Non-linear least squares

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.

Consider a set of ${\displaystyle m}$ data points, ${\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),\dots ,(x_{m},y_{m}),}$ and a curve (model function) ${\displaystyle y=f(x,{\boldsymbol {\beta }}),}$ that in addition to the variable ${\displaystyle x}$ also depends on ${\displaystyle n}$ parameters, ${\displaystyle {\boldsymbol {\beta }}=(\beta _{1},\beta _{2},\dots ,\beta _{n}),}$ with ${\displaystyle m\geq n.}$ It is desired to find the vector ${\displaystyle {\boldsymbol {\beta }}}$ of parameters such that the curve fits best the given data in the least squares sense, that is, the sum of squares

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