The least-squares triangle: a new method for evaluation of solid phase compositions from solubility measurements

1981 ◽  
Vol 59 (21) ◽  
pp. 3076-3083 ◽  
Author(s):  
John W. Lorimer
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Solid Phase ◽  
Extrapolation Method ◽  
New Method ◽  
Initial Composition ◽  
Hypothesis Tests ◽  
Error Analyses ◽  
Solid Phases

A least-squares method is described for calculating compositions of equilibrium solid phases from data on solubilities and either wet residues or initial compositions for systems of three or more thermodynamic components. The method minimizes the squares of areas of triangles formed by the solubility, wet residue (or initial composition), and solid composition points. Full descriptions of error analyses and hypothesis tests are given, along with an illustrative example and detailed comparisons with the traditional extrapolation method.

Accurate power series for eigenvalues of spheroidal angle functions and their convergence radii

2011 ◽  
Vol 89 (11) ◽  
pp. 1083-1099
Author(s):  
Tam Do-Nhat
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Complex Plane ◽  
Branch Point ◽  
Upper Bound ◽  
Least Squares Method ◽  
Radius Of Convergence ◽  
New Method ◽  
Recursion Relations

In this paper, the radius of convergence of the spheroidal power series associated with the eigenvalue is calculated without using the branch point approach. Studying the properties of the power series using the recursion relations among its coefficients in the new method offers some insights into the spheroidal power series and its associated eigenfunction. This study also used the least squares method to accurately compute the convergence radii to five or six significant digits. Within the circle of convergence in the complex plane of the parameter c = kF, where k is the wavenumber and F is the semifocal length of the spheroidal system, the extremely fast convergent spheroidal power series are computed with full precision. In addition, a formula for the magnitude of the upper bound of the error is obtained.

The least-squares intersection of a family of lines and its application to phase equilibria

1982 ◽  
Vol 60 (15) ◽  
pp. 1978-1981 ◽  
Author(s):  
John W. Lorimer
Keyword(s):  
Phase Equilibria ◽  
Least Squares ◽  
Least Squares Method ◽  
Solid Phase ◽  
Generalized Least Squares ◽  
First Approximation ◽  
Generalized Least Squares Method ◽  
Point Of Intersection ◽  
Straight Lines ◽  
General Method

A generalized least-squares method is described for finding the point of intersection of a family of straight lines, each of which is defined by two experimental points. It is shown that the method of the least-squares triangle (Can. J. Chem. 59, 3076 (1981)) is a good first approximation to the general method. An example demonstrates the method of iteration of both parameters and observations for a problem involving evaluation of solid phase compositions from solubility measurements.

scholarly journals A Least Squares Estimation Method for the Linear Learning Model

1978 ◽  
Vol 15 (1) ◽  
pp. 145-153
Author(s):  
Berend Wierenga
Keyword(s):  
Least Squares ◽  
Simulation Study ◽  
Goodness Of Fit ◽  
Least Squares Method ◽  
Estimation Method ◽  
Learning Model ◽  
New Method ◽  
Data Sets ◽  
Goodness Of Fit Test ◽  
Chi Square

The author presents a new method for estimating the parameters of the linear learning model. The procedure, essentially a least squares method, is easy to carry out and avoids certain difficulties of earlier estimation procedures. Applications to three different data sets are reported, as well as results from a goodness-of-fit test. A simulation study was carried out to validate the method. The outcomes are compared with those obtained from the minimum chi square estimation method. The results of the new method appear to be satisfactory.

A New Method to Identify the Preisach Distribution Function of Hysteresis

2005 ◽  
Vol 475-479 ◽  
pp. 2107-2110 ◽  
Author(s):  
Fan Li ◽  
Jian Qin Mao ◽  
Hai Shan Ding ◽  
Wen Bo Zhang ◽  
Hui Bin Xu ◽  
...  
Keyword(s):  
Distribution Function ◽  
Least Squares ◽  
Fuzzy Inference System ◽  
Fuzzy Inference ◽  
Least Squares Method ◽  
Input Sequence ◽  
Least Square Method ◽  
Least Square ◽  
New Method ◽  
Inference System

In this paper, a new method which combines the least square method with Tree-Structured fuzzy inference system is presented to approximate the Preisach distribution function. Firstly, by devising the input sequence and measure the output, discrete Preisach measure can be identified by the use of the least squares method. Then, the Preisach function can be obtained with Tree-Structured fuzzy inference system without any special smoothing means. So, this new method is not sensitive to noise, and is a universal approximator of the Preisach function. It collect the merit and overcome the deficiency of the existing methods.

A least-squares method for estimating the correlated error of GRACE models

2020 ◽  
Vol 221 (3) ◽  
pp. 1736-1749
Author(s):  
John W Crowley ◽  
Jianliang Huang
Keyword(s):  
Least Squares ◽  
Spherical Harmonic ◽  
Climate Model ◽  
Least Squares Method ◽  
A Priori ◽  
Temporal Trends ◽  
Standard Form ◽  
New Method ◽  
Correlated Errors ◽  
Global Trends

