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Concept
Non-linear least squares
Summary
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. There are many similarities to linear least squares, but also some significant differences. In economic theory, the non-linear least squares method is applied in (i) the probit regression, (ii) threshold regression, (iii) smooth regression, (iv) logistic link regression, (v) Box–Cox transformed regressors ().
Consider a set of data points, and a curve (model function) that in addition to the variable also depends on parameters, with It is desired to find the vector of parameters such that the curve fits best the given data in the least squares sense, that is, the sum of squares
is minimized, where the residuals (in-sample prediction errors) ri are given by
for
The minimum value of S occurs when the gradient is zero. Since the model contains n parameters there are n gradient equations:
In a nonlinear system, the derivatives are functions of both the independent variable and the parameters, so in general these gradient equations do not have a closed solution. Instead, initial values must be chosen for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation,
Here, k is an iteration number and the vector of increments, is known as the shift vector. At each iteration the model is linearized by approximation to a first-order Taylor polynomial expansion about
The Jacobian matrix, J, is a function of constants, the independent variable and the parameters, so it changes from one iteration to the next. Thus, in terms of the linearized model,
and the residuals are given by
Substituting these expressions into the gradient equations, they become
which, on rearrangement, become n simultaneous linear equations, the normal equations
The normal equations are written in matrix notation as
These equations form the basis for the Gauss–Newton algorithm for a non-linear least squares problem.
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