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Results for “generalized least squared”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

Multiparameter linear least-squares fitting to Poisson data one count at a time

A standard problem in gamma-ray astronomy data analysis is the decomposition of a set of observed counts, described by Poisson statistics, according to a given multicomponent linear model, with underlying physical count rates or fluxes which are to be estimated from the data. Despite its conceptual simplicity, the linear least-squares (LLSQ) method for solving this problem has generally been limited to situations in which the number n(sub i) of counts in each bin i is not too small, conventionally more than 5-30. It seems to be widely believed that the failure of the LLSQ method for small counts is due to the failure of the Poisson distribution to be even approximately normal for small numbers. The cause is more accurately the strong anticorrelation between the data and the wieghts w(sub i) in the weighted LLSQ method when square root of n(sub i) instead of square root of bar-n(sub i) is used to approximate the uncertainties, sigma(sub i), in the data, where bar-n(sub i) = E(n(sub i)), the expected value of N(sub i). We show in an appendix that, avoiding this approximation, the correct equations for the Poisson LLSQ (PLLSQ) problems are actually identical to those for the maximum likelihood estimate using the exact Poisson distribution. We apply the method to solve a problem in high-resolution gamma-ray spectroscopy for the JPL High-Resolution Gamma-Ray Spectrometer flown on HEAO 3. Systematic error in subtracting the strong, highly variable background encountered in the low-energy gamma-ray region can be significantly reduced by closely pairing source and background data in short segments. Significant results can be built up by weighted averaging of the net fluxes obtained from the subtraction of many individual source/background pairs. Extension of the approach to complex situations, with multiple cosmic sources and realistic background parameterizations, requires a means of efficiently fitting to data from single scans in the narrow (approximately = 1.2 keV, HEAO 3) energy channels of a Ge spectrometer, where the expected number of counts obtained per scan may be very low. Such an analysis system is discussed and compared to the method previously used.

Wheaton, Wm. A.↗
NASA NTRS2012

Linear Covariance Analysis and Epoch State Estimators

This paper extends in two directions the results of prior work on generalized linear covariance analysis of both batch least-squares and sequential estimators. The first is an improved treatment of process noise in the batch, or epoch state, estimator with an epoch time that may be later than some or all of the measurements in the batch. The second is to account for process noise in specifying the gains in the epoch state estimator. We establish the conditions under which the latter estimator is equivalent to the Kalman filter.

Markley, F. Landis↗
NASA NTRS2014

Linear Covariance Analysis and Epoch State Estimators

This paper extends in two directions the results of prior work on generalized linear covariance analysis of both batch least-squares and sequential estimators. The first is an improved treatment of process noise in the batch, or epoch state, estimator with an epoch time that may be later than some or all of the measurements in the batch. The second is to account for process noise in specifying the gains in the epoch state estimator. We establish the conditions under which the latter estimator is equivalent to the Kalman filter.

Linear covariance analysis↗
NASA NTRS2005

Using Remote Sensing Data to Evaluate Surface Soil Properties in Alabama Ultisols

Evaluation of surface soil properties via remote sensing could facilitate soil survey mapping, erosion prediction and allocation of agrochemicals for precision management. The objective of this study was to evaluate the relationship between soil spectral signature and surface soil properties in conventionally managed row crop systems. High-resolution RS data were acquired over bare fields in the Coastal Plain, Appalachian Plateau, and Ridge and Valley provinces of Alabama using the Airborne Terrestrial Applications Sensor multispectral scanner. Soils ranged from sandy Kandiudults to fine textured Rhodudults. Surface soil samples (0-1 cm) were collected from 163 sampling points for soil organic carbon, particle size distribution, and citrate dithionite extractable iron content. Surface roughness, soil water content, and crusting were also measured during sampling. Two methods of analysis were evaluated: 1) multiple linear regression using common spectral band ratios, and 2) partial least squares regression. Our data show that thermal infrared spectra are highly, linearly related to soil organic carbon, sand and clay content. Soil organic carbon content was the most difficult to quantify in these highly weathered systems, where soil organic carbon was generally less than 1.2%. Estimates of sand and clay content were best using partial least squares regression at the Valley site, explaining 42-59% of the variability. In the Coastal Plain, sandy surfaces prone to crusting limited estimates of sand and clay content via partial least squares and regression with common band ratios. Estimates of iron oxide content were a function of mineralogy and best accomplished using specific band ratios, with regression explaining 36-65% of the variability at the Valley and Coastal Plain sites, respectively.

