Showing papers on "Non-linear least squares published in 1969"

Extensions and applications of the Householder algorithm for solving linear least squares problems

Abstract: Mathematical and numerical least squares solution of linear equations, using Householder algorithm

The Exact Sampling Distribution of Ordinary Least Squares and Two-Stage Least Squares Estimators

Abstract: This paper presents the exact sampling distributions of the ordinary and the two-stage least squares estimators of a structural parameter in a structural equation with two endogenous variables in a complete system of stochastic equations. The results show that the distributions of the two estimators are essentially similar to each other. It can also be seen that both distributions depend crucially upon the deviation of a regression coefficient of disturbance terms of two endogenous variables from a structural parameter, and that the first estimator possesses moments up to the order N-2, while the second possesses them up to the order K-1, where N is the sample size and K is the number of exogenous variables excluded from the equation to be estimated. The small sample properties of the estimators are investigated by numerical evaluations of the density functions.

Linear least squares and quadratic programming

Abstract: Several algorithms are presented for solving linear least squares problems; the basic tool is orthogonalization techniques A highly accurate algorithm is presented for solving least squares problems with linear inequality constraints A method is also given for finding the least squares solution when there is a quadratic constraint on the solution

Parametric Augmentations and Error Structures under which Certain Simple Least Squares and Analysis of Variance Procedures are Also Best

Abstract: Conditions are presented under which in all linear models with a common specified systematic part every linear simple least squares estimator is also best. The behaviour of mean squares and test ratios in ordinary analysis of variance tables under these and also less stringent conditions is discussed with illustrative examples given.

Restricted Least Squares Regression and Convex Quadratic Programming

Abstract: A parsimonious stepwise procedure for obtaining least squares solutions of multiple regression eqllntions when the regression coefficients are subject to arbitrary but consistent linear restraints is presented. The method is also applicable to the minimization of positive definite quadratic functions. Key to the method is the use of the elements of the appropriate inverse matrix for determining the standardized distance from any unrestricted, or conditionally unrestricted, solution to any boundary or boundary intersection of the permissible region for the regression coefficients. Various other aspects of the problem are discussed.

Computation of zellner-theil's three stage least squares estimates'

Abstract: In their paper on the three stage least squares method of estimation of a simultaneous equation system, Zellner and Theil [5] make the interesting observation that the large sample efficiency of the estimates of the parameters in the group of over-identified equations is unaffected if the three stage method is applied to this subsystem alone, ignoring all the exactly identified equations. It is shown in this paper that the estimates themselvesnot just their large sample efficiency-are unaffected if one follows the above mentioned simplified procedure. This result is established for the general case of a simultaneous equation system subject to linear homogeneous a priori restrictions, whereas many other known properties of the three stage least squares method have been demonstrated only for the special case of "simple restrictions" on the structural coefficients. An error in the expression giving the estimates for the exactly identified equations in Zellner-Theil's paper is also corrected. The results of this paper have obvious significance for the problem of developing an efficient computer program for finding the three stage least squares estimates. ZELLNER AND THEIL [5] have proposed a three stage least squares method of estimation of the parameters of a simultaneous equation system as an alternative to the full information maximum likelihood method. The three stage least squares method has been shown to possess a number of attractive properties. Rothenberg and Leenders [2] have shown that when the variance-covariance matrix Z of the disturbance terms is unrestricted, the estimates have the same asymptotic variancecovariance matrix as the full information maximum likelihood method. Under the same condition of unrestricted Z matrix, Sargan [3] has shown that the difference between the estimates by the three stage least squares method and the estimates by the full information maximum likelihood method is of stochastic order 1/T, where T is the sample size. Though the three stage least squares method is much simpler computationally than the full information maximum likelihood method, it still involves the inversion of a moment matrix of a very large order and the difficulties of computing the inverse matrix accurately, especially when it may be near singular due to the presence of intercorrelation among the explanatory

Closed-Form Inversion of the Gram Matrix Arising in Certain Least-Squares Problems

Abstract: This correspondence describes a method for finding a closed-form expression for the inverse of some Gram matrices that occur in least-square theory. The method is applicable if an analytic expression for an appropriate set of orthonormal functions can be found. In particular, the method provides a new derivation for the inverse of the generalized Hilbert matrix. An explicit formula is also given for a class of polynomials that is orthonormal over any finite interval with respect to the uniform weight function.

