Generalized Linear Least-Squares Adjustment, Revisited

2006 ◽  
Vol 3 (7) ◽  
pp. 13461 ◽  
Author(s):  
BL Broadhead ◽  
ML Williams ◽  
JJ Wagschal ◽  
SW Dean
Keyword(s):  
Least Squares ◽  
Linear Least Squares ◽  
Least Squares Adjustment

Generalized Linear Least-Squares Adjustment, Revisited

2009 ◽  
pp. 340-340-8
Author(s):  
B. L. Broadhead ◽  
M. L. Williams ◽  
J. J. Wagschal
Keyword(s):  
Least Squares ◽  
Linear Least Squares ◽  
Least Squares Adjustment

Linear least-squares adjustment of UB matrix elements and the prediction of reflection positions

1990 ◽  
Vol 23 (2) ◽  
pp. 111-114 ◽  
Author(s):  
C. Wilkinson
Keyword(s):  
Single Crystal ◽  
Least Squares ◽  
Rapid Method ◽  
Matrix Elements ◽  
Linear Least Squares ◽  
Least Squares Adjustment

A method is presented for the linear least-squares refinement of small changes in the UB matrix elements and diffractometer offsets in single-crystal diffractometer experiments. It thus affords a rapid method for calculating the consequent changes in the positions of the centres of Bragg reflections.

Spin-Echo Experiment with NMR Spectra only Partially Narrowed by an Insufficiently Fast Motion

2003 ◽  
Vol 68 (7) ◽  
pp. 1193-1205
Author(s):  
Jaromír Jakeš
Keyword(s):  
Time Delay ◽  
Autocorrelation Function ◽  
Nmr Spectra ◽  
Spin Echo ◽  
Slow Motion ◽  
Linear Least Squares ◽  
Least Squares Adjustment ◽  
Fast Motion ◽  
Motional Narrowing ◽  
Gaussian Line

The spin-echo experiment on a spin system with only partial motional narrowing and an exponential field autocorrelation function is considered. The pattern of the intensity decrease in the echo spectra depends on the ratio τ/τc of the time delay τ in the echo experiment to the correlation time τc of the narrowing motion. With the large ratios (fast motion), the decrease is the same as in the case of extreme narrowing; to obtain undistorted T2 values, the ratio should be at least several units in the single-echo experiment and at least few decades in the multiple-echo experiment. With the small ratios (slow motion), the logarithmic decrease depends non-linearly on τ, and the T2 value found by the linear least-squares adjustment is much longer than that obtained from the extreme narrowing approximation. At very small ratios, the multiple echo yields about 3τc/(ωpτ)2 for T2 as compared with 1/(ωp2rc) obtained from the extreme narrowing approximation; ωp2 is the second moment of the Gaussian line being narrowed. The expression for T2 in the multiple spin echo is similar to that previously found for T2e in the solid multiple spin echo. The echo experiment changes the line shape, which at large τ/τc approaches the Lorentzian one. The case of a multiexponential field autocorrelation function is also briefly considered.

scholarly journals Using Least-Squares Residuals to Assess the Stochasticity of Measurements—Example: Terrestrial Laser Scanner and Surface Modeling

2021 ◽  
Vol 5 (1) ◽  
pp. 59
Author(s):  
Gaël Kermarrec ◽  
Niklas Schild ◽  
Jan Hartmann
Keyword(s):  
Least Squares ◽  
Laser Scanner ◽  
Real Data ◽  
Point Clouds ◽  
Statistical Testing ◽  
Normal Vector ◽  
Engineering Structures ◽  
3D Point Clouds ◽  
Least Squares Adjustment ◽  
Correlation Parameters

Terrestrial laser scanners (TLS) capture a large number of 3D points rapidly, with high precision and spatial resolution. These scanners are used for applications as diverse as modeling architectural or engineering structures, but also high-resolution mapping of terrain. The noise of the observations cannot be assumed to be strictly corresponding to white noise: besides being heteroscedastic, correlations between observations are likely to appear due to the high scanning rate. Unfortunately, if the variance can sometimes be modeled based on physical or empirical considerations, the latter are more often neglected. Trustworthy knowledge is, however, mandatory to avoid the overestimation of the precision of the point cloud and, potentially, the non-detection of deformation between scans recorded at different epochs using statistical testing strategies. The TLS point clouds can be approximated with parametric surfaces, such as planes, using the Gauss–Helmert model, or the newly introduced T-splines surfaces. In both cases, the goal is to minimize the squared distance between the observations and the approximated surfaces in order to estimate parameters, such as normal vector or control points. In this contribution, we will show how the residuals of the surface approximation can be used to derive the correlation structure of the noise of the observations. We will estimate the correlation parameters using the Whittle maximum likelihood and use comparable simulations and real data to validate our methodology. Using the least-squares adjustment as a “filter of the geometry” paves the way for the determination of a correlation model for many sensors recording 3D point clouds.

scholarly journals Adaptive Robust Kernels for Non-Linear Least Squares Problems

2021 ◽  
pp. 1-1
Author(s):  
Nived Chebrolu ◽  
Thomas Labe ◽  
Olga Vysotska ◽  
Jens Behley ◽  
Cyrill Stachniss
Keyword(s):  
Least Squares ◽  
Linear Least Squares ◽  
Least Squares Problems ◽  
Non Linear ◽  
Linear Least Squares Problems
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