Degeneracy, Singularity, and Multiplicity in Least-Squares Design of a Function-Generating Mechanism

1983 ◽  
Vol 105 (1) ◽  
pp. 104-107 ◽  
Author(s):  
D. J. Wilde
Keyword(s):  
Least Squares ◽  
Curve Fitting ◽  
Design Variable ◽  
Global Minimum ◽  
Degenerate Case ◽  
Fitting Procedure ◽  
Squared Error ◽  
Independent Design ◽  
Minimal Designs

Error Linearization (EL), an iterative curve-fitting procedure recently proposed for designing minimum squared error four-bar function generating mechanisms, suffers from frequent instability. The cause seems to be the near singularity of a certain 3×3 matrix, which produces artificially large steps, usually toward designs with unrealistically short driver and follower. This degenerate case proves unfortunately to be the true global minimum. To bring this behavior under control, the coupler length, formerly regarded as an independent design variable, is made to depend on the driver and follower lengths. They are determined by solving a now well-conditioned 2×2 set of error linearization equations. In an example this Stabilized EL procedure (SEL) located five reasonable locally minimal designs which would have been missed by the unstabilized version.

Curve Fitting of Heterodyne Data

1985 ◽  
Vol 39 (3) ◽  
pp. 480-484 ◽  
Author(s):  
B. R. Reddy
Keyword(s):  
Experimental Data ◽  
Maximum Likelihood ◽  
Least Squares ◽  
Curve Fitting ◽  
Maximum Likelihood Method ◽  
Heterodyne Detection ◽  
Likelihood Method ◽  
Fitting Procedure ◽  
Least Squares Curve Fitting

The least-squares curve fitting procedure of experimental data is described for heterodyne detection. Two independent methods, the Maximum Likelihood method of Fisher and SIMPLEX, have been tried. The relative merits and limitations are discussed in detail.

scholarly journals An Autocorrelation Term Method for Curve Fitting

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Louis M. Houston
Keyword(s):  
Least Squares ◽  
Curve Fitting ◽  
Least Squares Method ◽  
Total Error ◽  
Linear Least Squares ◽  
Visual Examination ◽  
Popular Method ◽  
Quadratic Equations ◽  
Squared Error ◽  
Polynomial Curve

The least-squares method is the most popular method for fitting a polynomial curve to data. It is based on minimizing the total squared error between a polynomial model and the data. In this paper we develop a different approach that exploits the autocorrelation function. In particular, we use the nonzero lag autocorrelation terms to produce a system of quadratic equations that can be solved together with a linear equation derived from summing the data. There is a maximum of solutions when the polynomial is of degree . For the linear case, there are generally two solutions. Each solution is consistent with a total error of zero. Either visual examination or measurement of the total squared error is required to determine which solution fits the data. A comparison between the comparable autocorrelation term solution and linear least squares shows negligible difference.

Use of reflectance spectroscopy to estimate the organic carbon and CaCO3 contents of soils

2012 ◽  
Vol 61 (2) ◽  
pp. 277-290 ◽  
Author(s):  
Ádám Csorba ◽  
Vince Láng ◽  
László Fenyvesi ◽  
Erika Michéli
Keyword(s):  
Organic Carbon ◽  
Least Squares ◽  
Partial Least Squares ◽  
Partial Least Squares Regression ◽  
Mean Squared Error ◽  
Reflectance Spectroscopy ◽  
Least Squares Regression ◽  
Root Mean Squared Error ◽  
Squared Error

Napjainkban egyre nagyobb igény mutatkozik olyan technológiák és módszerek kidolgozására és alkalmazására, melyek lehetővé teszik a gyors, költséghatékony és környezetbarát talajadat-felvételezést és kiértékelést. Ezeknek az igényeknek felel meg a reflektancia spektroszkópia, mely az elektromágneses spektrum látható (VIS) és közeli infravörös (NIR) tartományában (350–2500 nm) végzett reflektancia-mérésekre épül. Figyelembe véve, hogy a talajokról felvett reflektancia spektrum információban nagyon gazdag, és a vizsgált tartományban számos talajalkotó rendelkezik karakterisztikus spektrális „ujjlenyomattal”, egyetlen görbéből lehetővé válik nagyszámú, kulcsfontosságú talajparaméter egyidejű meghatározása. Dolgozatunkban, a reflektancia spektroszkópia alapjaira helyezett, a talajok ösz-szetételének meghatározását célzó módszertani fejlesztés első lépéseit mutatjuk be. Munkánk során talajok szervesszén- és CaCO3-tartalmának megbecslését lehetővé tévő többváltozós matematikai-statisztikai módszerekre (részleges legkisebb négyzetek módszere, partial least squares regression – PLSR) épülő prediktív modellek létrehozását és tesztelését végeztük el. A létrehozott modellek tesztelése során megállapítottuk, hogy az eljárás mindkét talajparaméter esetében magas R2értéket [R2(szerves szén) = 0,815; R2(CaCO3) = 0,907] adott. A becslés pontosságát jelző közepes négyzetes eltérés (root mean squared error – RMSE) érték mindkét paraméter esetében közepesnek mondható [RMSE (szerves szén) = 0,467; RMSE (CaCO3) = 3,508], mely a reflektancia mérési előírások standardizálásával jelentősen javítható. Vizsgálataink alapján arra a következtetésre jutottunk, hogy a reflektancia spektroszkópia és a többváltozós kemometriai eljárások együttes alkalmazásával, gyors és költséghatékony adatfelvételezési és -értékelési módszerhez juthatunk.

