Least squares, method alternatives

Terrado M, Barcelo D, Tauler R (2009) Quality assessment of the multivariate curve resolution alternating least squares method for the investigation of environmental pollution patterns in surface water. Environ Sci Technol 43 5321-5326... 

The constrained least-square method is developed in Section 5.3 and a numerical example treated in detail. Efficient specific algorithms taking errors into account have been developed by Provost and Allegre (1979). Literature abounds in alternative methods. Wright and Doherty (1970) use linear programming methods that are fast and offer an easy implementation of linear constraints but the structure of the data is not easily perceived and error assessment inefficiently handled. Principal component analysis (Section 4.4) is more efficient when the end-members are unknown. 

To solve this problem, some assumptions should be made on the relationship between the error of the regression line and the concentration. As a rule, one assumes that the error of the regression line is proportional to the concentration. The variance function Var(X) is obtained by plotting the standard error vs. the concentration. The function is consequently estimated with the least-squares method Var(X) = Sl = (c -T d cone)2. An alternative approach is described in the ISO 11483-2 standard, which uses an iterative procedure to estimate the variance function [18]. 

Other methods may be more appropriate for equations with particular mathematical characteristics or when more accurate, robust, stable and efficient solutions are required. The alternative spectral methods can be classified as sub-groups of the general approximation technique for solving differential equations named the method of weighted residuals (MWR) [51]. The relevant spectral methods are called the collocation Galerkin, Tan- and Least squares methods. These methods can also be applied to subdomains. The subdomain... 

Molecular structures determined directly from the observed groimd state moments of inertia, /, are called tq structures. They depend strongly on the specific set of isotopomer data used because the isotopic dqiendence of the vibration-rotation interaction terms is different from the dependence of the moments of inertia. This quickly leads to contradictions in the results from different isotopic species. Alternately, data from many isotopic molecules can be averaged by the least-squares method. The results still depend somewhat on the isotopic data set and they have low relative precision typically in the range of 1%. 

In general, there is an art and a science to molecular mechanics parameterization. On one extreme, least-squares methods can be used to optimize the parameters to best fit the available data set, and reviews on this topic are avail-able. Alternatively, parameters can be determined on a trial-and-error basis. The situation in either case is far from straightforward because the data usually available come from a variety of sources, are measured by different kinds of experiments in different units, and have relative importances that need subjective assessment. Therefore, straightforward applications of least-squares methods are not expected to give optimum results. 

Takane, Y. Young, F. W. de Leeuw, J. Nonmetric individual differences multidimensional scaling An alternating least squares method with optimum scaling features. Psychometrika, 1977, 42, 6-67. 

Independently of the approach used to get a constitutive law, parameters of the latter must be identified in laboratory. To simphfy the identification process, creep tests are preferred (Fafard et al., 2001 Picard et al., 2008) over any other tests to identify the constitutive laws parameters discussed in this work. In such case where the number of parameters to identify is small, methods such as Newton-Raphson algorithm could be used to minimize the error of an objective function through use of the least squares method. If the number of parameters is large, an alternative method like genetic algorithm may be more suitable. 

The coefficients of equations of the type (0.7) or (0.8) in previously cited works were determined by the least-squares method in the initial basis of expansion To satisfy the critical condition, the method of Lagrange s undetermined multipliers was used. In Ref. [3.1], and p r were fixed and was determined. An alternative calculation algorithm is also possible fix and and determine the critical density Ihe point at which critical conditions are... 

Goicoechea HC, Olivieri AC, Tauler R. Application of the correlation constrained multivariate curve resolution alternating least-squares method for analyte quantitation in the presence of unexpected interferences using first-order instrumental data. Analyst 2010 135 636-42. 

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

A straight line may be fitted to the (X, Y) or (X, Y) pairs of data when plotted on log-log graph paper from which the slope N and the intercept log K with X = 1 may be read. Alternatively, the method of least squares may be used to estimate the values of K and N, giving the best fit to the available data. 

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