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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).
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 mathematics and computing, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. These minimization problems arise especially in least squares curve fitting .
Many improved algorithms have been suggested since 1974. [1] Fast NNLS (FNNLS) is an optimized version of the Lawson–Hanson algorithm. [ 2 ] Other algorithms include variants of Landweber 's gradient descent method [ 10 ] and coordinate-wise optimization based on the quadratic programming problem above.
Non-linear least squares problems arise, for instance, in non-linear regression, where parameters in a model are sought such that the model is in good agreement with available observations. The method is named after the mathematicians Carl Friedrich Gauss and Isaac Newton , and first appeared in Gauss's 1809 work Theoria motus corporum ...
The earliest regression form was seen in Isaac Newton's work in 1700 while studying equinoxes, being credited with introducing "an embryonic linear aggression analysis" as "Not only did he perform the averaging of a set of data, 50 years before Tobias Mayer, but summing the residuals to zero he forced the regression line to pass through the ...
IRLS can be used for ℓ 1 minimization and smoothed ℓ p minimization, p < 1, in compressed sensing problems. It has been proved that the algorithm has a linear rate of convergence for ℓ 1 norm and superlinear for ℓ t with t < 1, under the restricted isometry property, which is generally a sufficient condition for sparse solutions.
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These methods involve using linear classifiers to solve nonlinear problems. [1]
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