A Decomposition Theorem for the Least Squares Piecewise Monotonic Data Approximation Problem

2019 ◽  
pp. 119-134 ◽  
Author(s):  
Ioannis C. Demetriou
Keyword(s):  
Least Squares ◽  
Decomposition Theorem ◽  
Approximation Problem ◽  
Data Approximation ◽  
Piecewise Monotonic

scholarly journals Peak Estimation of Univariate Spectraby the Best L1 Piecewise Monotonic Approximation Method

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Magnetic Resonance ◽  
Least Squares ◽  
Approximation Method ◽  
Nmr Spectra ◽  
Inorganic Compound ◽  
Piecewise Monotonic ◽  
Inherent Problem ◽  
New Algorithms ◽  
Nuclear Magnetic

We consider applications of the best L1 piecewise monotonic approximation method for the peak estimation of three sets of up to 2500 measurements of Raman, Infrared and Nuclear Magnetic Resonance (NMR)spectra. Peak estimation is an inherent problem of spectroscopy. The location of peaks and their intensities arethe signature of a sample of an organic or an inorganic compound. The diversity and the complexity of our measurements makes it a difficult test of the effectiveness of the method. We find that the method identifies efficientlypeaks and we compare to the results obtained by the analogous least squares calculations. These results havemany similarities and occasionally considerable differences due to both properties of the norms employed in theoptimization calculations and nature of the spectra. Our results may be helpful to subject analysts as part of theinformation on which decisions will be made for estimating peaks in sequences of spectra and to the developmentof new algorithms that are particularly suitable for peak estimation calculations.

Form Error Evaluation: An Iterative Reweighted Least Squares Algorithm*

2004 ◽  
Vol 126 (3) ◽  
pp. 535-541 ◽  
Author(s):  
Xiangyang Zhu ◽  
Han Ding ◽  
Michael Y. Wang
Keyword(s):  
Least Squares ◽  
Weighted Least Squares ◽  
Chebyshev Approximation ◽  
Approximation Problem ◽  
General Purpose ◽  
Geometric Features ◽  
Error Evaluation ◽  
Form Error ◽  
Minimum Zone ◽  
Effectiveness And Efficiency

This paper establishes the equivalence between the solution to a linear Chebyshev approximation problem and that of a weighted least squares (WLS) problem with the weighting parameters being appropriately defined. On this basis, we present an algorithm for form error evaluation of geometric features. The algorithm is implemented as an iterative procedure. At each iteration, a WLS problem is solved and the weighting parameters are updated. The proposed algorithm is of general-purpose, it can be used to evaluate the exact minimum zone error of various geometric features including flatness, circularity, sphericity, cylindericity and spatial straightness. Numerical examples are presented to show the effectiveness and efficiency of the algorithm.

Integration of Moving Least Squares Method and Multilevel B-Spline Method for Scattered Data Approximation

2011 ◽  
Vol 291-294 ◽  
pp. 2245-2249
Author(s):  
Shi Ju Yan ◽  
Bin Ge
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Scattered Data ◽  
Moving Least Squares ◽  
Scattered Data Approximation ◽  
Approximation Accuracy ◽  
Data Approximation ◽  
B Spline ◽  
Moving Least Squares Method ◽  
Method Accuracy

For scattered data approximation with multilevel B-spline(MBS) method, accuracy could be enhanced by densifying control lattice. Nevertheless, when control lattice density reaches to some extent, approximation accuracy could not be enhanced further. A strategy based on integration of moving least squares(MLS) and multilevel B-spline(MBS) is presented. Experimental results demonstrate that the presented strategy has higher approximation accuracy.

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