Editing Nonlinear regression (section)

==Linearization==
===Transformation===
{{further|Data transformation (statistics)}}

Some nonlinear regression problems can be moved to a linear domain by a suitable transformation of the model formulation.

For example, consider the nonlinear regression problem

<math display="block"> y = a e^{b x}U </math>

with parameters ''a'' and ''b'' and with multiplicative error term ''U''. If we take the logarithm of both sides, this becomes

<math display="block"> \ln{(y)} = \ln{(a)} + b x + u, </math>

where ''u'' = ln(''U''), suggesting estimation of the unknown parameters by a linear regression of ln(''y'') on ''x'', a computation that does not require iterative optimization. However, use of a nonlinear transformation requires caution.  The influences of the data values will change, as will the error structure of the model and the interpretation of any inferential results.  These may not be desired effects.  On the other hand, depending on what the largest source of error is, a nonlinear transformation may distribute the errors in a Gaussian fashion, so the choice to perform a nonlinear transformation must be informed by modeling considerations.

For [[Michaelis–Menten kinetics]], the linear [[Lineweaver–Burk plot]]

<math display="block"> \frac{1}{v} = \frac{1}{V_\max} + \frac{K_m}{V_{\max}[S]}</math>

of 1/''v'' against 1/[''S''] has been much used. However, since it is very sensitive to data error and is strongly biased toward fitting the data in a particular range of the independent variable, [''S''], its use is strongly discouraged.

For error distributions that belong to the [[exponential family]], a link function may be used to transform the parameters under the [[Generalized linear model]] framework.

===Segmentation===
[[Image:MUSTARD.JPG|thumb|175 px|right|Yield of mustard and soil salinity]]
{{main|Segmented regression}}

The [[Independent variable|''independent'' or ''explanatory variable'']] (say X) can be split up into classes or segments and [[linear regression]] can be performed per segment. Segmented regression with [[Confidence interval|confidence analysis]] may yield the result that the [[Dependent variable|''dependent'' or ''response'' variable]] (say Y) behaves differently in the various segments.<ref>R.J.Oosterbaan, 1994, Frequency and Regression Analysis. In: H.P.Ritzema (ed.), Drainage Principles and Applications, Publ. 16, pp. 175-224, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. {{ISBN|90-70754-33-9}} . Download as PDF : [http://www.waterlog.info/pdf/regtxt.pdf]</ref>

The figure shows that the [[soil salinity]] (X) initially exerts no influence on the [[crop yield]] (Y) of mustard, until a ''critical'' or ''threshold'' value (''breakpoint''), after which the yield is affected negatively.<ref>R.J.Oosterbaan, 2002. Drainage research in farmers' fields: analysis of data. Part of project “Liquid Gold” of the
International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. Download as PDF : [http://www.waterlog.info/pdf/analysis.pdf]. The figure was made with the [[SegReg]] program, which can be downloaded freely from [http://www.waterlog.info/segreg.htm]</ref>