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Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations.
In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations (iterations).
Top: Raw data and model. Bottom: Evolution of the normalised sum of the squares of the errors. The Gauss–Newton algorithm is used to solve non-linear least squares problems, which is equivalent to minimizing a sum of squared function values. It is an extension of Newton's method for finding a minimum of a non-linear function.
Scheme of random search using a non-linear regression problem as an example. The goal is to minimize the value of the penalty function. The right bottom shows a few example methods: 1. Non-structured random search, 2. structured random search, 3. Gauss-Newton algorithm, and 4.
Thus, for example, MARS models can incorporate logistic regression to predict probabilities. Non-linear regression is used when the underlying form of the function is known and regression is used only to estimate the parameters of that function. MARS, on the other hand, estimates the functions themselves, albeit with severe constraints on the ...
A tutorial by Ananth Ranganathan; K. Madsen, H. B. Nielsen, O. Tingleff, Methods for Non-Linear Least Squares Problems (nonlinear least-squares tutorial; L-M code: analytic Jacobian secant) T. Strutz: Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond). 2nd edition, Springer Vieweg, 2016, ISBN 978-3-658 ...
Powell's dog leg method, also called Powell's hybrid method, is an iterative optimisation algorithm for the solution of non-linear least squares problems, introduced in 1970 by Michael J. D. Powell. [1] Similarly to the Levenberg–Marquardt algorithm, it combines the Gauss–Newton algorithm with gradient descent, but it uses an explicit trust ...
The newer nonlinear modelling approaches include non-parametric methods, such as feedforward neural networks, kernel regression, multivariate splines, etc., which do not require a priori knowledge of the nonlinearities in the relations. Thus the nonlinear modelling can utilize production data or experimental results while taking into account ...