squares solution where both\(\hat x\) and\(\sum _{\hat x} \) are indeterminate corresponds tow=arbitrary random vector. 2) The minimum trace (least-squares) solution where\(\hat x\) is indeterminate but\(\sum _{\hat x} \) is detemined (and trace\(\sum _{\hat x} \) corresponds tow=arbitrary constant vector. 3) The minimum norm (least-squares) solution where both\(\hat x\) and\(\sum _{\hat x} \) are determined (and norm\(\hat x\), trace\(\sum _{\hat x} \) corresponds tow−0

Notes on equivalent forms of the general least-squares solution

The general least-squares solution for a rank-deficient system is expressed in several completely equivalent forms. The simplest form is written in terms of A+, the pseudo-inverse of the design matrix A, and of an arbitrary vector whose presence removes the need for more general inverse operators.

The galactic rotation effect in tenth magnitude K stars.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The galactic rotation effect in tenth magnitude K stars. Edmondson, Frank K. Abstract Radial velocities of 708 tenth magnitude K stars contained in the McCormick proper motion catalogues have been measured. The majority of the observations have been made with the 8 -inch McDonald reflector, and the remainder with the Mount Wilson 6o-inch. Of these stars 125 have been observed two or more times. Standard stars have been observed frequently in order to check the systematic accuracy of the velocities. The present paper gives first a preliminary discussion of 372 K stars between galactic latitudes $300 57' and - 300 57 , and between photovisual magnitudes 10.0 and 11.0. Of these 372 stars, 60 are dwarfs and 213 are giants, according to luminosity classifications made by Miss Edith Jaussen working under the supervision of Dr. Vyssotsky. Ninety-nine stars are as yet unclassified. The dwarfs were grouped into I I normal places, and the giants into 17. Using the unsmoothed normals, separate least-squares solutions were made for solar motion and galactic rotation. The results are shown in Table I, and some of Redman's results are quoted for comparison. It is interesting to note that Redman's giants were no further away than the dwarfs in the present investigation. The negative K-term found by Redman is also found for the giants here, whereas the dwarfs give a positive value. An absorption of one magnitude per kiloparsec was assumed in estimating the distances, leading to A = 18.2 for the giants. R. E. Wilson's discussion gave 17.7 and Vyssotsky's recent value is 19.0. TABLE I. GALACTIC ROTATION EFFECT FOR K STARS Red- Redman man FKE FKE .A.0. vol. 4 D.A.0. (dwarfs) (giants) Sol. 2 SnI. 4 vol. 6 ? 210 210 230 200 850 K -5.9 -5.2 +2.9 - .8 V 15.6 (20) 21.5 18.9 20 5 L (22 ) (22a) 32 17 19 rA 3.8 5.4 1.9 .6 15.4 1 350 358 17 ~ 328 The large velocity dispersion reported at the Th adison meeting has been confirmed by the additional data now at hand. Table II gives the results. The dispersions have been computed both with and without the high velocity stars. The figures for all stars, high velocity included, may be compared with the values given at Madison ((Ta = 29.5 1.7, b = 27.2 + 1.6, = + 32.1 + 2.2). The general least-squares solution for the entire group of stars is now being set up, and will be completed as soon as Miss Janssen and Dr. Vyssotsky have finished the luminosity classifications. TABLE II. VELOCITY DISPERSION FOR TENTH MAGNITUDE K STARS a-axis b-axis c-axis Giants +29.9+2.6 (71) +36.3+5.0(29) +29.8+3.2 (48) 25.7 2.4 (67) 25.1 3.5 (26) 20.3 2.4 (44) Dwarfs +24.8+4.5 (i8) +31.9+6.3 (14) +28.4+3.9 (30) 22.0 3.9 (28) All +30.9+2.2 (io6) +28.4+2.4(78) +28.5+2.2 (91) 25.2 1.9 (98) 22.4 2.0 (75) 21.5 1.7 (8s) Kirkwood Observatory, Indiana University, Bloomington, Ind. Publication: The Astronomical Journal Pub Date: 1948 DOI: 10.1086/106152 Bibcode: 1948AJ.....54Q..35E full text sources ADS |