On the Enduring Appeal of Least-Squares Fitting in Computational Coordinate Metrology

2011 ◽  
Vol 12 (1) ◽  
Author(s):  
Vijay Srinivasan ◽  
Craig M. Shakarji ◽  
Edward P. Morse
Keyword(s):  
Least Squares ◽  
Geometric Features ◽  
Coordinate Measuring Machines ◽  
Historical Survey ◽  
Least Squares Fitting ◽  
Coordinate Metrology ◽  
International Standardization ◽  
Manufacturing Applications ◽  
Fitting In ◽  
Current Practices

The vast majority of points collected with coordinate measuring machines are not used in isolation; rather, collections of these points are associated with geometric features through fitting routines. In manufacturing applications, there are two fundamental questions that persist about the efficacy of this fitting—first, do the points collected adequately represent the surface under inspection; and second, does the association of substitute (fitted) geometry with the points meet criteria consistent with the standardized geometric specification of the product. This paper addresses the second question for least-squares fitting both as a historical survey of past and current practices, and as a harbinger of the influence of new specification criteria under consideration for international standardization. It also touches upon a set of new issues posed by the international standardization on the first question as related to sampling and least-squares fitting.

Fitting Weighted Total Least-Squares Planes and Parallel Planes to Support Tolerancing Standards

2012 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Total Least Squares ◽  
Point Sampling ◽  
Least Squares Fitting ◽  
Weighted Total Least Squares ◽  
Measuring Machine ◽  
Fitting In ◽  
Value Decomposition

We present elegant algorithms for fitting a plane, two parallel planes (corresponding to a slot or a slab) or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3×3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact Coordinate Measuring Machine). However, we demonstrate that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

scholarly journals Polynomial least squares fitting in the Bernstein basis

2010 ◽  
Vol 433 (7) ◽  
pp. 1254-1264 ◽  
Author(s):  
Ana Marco ◽  
José-Javier Martı´nez
Keyword(s):  
Least Squares ◽  
Bernstein Basis ◽  
Least Squares Fitting ◽  
Fitting In

scholarly journals A Recursive Bifurcation Model for Predicting the Peak of COVID-19 Virus Spread in United States and Germany

2020 ◽  
Author(s):  
Julia Shen
Keyword(s):  
United States ◽  
Decision Making ◽  
South Korea ◽  
Least Squares ◽  
Nonlinear Least Squares ◽  
Peak Time ◽  
Virus Spread ◽  
Linear Least Squares ◽  
Least Squares Fitting ◽  
Fitting In

AbstractPrediction on the peak time of COVID-19 virus spread is crucial to decision making on lockdown or closure of cities and states. In this paper we design a recursive bifurcation model for analyzing COVID-19 virus spread in different countries. The bifurcation facilitates a recursive processing of infected population through linear least-squares fitting. In addition, a nonlinear least-squares fitting is utilized to predict the future values of infected populations. Numerical results on the data from three countries (South Korea, United States and Germany) indicate the effectiveness of our approach.

Inversion of fracture density from field seismic velocities using artificial neural networks

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 534-545 ◽  
Author(s):  
Fred K. Boadu
Keyword(s):  
Neural Network ◽  
Least Squares ◽  
Displacement Discontinuity ◽  
Fracture Model ◽  
Seismic Velocities ◽  
Fracture Density ◽  
S Wave ◽  
Least Squares Fitting ◽  
The Neural Network ◽  
Fitting In

The inversion of fracture density from field measured P- and S-wave seismic velocities is performed using a neural network trained with an output from the modified displacement discontinuity fracture model. The basic idea is to use input‐output pairs generated by the fracture model to train the neural network. Once the neural network is trained, inversion of fracture density from field‐measured seismic velocities is performed very quickly. The overall performance of the neural network in the inversion process is assessed by means of a loss function. The results indicate that both sources of field information (P- and S-wave velocities) predict the field fracture density with reasonable accuracy. The performance of the neural network was compared to the prediction from least‐squares fitting. It is shown that the neural network out performs the least‐squares fitting in predicting the field‐fracture density values.

scholarly journals A recursive bifurcation model for early forecasting of COVID-19 virus spread in South Korea and Germany

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Julia Shen
Keyword(s):  
South Korea ◽  
Least Squares ◽  
Nonlinear Least Squares ◽  
Future Research ◽  
Logistic Growth ◽  
Virus Spread ◽  
Least Squares Fitting ◽  
Fitting Procedure ◽  
Logistic Growth Model ◽  
Fitting In

AbstractEarly forecasting of COVID-19 virus spread is crucial to decision making on lockdown or closure of cities, states or countries. In this paper we design a recursive bifurcation model for analyzing COVID-19 virus spread in different countries. The bifurcation facilitates recursive processing of infected population through linear least-squares fitting. In addition, a nonlinear least-squares fitting procedure is utilized to predict the future values of infected populations. Numerical results on the data from two countries (South Korea and Germany) indicate the effectiveness of our approach, compared to a logistic growth model and a Richards model in the context of early forecast. The limitation of our approach and future research are also mentioned at the end of this paper.

GEOMETRIC LEAST SQUARES FITTING OF CIRCLE AND ELLIPSE

1999 ◽  
Vol 13 (07) ◽  
pp. 987-996 ◽  
Author(s):  
SUNG JOON AHN ◽  
WOLFGANG RAUH
Keyword(s):  
Least Squares ◽  
Shortest Path ◽  
Geometric Feature ◽  
Geometric Features ◽  
Robust Algorithms ◽  
Geometric Fitting ◽  
Least Squares Fitting ◽  
The Given

The least squares fitting of geometric features to given points minimizes the squares sum of error-of-fit in predefined measures. By the geometric fitting, the error distances are defined with the orthogonal, or shortest, distances from the given points to the geometric feature to be fitted. For the geometric fitting of circle and ellipse, robust algorithms are proposed which are based on the coordinate descriptions of the corresponding point on the circle/ellipse for the given point, where the connecting line of the two points is the shortest path from the given point to the circle/ellipse.

Theory and Algorithms for Weighted Total Least-Squares Fitting of Lines, Planes, and Parallel Planes to Support Tolerancing Standards

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Total Least Squares ◽  
Point Sampling ◽  
Least Squares Fitting ◽  
Weighted Total Least Squares ◽  
Measuring Machine ◽  
Fitting In ◽  
Value Decomposition

We present the theory and algorithms for fitting a line, a plane, two parallel planes (corresponding to a slot or a slab), or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3 × 3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact coordinate measuring machine). However, we show by example that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

Sign in / Sign up

Export Citation Format

Share Document