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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).
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 ...
In this example we try to fit the function = + using the Levenberg–Marquardt algorithm implemented in GNU Octave as the leasqr function. The graphs show progressively better fitting for the parameters a = 100 {\displaystyle a=100} , b = 102 {\displaystyle b=102} used in the initial curve.
It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models. The total least squares approximation of the data is generically equivalent to the best, in the Frobenius norm, low-rank approximation of the data matrix. [1]
In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares problem where the coefficients are not allowed to become negative.
In statistics, M-estimators are a broad class of extremum estimators for which the objective function is a sample average. [1] Both non-linear least squares and maximum likelihood estimation are special cases of M-estimators.
The function F is some nonlinear function, such as a polynomial. F can be a neural network , a wavelet network , a sigmoid network and so on. To test for non-linearity in a time series, the BDS test (Brock-Dechert-Scheinkman test) developed for econometrics can be used.