Reply by authors to Discussion by Yoich Ohta and Masanori Saito.

Geophysics ◽  
1979 ◽  
Vol 44 (9) ◽  
pp. 1589-1591
Author(s):  
A. F. Gangi ◽  
J. N. Shapiro
Keyword(s):  
Least Squares ◽  
Odd Order ◽  
Data Points ◽  
Independent Variable ◽  
Least Squares Algorithm

We are pleased to hear that Ohta and Saito found our “Propagating‐least‐Squares” algorithm (PROLSQ) for fitting polynomials, [Formula: see text], (1) simple to use and efficient in execution. We appreciate their pointing out that there can be a difficulty with the algorithm under very special (but easily determined) circumstances; that is, when the independent‐variable values [Formula: see text] at the data points are so distributed that the odd‐order moments [Formula: see text] are zero. We did not experience this difficulty because we never treated a case in which all these odd moments were zero.

Characteristics of statistical parameters used to interpret least-squares results.

1978 ◽  
Vol 24 (4) ◽  
pp. 611-620 ◽  
Author(s):  
R B Davis ◽  
J E Thompson ◽  
H L Pardue
Keyword(s):  
Experimental Data ◽  
Least Squares ◽  
Confidence Intervals ◽  
Standard Error ◽  
Statistical Parameters ◽  
Average Value ◽  
Data Points ◽  
Independent Variable ◽  
Highly Correlated

Abstract This paper discusses properties of several statistical parameters that are useful in judging the quality of least-squares fits of experimental data and in interpreting least-squares results. The presentation includes simplified equations that emphasize similarities and dissimilarities among the standard error of estimate, the standard deviations of slopes and intercepts, the correlation coefficient, and the degree of correlation between the least-squares slope and intercept. The equations are used to illustrate dependencies of these parameters upon experimentally controlled variables such as the number of data points and the range and average value of the independent variable. Results are interpreted in terms of which parameters are most useful for different kinds of applications. The paper also includes a discussion of joint confidence intervals that should be used when slopes and intercepts are highly correlated and presents equations that can be used to judge the degree of correlation between these coefficients and to compute the elliptical joint confidence intervals. The parabolic confidence intervals for calibration cures are also discussed briefly.

A least squares algorithm for fitting data points to a circular arc cam

Measurement ◽  
2017 ◽  
Vol 102 ◽  
pp. 170-178 ◽  
Author(s):  
Shan Lin ◽  
Otto Jusko ◽  
Frank Härtig ◽  
Jörg Seewig
Keyword(s):  
Least Squares ◽  
Circular Arc ◽  
Data Points ◽  
Least Squares Algorithm

EXPRESS: Baseline Correction Based on a Search Algorithm from Artificial Intelligence

2020 ◽  
pp. 000370282097751
Author(s):  
Xin Wang ◽  
Xia Chen
Keyword(s):  
Artificial Intelligence ◽  
Objective Function ◽  
Least Squares ◽  
Least Squares Method ◽  
Search Algorithm ◽  
Absolute Error ◽  
Search Problem ◽  
Baseline Correction ◽  
Current Spectrum ◽  
Data Points

Many spectra have a polynomial-like baseline. Iterative polynomial fitting (IPF) is one of the most popular methods for baseline correction of these spectra. However, the baseline estimated by IPF may have substantially error when the spectrum contains significantly strong peaks or have strong peaks located at the endpoints. First, IPF uses temporary baseline estimated from the current spectrum to identify peak data points. If the current spectrum contains strong peaks, then the temporary baseline substantially deviates from the true baseline. Some good baseline data points of the spectrum might be mistakenly identified as peak data points and are artificially re-assigned with a low value. Second, if a strong peak is located at the endpoint of the spectrum, then the endpoint region of the estimated baseline might have significant error due to overfitting. This study proposes a search algorithm-based baseline correction method (SA) that aims to compress sample the raw spectrum to a dataset with small number of data points and then convert the peak removal process into solving a search problem in artificial intelligence (AI) to minimize an objective function by deleting peak data points. First, the raw spectrum is smoothened out by the moving average method to reduce noise and then divided into dozens of unequally spaced sections on the basis of Chebyshev nodes. Finally, the minimal points of each section are collected to form a dataset for peak removal through search algorithm. SA selects the mean absolute error (MAE) as the objective function because of its sensitivity to overfitting and rapid calculation. The baseline correction performance of SA is compared with those of three baseline correction methods: Lieber and Mahadevan–Jansen method, adaptive iteratively reweighted penalized least squares method, and improved asymmetric least squares method. Simulated and real FTIR and Raman spectra with polynomial-like baselines are employed in the experiments. Results show that for these spectra, the baseline estimated by SA has fewer error than those by the three other methods.

Deep kernel recursive least-squares algorithm

2021 ◽  
Author(s):  
Hossein Mohamadipanah ◽  
Mahdi Heydari ◽  
Girish Chowdhary
Keyword(s):  
Least Squares ◽  
Recursive Least Squares ◽  
Recursive Least Squares Algorithm ◽  
Least Squares Algorithm
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