Nonlinear regression: Difference between revisions

Revision as of 22:30, 27 March 2024

Revision as of 02:15, 28 March 2024

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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 | eprintarxiv=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.

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| year = 1987

| doi = 10.1096/fasebj.1.5.3315805

| doi-access = free

| url = https://doi.org/10.1096/fasebj.1.5.3315805

| pmid = 3315805

}}</ref>

, but weights may be recomputed on each iteration, in an iteratively weighted least squares algorithm.

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==Further reading==

*{{cite book |firstfirst1=R. M. |lastlast1=Bethea |first2=B. S. |last2=Duran |first3=T. L. |last3=Boullion |title=Statistical Methods for Engineers and Scientists |___location=New York |publisher=Marcel Dekker |year=1985 |isbn=0-8247-7227-X }}

*{{cite journal |lastlast1=Meade |firstfirst1=N. |first2=T. |last2=Islam |year=1995 |title=Prediction Intervals for Growth Curve Forecasts |journal=Journal of Forecasting |volume=14 |issue=5 |pages=413–430 |doi=10.1002/for.3980140502 }}

*{{cite book |first=K. |last=Schittkowski |title=Data Fitting in Dynamical Systems |publisher=Kluwer |___location=Boston |year=2002 |isbn=1402010796 }}

*{{cite book |firstfirst1=G. A. F. |lastlast1=Seber |first2=C. J. |last2=Wild |title=Nonlinear Regression |___location=New York |publisher=John Wiley and Sons |year=1989 |isbn=0471617601 }}

{{Statistics}}