On Least Squares Fitting Nonlinear Submodels.

Three simplifying conditions are given for obtaining least squares (LS) estimates for a nonlinear submodel of a linear model. If these are satisfied, and if the subset of nonlinear parameters may be LS fit to the corresponding LS estimates of the linear model, then one attains the desired LS estimates for the entire submodel. Two illustrative analyses employing this method are given, each involving an Eckart-Young (LS) decomposition of a matrix of linear LS estimates. In each case the factors provide an LS fit of the nonlinear submodel to the original data. The minimum error sum of squares for this fit is the error sum of squares for the corresponding linear model plus a function of the eigenvalues involved in the factorization. An Eckart-Young factorization, however, is only a special case of an LS decomposition of LS estimates. The present method is more generally applicable (under the three simplifying conditions) whenever any LS procedure may be found for fitting certain parameters of a nonlinear submodel to the corresponding LS estimates of a linear model. (Author)