scholarly journals SIMULTANEOUS SPECTROPHOTOMETRIC DETERMINATION AND CLASSICAL LEAST SQUARES METHOD: A SIMPLE EXPERIMENT TO INTRODUCE THE CONCEPT OF MULTIVARIATE CALIBRATION

Química Nova ◽  
2020 ◽  
Author(s):  
Jhonatas Carvalho ◽  
Larissa Lopes ◽  
Luciano Vidal ◽  
Poliana Santos
Keyword(s):  
Least Squares ◽  
Multivariate Calibration ◽  
Least Squares Method ◽  
Linear Equations ◽  
Natural Extension ◽  
Sunset Yellow ◽  
Absorption Data ◽  
Calibration Models ◽  
Calibration Methods ◽  
Food Dyes

We herein present an experiment where the concentrations of tartrazine, sunset yellow and amaranth in samples containing these three food dyes are determined by system of equations (SE) and classical least squares (CLS) multivariate calibration methods using light absorption data. Firstly, concentrations are obtained by means of the well-known SE method, that is, by solving a set of three linear equations in which the Beer-Lambert’s proportionality coefficients are obtained from analytical curves. Then, it is shown that the CLS method is a natural extension to SE, with an arbitrarily large number of equations. Nevertheless, within the CLS method, the unknown coefficients are found using mixtures with known concentrations of each dye. In order to introduce the students to the basics of algorithms and numerical computations, data treatment is performed in a command-line fashion using a freely available software. Advantages of multivariate calibration models over univariate ones are made clear, and the performance of the CLS and SE methods is compared based on the root-mean-square error.

A Nonlinear Least-Squares Graphical Tool (‘Gaussfit’) for Educational Purposes

2010 ◽  
Vol 47 (1) ◽  
pp. 11-22 ◽  
Author(s):  
Krešimir Malarić ◽  
Roman Malarić ◽  
Hrvoje Hegeduš
Keyword(s):  
Least Squares ◽  
Input Data ◽  
Least Squares Method ◽  
Linear Equations ◽  
Computer Calculation ◽  
Nonlinear Least Squares ◽  
Undergraduate Level ◽  
Graphical Tool ◽  
Systems Of Linear Equations ◽  
Graphical Presentation

This paper describes a computer program that finds a function which closely approximates experimental data using the least-squares method. The program finds parameters of the function as well as their corresponding uncertainties. It also has a subroutine for graphical presentation of the input data and the function. The program is used for educational purposes at undergraduate level for students who are learning least-squares fitting, how to solve systems of linear equations and about computer calculation errors.

New Hybrid Algorithm for Transferring Multivariate Quantitative Calibrations of Intra-Vendor Near-Infrared Spectrometers

2002 ◽  
Vol 56 (7) ◽  
pp. 877-886 ◽  
Author(s):  
Christine M. Wehlburg ◽  
David M. Haaland ◽  
David K. Melgaard
Keyword(s):  
Least Squares ◽  
Hybrid Algorithm ◽  
Near Infrared ◽  
Multivariate Calibration ◽  
Correct Prediction ◽  
Calibration Transfer ◽  
Calibration Models ◽  
Secondary Instrument ◽  
Instrument Response ◽  
Instrumental Drift

A new prediction-augmented classical least-squares/partial least-squares (PACLS/PLS) hybrid algorithm is ideally suited for use in transferring multivariate calibrations between spectrometers. Spectral variations such as instrument response differences can be explicitly incorporated into the algorithm through the use of subset sample spectra collected on both spectrometers. Two current calibration transfer methods, subset recalibration and piecewise direct standardization (PDS), also utilize subset sample spectra to facilitate transfer of calibration. The three methods were applied to the transfer of quantitative multivariate calibration models for near-infrared (NIR) data of organic samples containing chlorobenzene, heptane, and toluene between a primary and three secondary spectrometers that were all the same model, called intra-vendor transfer of calibration. The hybrid PACLS/PLS method outperformed subset recalibration and provided predictions equivalent to PDS with additive background correction on the two secondary spectrometers whose instrument drift appeared to be dominated by simple linear baseline variations. One of the secondary spectrometers had complex instrument drift that was captured by repeatedly measuring the spectrum of a single repeat sample. In calculating a transfer function to correct prediction spectra, PDS assumes no instrumental drift on the secondary spectrometer. Therefore, PDS was unable to directly accommodate both the subset samples and the use of a single repeat sample to transfer and maintain a calibration on that secondary instrument. In order to implement the transfer of calibration with PDS in the presence of complex instrument drift, recalibrated PLS models that included the repeat spectra from the secondary spectrometer were used to predict the spectra transformed by PDS. The importance of correcting for drift on the secondary spectrometer during calibration transfer was illustrated by the improvements in prediction for all three methods vs. using only the instrument response differences derived from the subset sample spectra. When the effects of instrument drift were complex on the secondary spectrometer, the PACLS/PLS hybrid algorithm outperformed both PDS and subset recalibration. Through the explicit incorporation of spectral variations, due to instrument response differences and drift on the secondary spectrometer, the PACLS/PLS algorithm was successful at intra-vendor transfer of calibrations between NIR spectrometers.

