Editing Nonlinear regression (section)

==Regression statistics==
The assumption underlying this procedure is that the model can be approximated by a linear function, namely a first-order [[Taylor series]]:

<math display="block"> f(x_i,\boldsymbol\beta) \approx f(x_i,0) + \sum_j J_{ij} \beta_j </math>

where <math>J_{ij} = \frac{\partial f(x_i,\boldsymbol\beta)}{\partial \beta_j}</math> are Jacobian matrix elements. It follows from this that the least squares estimators are given by

<math display="block">\hat{\boldsymbol{\beta}} \approx \mathbf { (J^TJ)^{-1}J^Ty},</math>
compare [[generalized least squares]] with covariance matrix proportional to the unit matrix. The nonlinear regression statistics are computed and used as in linear regression statistics, but using '''J''' in place of '''X''' in the formulas. 

When the function <math>f(x_i,\boldsymbol\beta)</math> itself is not known analytically, but needs to be [[Linear regression|linearly approximated]] from <math>n+1</math>, or more, known values (where <math>n</math> is the number of estimators), the best estimator is obtained directly from the [[Linear Template Fit]] as <ref>{{cite journal | title=The Linear Template Fit | last=Britzger | first=Daniel | journal=Eur. Phys. J. C | volume=82 | year=2022 | issue=8 |pages=731 | doi=10.1140/epjc/s10052-022-10581-w | arxiv=2112.01548| bibcode=2022EPJC...82..731B }}</ref><math display="block"> \hat{\boldsymbol\beta} = ((\mathbf{Y\tilde{M}})^\mathsf{T} \boldsymbol\Omega^{-1} \mathbf{Y\tilde{M}})^{-1}(\mathbf{Y\tilde{M}})^\mathsf{T}\boldsymbol\Omega^{-1}(\mathbf{d}-\mathbf{Y\bar{m})}</math> (see also [[Linear_least_squares#Alternative_formulations|linear least squares]]). 

The linear approximation introduces [[bias (statistics)|bias]] into the statistics. Therefore, more caution than usual is required in interpreting statistics derived from a nonlinear model.