Least Squares Fitting for Linear Models

2020 ◽  
pp. 81-97
Keyword(s):  
Least Squares ◽  
Linear Models ◽  
Least Squares Fitting

The use of linear models and matrix least squares in clinical chemistry.

1979 ◽  
Vol 25 (6) ◽  
pp. 840-855 ◽  
Author(s):  
S N Deming ◽  
S L Morgan
Keyword(s):  
Least Squares ◽  
Precise Measurement ◽  
Linear Models ◽  
Clinical Chemistry ◽  
Experimental Designs ◽  
Unified Approach

Abstract We present a unified approach to the use of linear models and matrix least squares with the intention of providing a better understanding of the techniques themselves and of the statistics that arise from these techniques as they are used in clinical chemistry. Emphasis is placed on the importance of appropriate experimental designs and adequately precise measurement processes for efficiently obtaining the desired information.

Fuzzy C-Regression Models

2013 ◽  
Vol 278-280 ◽  
pp. 1323-1326
Author(s):  
Yan Hua Yu ◽  
Li Xia Song ◽  
Kun Lun Zhang
Keyword(s):  
Linear Regression ◽  
Least Squares ◽  
Linear Models ◽  
Least Squares Method ◽  
Least Square Method ◽  
Least Square ◽  
Estimation Methods ◽  
Statistical Regression ◽  
Fuzzy Linear Regression ◽  
Fuzzy Degree

Fuzzy linear regression has been extensively studied since its inception symbolized by the work of Tanaka et al. in 1982. As one of the main estimation methods, fuzzy least squares approach is appealing because it corresponds, to some extent, to the well known statistical regression analysis. In this article, a restricted least squares method is proposed to fit fuzzy linear models with crisp inputs and symmetric fuzzy output. The paper puts forward a kind of fuzzy linear regression model based on structured element, This model has precise input data and fuzzy output data, Gives the regression coefficient and the fuzzy degree function determination method by using the least square method, studies the imitation degree question between the observed value and the forecast value.

scholarly journals An Error-Reduction Method for Temporal Phase Unwrapping Based on Double Three-Step Phase-Shifting and Least-Squares Fitting

2012 ◽  
Vol 6-7 ◽  
pp. 76-81
Author(s):  
Yong Liu ◽  
Ding Fa Huang ◽  
Yong Jiang
Keyword(s):  
Least Squares ◽  
Reduction Method ◽  
Intermediate Phase ◽  
Error Reduction ◽  
Phase Unwrapping ◽  
Phase Shifting ◽  
Surface Measurement ◽  
Acquisition Time ◽  
Least Squares Fitting ◽  
Temporal Phase

Phase-shifting interferometry on structured light projection is widely used in 3-D surface measurement. An investigation shows that least-squares fitting can significantly decrease random error by incorporating data from the intermediate phase values, but it cannot completely eliminate nonlinear error. This paper proposes an error-reduction method based on double three-step phase-shifting algorithm and least-squares fitting, and applies it on the temporal phase unwrapping algorithm using three-frequency heterodyne principle. Theoretical analyses and experiment results show that this method can greatly save data acquisition time and improve the precision.

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.

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