Remark AS R75: Some Remarks on Algorithm AS 164: Least Squares Subject to Linear Constraints

The inverse gravity problem is posed as a linear least‐squares problem with the variables being densities of two‐dimensional prisms. Upper and lower bounds on the densities are prescribed so that the problem becomes a linearly constrained least‐squares problem, which is solved using a quadratic programming algorithm designed for upper and lower bound‐type constraints. The solution to any problem is smoothed by damping, using the singular value decomposition of the matrix of gravitational attractions. If the solution is required to be monotonically increasing with depth, then this feature can be incorporated. The method is applied to both field and theoretical data. The results are plotted for (1) undamped, nonmonotonic, (2) damped, nonmonotonic, and (3) damped, monotonic solutions; these conditions illustrate the composite approach of interpretation where both damping techniques and linear constraints are used in refining a solution which at first is unacceptable on geologic grounds while fitting the observed data well.

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Research of New Criteria for Detecting First-order Residuals Autocorrelation in Regression Models

Mathematics and Mathematical Modeling ◽

10.24108/mathm.0318.0000102 ◽

2018 ◽

pp. 13-25

Author(s):

M. P. Bazilevsky

Keyword(s):

Mathematical Programming ◽

Regression Model ◽

Least Squares ◽

Regression Models ◽

Linear Constraints ◽

Mathematical Programming Problem ◽

First Order ◽

Watson Statistic ◽

Auto Regression ◽

New Criteria

When estimating regression models using the least squares method, one of its prerequisites is the lack of autocorrelation in the regression residuals. The presence of autocorrelation in the residuals makes the least-squares regression estimates to be ineffective, and the standard errors of these estimates to be untenable. Quantitatively, autocorrelation in the residuals of the regression model has traditionally been estimated using the Durbin-Watson statistic, which is the ratio of the sum of the squares of differences of consecutive residual values to the sum of squares of the residuals. Unfortunately, such an analytical form of the Durbin-Watson statistic does not allow it to be integrated, as linear constraints, into the problem of selecting informative regressors, which is, in fact, a mathematical programming problem in the regression model. The task of selecting informative regressors is to extract from the given number of possible regressors a given number of variables based on a certain quality criterion.The aim of the paper is to develop and study new criteria for detecting first-order autocorrelation in the residuals in regression models that can later be integrated into the problem of selecting informative regressors in the form of linear constraints. To do this, the paper proposes modular autocorrelation statistic for which, using the Gretl package, the ranges of their possible values and limit values were first determined experimentally, depending on the value of the selective coefficient of auto-regression. Then the results obtained were proved by model experiments using the Monte Carlo method. The disadvantage of the proposed modular statistic of adequacy is that their dependencies on the selective coefficient of auto-regression are not even functions. For this, double modular autocorrelation criteria are proposed, which, using special methods, can be used as linear constraints in mathematical programming problems to select informative regressors in regression models.

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Clarification of discussion on constrained refinement

Acta Crystallographica Section A ◽

10.1107/s0567739478002077 ◽

1978 ◽

Vol 34 (6) ◽

pp. 1020-1021

Author(s):

P. F. Price

Keyword(s):

Least Squares ◽

Linear Constraints ◽

Linear Case ◽

Linear Least Squares ◽

Acta Cryst ◽

Non Linear

Pawley's [Acta Cryst. (1976). A32, 921-922] disproof of the C & D method of applying constraints in crystallographic refinements [Chesick & Davidon (1975). Acta Cryst. A31, 586-591] is incorrect. Nevertheless, the C & D method is incorrect. Both the method and its disproof clearly fail for the particular case of linear least-squares with linear constraints. The method can be corrected in this case but not for the non-linear case.

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A Fuzzy Linear Regression Analysis for Fuzzy Input-Output Data Using the Least Squares Method under Linear Constraints and Its Application to Fuzzy Rating Data

Journal of Advanced Computational Intelligence and Intelligent Informatics ◽

10.20965/jaciii.1999.p0036 ◽

1999 ◽

Vol 3 (1) ◽

pp. 36-41 ◽

Cited By ~ 5

Author(s):

Kazuhisa Takemura ◽

Keyword(s):

Regression Analysis ◽

Linear Regression ◽

Least Squares ◽

Least Squares Method ◽

Linear Regression Analysis ◽

Linear Constraints ◽

Fuzzy Linear Regression ◽

Output Data ◽

Rating Data ◽

Fuzzy Input

Fuzzy linear regression analysis using the least squares method under linear constraint, where input data, output data, and coefficients are represented by triangular fuzzy numbers, was proposed and compared to possibilistic linear regression analysis proposed by Sakawa and Yano (1992) using fuzzy rating data in a psychological study. Major findings of the comparison were as follows: (1) Under the proposed analysis, the width between the maximum and minimum of the predicted model was nearer to the width of the dependent variable than that of possibilistic linear regression analysis, (2) the representative prediction by the proposed analysis was also nearer to that of the dependent variable, compared to that of possibilistic linear regression analysis.

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