scholarly journals The Accuracy of Respiratory Calibration Methods for Estimating Lung Volume During Speech Breathing: A Comparison of Four Methods Across Three Adult Cohorts

2019 ◽  
Vol 62 (8) ◽  
pp. 2632-2644 ◽  
Author(s):  
Victoria S. McKenna ◽  
Jessica E. Huber
Keyword(s):  
Older Adults ◽  
Young Adults ◽  
Least Squares ◽  
Lung Volume ◽  
Least Squares Method ◽  
Absolute Difference ◽  
Kinematic Data ◽  
Percent Error ◽  
Calibration Methods ◽  
Speech Breathing

Purpose This study evaluated the accuracy of respiratory calibration methods for estimating lung volume during speech breathing. Method Respiratory kinematic data were acquired via inductance plethysmography in 32 young adults, 22 older adults, and 13 older adults with Parkinson's disease (PD). Raw rib cage (RC) and abdomen (AB) signals (V) were calibrated to liters using 4 correction methods: (a) isovolume maneuvers, (b) a constant 2:1 RC-to-AB ratio, (c) least squares method with RC correction only (LsqRC), and (d) least squares method with both RC and AB corrections (LsqRC/AB). Mean percent error, the absolute difference between estimated and actual lung volumes then normalized to each speaker's vital capacity, was calculated for each method. Results For young adults, the LsqRC/AB method significantly reduced mean percent error compared to all other methods. Although LsqRC/AB also resulted in smaller errors for older adults and adults with PD, LsqRC/AB and LsqRC were not significantly different from one another in these groups. Conclusion The LsqRC/AB method reduces errors across all cohorts, but older adults and adults with PD also have reduced errors when using LsqRC. Further research should investigate both least squares methods across larger age and disease severity ranges.

scholarly journals Multi-Window Classical Least-Squares Multivariate Calibration Methods for Quantitative ICP-AES Analyses

2000 ◽  
Vol 54 (9) ◽  
pp. 1291-1302 ◽  
Author(s):  
David M. Haaland ◽  
William B. Chambers ◽  
Michael R. Keenan ◽  
David K. Melgaard
Keyword(s):  
Least Squares ◽  
Multivariate Calibration ◽  
Calibration Methods

Effect of Random Noise on the Performance of NIR Calibrations

1998 ◽  
Vol 6 (1) ◽  
pp. 77-87 ◽  
Author(s):  
Jing Lu ◽  
W.F. McClure ◽  
F.E. Barton ◽  
D.S. Himmelsbach
Keyword(s):  
Least Squares ◽  
Near Infrared ◽  
Multivariate Calibration ◽  
Simple Procedure ◽  
Random Noise ◽  
Nir Spectroscopy ◽  
Principal Component ◽  
Optical Data ◽  
Spectra Analysis ◽  
Calibration Methods

The proliferation of applications for near infrared (NIR) spectroscopy has been fostered by advances in instrumentation and statistics. NIR analytical instrumentation is becoming more stable and reliable. Chemometrics is playing an important role in qualitative and quantitative NIR spectra analysis. The objective of this study was to evaluate the performances of four commonly used calibration models: (1) stepwise multiple linear regression (SMLR); (2) classical least-squares (CLS); (3) principal component regression (PCR); and (4) partial least-squares (PLS) in NIR spectroscopy analysis when random noise is present in the optical data. A conceptually simple procedure for comparing the performance of the four calibration methods in the presence of different levels of random noise in spectra data has been introduced here. This procedure, using the computer simulation data and real spectra of tobacco, has provided useful information for understanding the effects of random noise on the performance of multivariate calibration methods. Both numerical and graphical results will be shown.

Why we should give up the sin2ψ method

2009 ◽  
Vol 24 (S1) ◽  
pp. S16-S21 ◽  
Author(s):  
Balder Ortner
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Linear Equations ◽  
Single System ◽  
Stress Determination ◽  
Method Of Least Squares ◽  
X Ray ◽  
Stress Tensors ◽  
Course Of Action ◽  
Simultaneous Linear Equations

The sin2ψ method can be formulated as a single system of simultaneous linear equations. Using this it is easy to show that the sin2ψ method is not a least-squares method. It further helps to compare the accuracies of the stress tensors obtained by the sin2ψ method and the method of least squares. Quantitative comparisons have been made for different fictitious measurements. It is shown that the unnecessary loss in accuracy by using the sin2ψ method is quite significant and by no means negligible. The same course of action has been applied to compare the so-called Dölle-Hauk method with a least-squares method; the result is similar. Some other methods for X-ray stress determination, most often similar to the sin2ψ method, and their shortcomings are also discussed briefly, together with the corresponding, more effective and more versatile least-squares method.

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