On a Restricted Least Squares Estimator

Abstract: The method of Sweep-Out is used to obtain computing formulas for calculating the least-squares estimator and its variance matrix in linear models, not necessarily of full rank, in which certain restrictions may hold on the actual parameters. A test for a specified value of parametric restrictions is also discussed.

More on least squares estimation of the transition matrix in a stationary first-order markov process from sample proportions data

Abstract: Miller suggested ordinary least squares estimation of a constant transition matrix; Madansky proposed a relatively more efficient weighted least squares estimator which corrects for heteroscedasticity. In this paper an efficient generalized least squares estimator is derived which utilizes the entire covariance matrix of the distrubances. This estimator satisfies the condition that each row of the transition matrix must sum to unity. Madansky noted that estimates of the variances could be negative; a method for obtaining consistent non-negative estimates of the variances is suggested in this paper. The technique is applied to the hypothetical sample data used by Miller and Madansky.

On Representing a Square as the Sum of Three Squares

Abstract: (1969). On Representing a Square as the Sum of Three Squares. The American Mathematical Monthly: Vol. 76, No. 8, pp. 922-923.

On Some Representations of Generalized Inverses

Abstract: Unitary matrix representing generalized inverses, proving weak method of steepest descent for least squares solution of equations

Application of a Nonlinear Least-Squares Method to Atmospheric Temperature Sounding

Abstract: An optimization program for least-squares estimation of nonlinear parameters is applied to the problem of atmospheric temperature sounding. From simulated sets of intensity measurements at 12 frequencies in the 15μ CO2 atmospheric band, this program is used to solve for the temperatures at 5 altitudes throughout the troposphere. Both temperature and altitude can be treated as variables in order to obtain the best match to the real profile.

A least squares iterative method for singular equations

Abstract: Least squares iterative method for solving simultaneous linear equations having singular coefficient matrix

Least Squares Made Easy.

Abstract: A very practical method for hand calculation of least-squares-adjusted slopes, acceleration coefficients, etc., is presented and compared with better known but less-satisfactory algorithms. Particular attention is given to easy methods for estimating the precision of the adjusted constants. The method, based on weighted successive differences, applies to the special case of equal increments in the independent variable.

Two numerical methods that converge to the method of least squares

Abstract: For approximation problems involving residuals that are linear functions of the parameters, it is shown that the collocation approximation approaches the least-squares approximation if the collocation equations are premultiplied by the transposed matrix and the number of collocation points becomes infinite. A similar conclusion is derived for the partition method. An example suggests that these principles furnish practical alternatives to the method of least squares in some cases.

Linear multidimensional scaling of choice1

Abstract: A multidimensional scaling analysis is presented for replicated layouts of pairwise choice responses. In most applications the replicates will represent individuals who respond to all pairs in some set of objects. The replicates and the objects are scaled in a joint space by means of a linear model which assigns weights to each of the dimensions of the space. Least squares estimates of the replicates' and objects' coordinates, and of unscalability parameters, are obtained through a manipulation of the error sum of squares for fitting the linear model. The solution involves the reduction of a three-way least squares problem to two subproblems, one trivial and the other solvable by classical least squares matrix factorization. The analytic technique is illustrated with political preference data and is contrasted with multidimensional unfolding in the domain of preferential choice.

Stationary and nonstationary characteristics of gyro drift rate

Abstract: The full text of the paper describes in detail: 1) the identification of the time series model from autocorrelation and partial correlation of the data; 2) the estimation of the , 6, ramp and random walk parameters using maximum likelihood and nonlinear least squares; and 3) the means by which one would deduce model adequacy through autocorrelation of the white noise residuals and confidence limit theory. As an example of the theory, consider Fig. 1, which shows a sample of normalized long term gyro drift rate. Since the process is nonstationary, the data is differenced as is shown in Fig. 2. The analysis indicated that the math model for this gyro drift rate sample is

An experimental comparison of several approaches to the linear least squares problem

Abstract: Several computational algarithms for obtaining least square solutions to a system of equations are made. The a l gorithms include both conventional and pseudo inversion methods. Comparisons of running time, computer storage and accuracy of recovery are made for each algorithm in both single and doub16 precision.