Sullivan, Dana G.↗
NASA NTRS1976

Tests of general relativity using astrometric and radiometric observations of the planets

Current least squares fits to solar system data, including transit circle observations of the terrestrial and giant planets, radar observations of the terrestrial planets, Mariner 9 range fixes to Mars, and Pioneer 10/11 range fixes to Jupiter, have yielded some new results of interest to experimental relativity. Solutions have been obtained for the parameterized post-Newtonian (PPN) parameters beta and gamma, the solar gravitational quadrupole moment J2, a time variation in the gravitational constant G, and four Nordtvedt parameters.

Anderson, J. D.↗
DOE OSTI2020

Windowed least-squares model reduction for dynamical systems

Here we propose a windowed least-squares (WLS) approach for model reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space–time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as particular instances of the methodology recover Galerkin, least-squares Petrov–Galerkin (LSPG), and space–time LSPG projection. In addition, the approach addresses key deficiencies in existing model reduction techniques, e.g., the dependence of LSPG and space–time LSPG projection on the time discretization and the exponential growth in time exhibited by a posteriori error bounds for both Galerkin and LSPG projection. We consider two types of space–time trial subspaces within the proposed approach: one that reduces only the spatial dimension of the full-order model, and one that reduces both the spatial and temporal dimensions of the full-order model. For each type of trial subspace, we consider two different solution techniques: direct (i.e., discretize then optimize) and indirect (i.e., optimize then discretize). Numerical experiments conducted using trial subspaces characterized by spatial dimension reduction demonstrate that the WLS approach can yield more accurate solutions with lower space–time residuals than Galerkin and LSPG projection.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A quasi-meshfree method for constructing boundary-aware reproducing bases on geometrically complex domains using manifold geodesics

In many applications, physical domains are geometrically complex making it challenging to perform coarse-scale approximation. A defeaturing process is often used to simplify the domain in preparation for approximation and analysis at the coarse scale. In this paper, a methodology is presented for constructing a coarse-scale reproducing basis on geometrically complex domains given an initial fine-scale mesh of the fully featured domain. The initial fine-scale mesh can be of poor quality and extremely refined. The construction of the basis functions begins with a coarse-scale covering of the domain and generation of weighting functions with local support. Manifold geodesics are used to define distances within the local support for general applicability to non-convex domains. Conventional moving least squares is used to construct the coarse-scale reproducing basis. Applications in quasi-interpolation and linear elasticity are presented.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1974

Surface fitting three-dimensional bodies

The geometry of general three-dimensional bodies is generated from coordinates of points in several cross sections. Since these points may not be smooth, they are divided into segments and general conic sections are curve fit in a least-squares sense to each segment of a cross section. The conic sections are then blended in the longitudinal direction by fitting parametric cubic-spline curves through coordinate points which define the conic sections in the cross-sectional planes. Both the cross-sectional and longitudinal curves may be modified by specifying particular segments as straight lines and slopes at selected points. Slopes may be continuous or discontinuous and finite or infinite. After a satisfactory surface fit has been obtained, cards may be punched with the data necessary to form a geometry subroutine package for use in other computer programs. At any position on the body, coordinates, slopes and second partial derivatives are calculated. The method is applied to a blunted 70 deg delta wing, and it was found to generate the geometry very well.

Dejarnette, F. R.↗
NASA NTRS1975

Surface fitting three-dimensional bodies

The geometry of general three-dimensional bodies was generated from coordinates of points in several cross sections. Since these points may not be on smooth curves, they are divided into groups forming segments and general conic sections are curve fit in a least-squares sense to each segment of a cross section. The conic sections are then blended in the longitudinal direction through longitudinal curves. Both the cross-sectional and longitudinal curves may be modified by specifying particular segments as straight lines or specifying slopes at selected points. This method was used to surface fit a 70 deg slab delta wing and the HL-10 Lifting Body. The results for the delta wing were very close to the exact geometry. Although there is no exact solution for the lifting body, the surface fit generated a smooth surface with cross-sectional planes very close to prescribed coordinate points.