Parametric Description of Vibration Using the Autocorrelation Function

2009 ◽  
Vol 147-149 ◽  
pp. 606-611
Author(s):  
Adam Kotowski
Keyword(s):  
Autocorrelation Function ◽  
Curve Fitting ◽  
Vibration Signal ◽  
Analytic Form ◽  
Parametric Representation ◽  
Rotating Machine ◽  
Fitting Procedure ◽  
Power Spectral ◽  
Parametric Description ◽  
Harmonic Components

The paper presents the use of the autocorrelation function for the description of vibrations and the problems connected with. The proposed method is based on the analysis of vibration signal recorded for machine during its operations using an analytic form of the autocorrelation function. The parameters are obtained using a curve fitting procedure. To keep a quality of parametric representation of considered vibration, only the curve fitting causes a determination coefficient over 0.90 is taken into consideration. Therefore, the autocorrelation functions are submitted for the fast Fourier transform to be helped, in determination of number of the dominant harmonic components. Also, the analytic form and parameters of power spectral density has been also calculated. Finally, the set of parameters has been collected to describe the selected fragment of vibration of the simple rotating machine. The influence of duration of analyzed vibration on the parameters values is also examined in this work.

Statistical characterization of the random errors in the radioimmunoassay dose--response variable.

1976 ◽  
Vol 22 (3) ◽  
pp. 350-358 ◽  
Author(s):  
D Rodbard ◽  
R H Lenox ◽  
H L Wray ◽  
D Ramseth
Keyword(s):  
Least Squares ◽  
Dose Response ◽  
Curve Fitting ◽  
Response Curve ◽  
Dose Response Curve ◽  
Random Errors ◽  
Statistical Characterization ◽  
Response Variable ◽  
Least Squares Curve Fitting ◽  
The Relationship

Abstract We have developed practical methods for evaluating the magnitude of the random errors in radioimmunoassay dose--response variables, and the relationship between this error and position on the dose--response curve. This is important: to obtain appropriate weights for each point on the dose--response curve when utilizing least-squares curve-fitting methods; to evaluate whether the standards and the unknowns are subject to error of the same magnitude; for quality-control purposes; and to study the sources of errors in radioimmunoassay. Both standards and unknowns in radioimmunoassays for cAMP and cGMP were analyzed in triplicate. The same mean (Y), sample standard deviation, sy, and variance (2-y) of the response variable were calculated for each dose level. The relationship between s 2-y and y was calculated utilizing several models. Results for standards and unknowns from several assays were pooled, and a curve smoothing procedure was used to minimize random sampling errors. This pooling increased the reliability of the analysis, and confirmed the presence of the theoretically predicted nonuniformity of variance. Thus, the calculation of results from these radioimmunoassays should utilize a weighted least-squares curve-fitting program. These analyses have been computerized, and can be used as a "pre-processor" for programs for routine analysis of results of radioimmunoassay.

A Fourier Transform Infrared Study of Mineral Matter in Coal: The Application of a Least Squares Curve-Fitting Program

1981 ◽  
Vol 35 (1) ◽  
pp. 102-106 ◽  
Author(s):  
Paul C. Painter ◽  
Susan M. Rimmer ◽  
Randy W. Snyder ◽  
Alan Davis
Keyword(s):  
Fourier Transform ◽  
Least Squares ◽  
Curve Fitting ◽  
Fourier Transform Infrared ◽  
Mineral Matter ◽  
Infrared Study ◽  
Least Squares Curve Fitting ◽  
Mineral Mixtures ◽  
Good Agreement

The application of Fourier transform infrared spectroscopy to the quantitative determination of mineral matter in coal is discussed. The use of a least squares curve-fitting program allows a choice between standards to be made. The results of an analysis of mineral mixtures and a coal low temperature ash are presented. The results are in good agreement with known concentrations and those obtained by other methods of analysis.

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