Accession Number : ADA130502

Title :   Constrained Nonlinear Least Squares.

Descriptive Note : Technical summary rept.,


Personal Author(s) : Box,George E P ; Kanemasu,Hiromitsu

PDF Url : ADA130502

Report Date : Jun 1983

Pagination or Media Count : 27

Abstract : The method of least squares is widely used for the fitting of data of functions containing unknown parameters. When these functions are non-linear in the parameters an iterative approach using successive local linearizations was suggested by Gauss. To encourage convergence various modifications and alternatives have been suggested. It has been proposed that, at the early stages, use of the method of steepest descent might speed convergence. A method might be used in which changes were made in the direction of, but not of the magnitude of, the Gauss adjustment. It is convenient to call this the Gauss direction method. Levenberg and later Marquardt proposed a spherically constrained procedure. The procedure could be made scale-invariant to render its behavior independent of the (arbitrary) units in which the parameters were measured. It was argued the resulting method offered an appropriate compromise between steepest descent and Gauss's method. In this report some of the geometry of these methods is discussed. As a consequence it is argued that the procedure should also be made invariant under linear transformation. With this modification the scale-invariant Levenberg-Marquardt method becomes inappropriate and the steepest descent direction and the Gauss direction method identical. It remains, therefore, to determine how far along the Gauss direction one should proceed. Two alternative methods for deciding this are suggested. Both methods use quantities already available from previous calculations.

Descriptors :   *Statistical decision theory, *Least squares method, *Nonlinear differential equations, *Convergence, Statistical analysis, Mathematical prediction, Forecasting, Parameters, Gaussian quadrature, Algorithms, Vector analysis, Computations, Management information systems, Resource management

Subject Categories : Administration and Management
      Statistics and Probability

Distribution Statement : APPROVED FOR PUBLIC RELEASE