Nonlinear regression: Difference between revisions

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

:<math> f(x_i,\boldsymbol\beta) \approx f(0; \boldsymbol\beta) + \sum_j J_{ij} \beta_j </math>

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

:<math>\hat{\boldsymbol{\beta}} \approx \mathbf { (J^TJ)^{-1}J^Ty}.</math>

The nonlinear regression statistics are computed and used as in linear regression statistics, but using '''J''' in place of '''X''' in the formulas. 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.

==Ordinary and weighted least squares==