Dejarnette, F. R.↗
NASA NTRS1992

Linear regression in astronomy. II

A wide variety of least-squares linear regression procedures used in observational astronomy, particularly investigations of the cosmic distance scale, are presented and discussed. The classes of linear models considered are (1) unweighted regression lines, with bootstrap and jackknife resampling; (2) regression solutions when measurement error, in one or both variables, dominates the scatter; (3) methods to apply a calibration line to new data; (4) truncated regression models, which apply to flux-limited data sets; and (5) censored regression models, which apply when nondetections are present. For the calibration problem we develop two new procedures: a formula for the intercept offset between two parallel data sets, which propagates slope errors from one regression to the other; and a generalization of the Working-Hotelling confidence bands to nonstandard least-squares lines. They can provide improved error analysis for Faber-Jackson, Tully-Fisher, and similar cosmic distance scale relations.

Feigelson, Eric D.↗
NASA NTRS1984

Optical systolic solutions of linear algebraic equations

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Neuman, C. P.↗
NASA NTRS1990

Multistage estimation of received carrier signal parameters under very high dynamic conditions of the receiver

A multistage estimator is provided for the parameters of a received carrier signal possibly phase-modulated by unknown data and experiencing very high Doppler, Doppler rate, etc., as may arise, for example, in the case of Global Positioning Systems (GPS) where the signal parameters are directly related to the position, velocity and jerk of the GPS ground-based receiver. In a two-stage embodiment of the more general multistage scheme, the first stage, selected to be a modified least squares algorithm referred to as differential least squares (DLS), operates as a coarse estimator resulting in higher rms estimation errors but with a relatively small probability of the frequency estimation error exceeding one-half of the sampling frequency, provides relatively coarse estimates of the frequency and its derivatives. The second stage of the estimator, an extended Kalman filter (EKF), operates on the error signal available from the first stage refining the overall estimates of the phase along with a more refined estimate of frequency as well and in the process also reduces the number of cycle slips.

Kumar, Rajendra↗
NASA NTRS1991

Multistage estimation of received carrier signal parameters under very high dynamic conditions of the receiver

A multistage estimator is provided for the parameters of a received carrier signal possibly phase-modulated by unknown data and experiencing very high Doppler, Doppler rate, etc., as may arise, for example, in the case of Global Positioning Systems (GPS) where the signal parameters are directly related to the position, velocity and jerk of the GPS ground-based receiver. In a two-stage embodiment of the more general multistage scheme, the first stage, selected to be a modified least squares algorithm referred to as differential least squares (DLS), operates as a coarse estimator resulting in higher rms estimation errors but with a relatively small probability of the frequency estimation error exceeding one-half of the sampling frequency, provides relatively coarse estimates of the frequency and its derivatives. The second stage of the estimator, an extended Kalman filter (EKF), operates on the error signal available from the first stage refining the overall estimates of the phase along with a more refined estimate of frequency as well and in the process also reduces the number of cycle slips.

Kumar, Rajendra↗
NASA NTRS2009

3D Lunar Terrain Reconstruction from Apollo Images

Generating accurate three dimensional planetary models is becoming increasingly important as NASA plans manned missions to return to the Moon in the next decade. This paper describes a 3D surface reconstruction system called the Ames Stereo Pipeline that is designed to produce such models automatically by processing orbital stereo imagery. We discuss two important core aspects of this system: (1) refinement of satellite station positions and pose estimates through least squares bundle adjustment; and (2) a stochastic plane fitting algorithm that generalizes the Lucas-Kanade method for optimal matching between stereo pair images.. These techniques allow us to automatically produce seamless, highly accurate digital elevation models from multiple stereo image pairs while significantly reducing the influence of image noise. Our technique is demonstrated on a set of 71 high resolution scanned images from the Apollo 15 mission

Broxton, Michael J.↗
NASA NTRS1981

Remarks on nonstandard least-squares problems

It is shown that after treating a least-squares problem in the standard fashion, the imposition of stochastic restrictions on certain parameters can be accomplished essentially by inverting a matrix of an order no larger than the total number of restricted parameters, and that generally there is an explicit term for correcting the unrestricted to the restricted solution. In addition, any known type of nonstandard least-squares adjustment problem can be reduced to an equivalent standard linear problem, that is, one in which there is exactly one observation per condition equation and in which the observations are not correlated.

Eichhorn, H.↗
NASA NTRS1980

On the method of least squares

A method for fitting models to observed data by least squares when the data appear nonlinearly in the equations of condition, where there are nonlinear constraints among model parameters, and where the observations may be correlated, is described. The method is independent of the form in which the equations of condition are expressed, and is a generalization of the classical technique. The method is nearly as simple to apply in practice as the classical method of least squares.

Jefferys, W. H.↗