SUMMARY A new least-squares method is developed for estimating and removing the correlated errors (stripes) from the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) mission data. This method is based on a joint parametric model of the correlated errors and temporal trends in the spherical harmonic coefficients of GRACE models. Three sets of simulation data are created from the Global Land Data Assimilation System (GLDAS), the Regional Atmospheric Climate Model 2.3 (RACMO2.3) and GRACE models and used to test it. The results show that the new method improves the decorrelation method by Swenson & Wahr significantly. Its application to the release 5 (RL05) and new release 6 (RL06) spherical harmonic solutions from the Center for Space Research (CSR) at The University of Texas at Austin demonstrates its effectiveness and provides a relative assessment of the two releases. A comparison to the Swenson & Wahr and Kusche et al. methods highlights the deficiencies in past destriping methods and shows how the inclusion and decoupling of temporal trends helps to overcome them. A comparison to the CSR mascon and JPL mascon solutions demonstrates that the new method yields global trends that have greater amplitude than those produced by the CSR RL05 mascon solution and are of comparable quality to the JPL RL06 mascon solution. Furthermore, these results are obtained without the need for a priori information, scale factors or complex regularization methods and the solutions remain in the standard form of spherical harmonics rather than discrete mascons. The latter could introduce additional discretization error when converting to the spherical harmonic model, upon which many post-processing methods and applications are built.

Analysis of Static Frequency Characteristics Based on Curve Fitting

2013 ◽  
Vol 339 ◽  
pp. 602-607
Author(s):  
Chun Li Song ◽  
Di Chen Liu ◽  
Jun Wu ◽  
Fei Fei Dong ◽  
Lian Tu ◽  
...  
Keyword(s):  
Power System ◽  
Least Squares ◽  
Curve Fitting ◽  
Least Squares Method ◽  
Frequency Characteristics ◽  
Frequency Deviation ◽  
New Method ◽  
Calculation Error ◽  
Restriction Point

Identification and calculation of static frequency characteristics is of great significance for power system to maintain its stability. In this paper, coefficient of static frequency characteristics is fitted by the least squares method. Frequency deviation restriction point under different capacitances is forecasted by the fitted trend of coefficient of static frequency characteristics. Moreover, the new method is simulated and its calculation error is also compared.

scholarly journals APPLICATION OF STATISTICAL TECHNIQUES IN THE PRACTICE OF BUILDING A DEFILIENCE SCALE

2021 ◽  
Vol 5 (40) ◽  
pp. 35-44
Author(s):  
Ye. Pavliuk ◽  
O. Pavliuk
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Default Probability ◽  
Extrapolation Method ◽  
Classification Model ◽  
Weibull Function ◽  
Slow Increase ◽  
Moderate Growth ◽  
S Curve ◽  
Number Of Classes

Abstract. The main substantial features of the PD curve (default probability) formed in practical modeling are substantiated in the articles. It is proved that the main characteristics of the PD curve are that it is based on data on the actually restored default rate in each of the risk classes over a period of time and has a shape that approximate for coincides with the exposure function. It is shown that the best aspect that affects the calibration is the number of rating classes and ways to build them. It is determined that the slope of the curve demonstrates the classification model of efficiency. It is determined that the slope of the curve demonstrates the classification efficiency of the model. Models with high discriminant properties are characterized by a curve shape that has a slow increase in the rating classes of the upper part of the scale and a significant acceleration of growth in the last risk classes. Two main approaches to determining the number of risk classes are analyzed: the percentile-based approach and the equal score range approach. It is shown that when forming classes, it is necessary to take into account the total amount of sample observations, the proportion of «good» and «bad», and choose the number of classes so that it is not too large and not too small. Calibration practice shave been shown to be influenced by data, purpose, and study limitations. The application of the least squares method and the extrapolation method is considered on practical examples. The least squares method and in particular the derived extrapolation method allow to build a calibration curve on the basis of data on the relative frequency of defaults. It is determined that the mathematical apparatus of the family of nonlinear curves allows to model the process of exponential growth with different levels of intensity. The exponential curve and related functions may be useful in modeling more conservative PD estimates or for models with highly discriminatory properties, while the Weibull function, S-curve, and power function may be better adapted to moderate growth processes. The application of practical methods of constructing the PD scale is important for many domestic banking professionals who deal with internal models of credit risk. Keywords: Calibration, Default, Probability, Curves, Probability of default curve calibration, Least squares method, Extrapolation method. JEL Classіfіcatіon С44 Formulas: 21; fig.: 1; tabl.: 7; bibl.: 10.

Global least squares method based on tensor form to solve linear systems in Kronecker format

2017 ◽  
Vol 40 (7) ◽  
pp. 2378-2386 ◽  
Author(s):  
Saeed Karimi ◽  
Maryam Dehghan
Keyword(s):  
Approximate Solution ◽  
Least Squares ◽  
Convergence Analysis ◽  
Linear Systems ◽  
Iterative Methods ◽  
Least Squares Method ◽  
New Method ◽  
Tensor Form ◽  
High Dimensions ◽  
Special Case

In this paper, we propose a new algorithm based on the global least squares method for solving linear systems in Kronecker format. Because of the inefficiency of iterative methods for solving linear systems in Kronecker format in high dimensions, we consider the tensor form of these systems and apply the global least squares method based on the tensor form to obtain an approximate solution. We use the new method to solve the Sylvester tensor equations in Kronecker format, as a special case of these systems. The convergence analysis of the new method is also investigated. Numerical results demonstrate the efficiency of the new method in comparison with some existing methods.

scholarly journals Approximate Analytical Solutions of the Fractional-Order Brusselator System Using the Polynomial Least Squares Method

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Least Squares Method ◽  
Nonlinear Differential Equations ◽  
Fractional Order System ◽  
New Method ◽  
Order System ◽  
Approximate Analytical Solutions

The paper presents a new method, called the Polynomial Least Squares Method (PLSM). PLSM allows us to compute approximate analytical solutions for the Brusselator system, which is a fractional-order system of nonlinear differential equations.

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