this function can be viewed as second-order approximation to general smooth prediction function ( )see also exercise polynomial regression models are also called response surface models the generalization of the above logistic prediction to rd is ( ( - )- response surface model ( this function will make its appearance in section and later on in and the first strategy for performing regression with nonlinear prediction functions is to extend the feature space to obtain simpler (ideally linearprediction function in the extended feature space we already saw an application of this strategy in example for the polynomial regression modelwhere the original feature was extended to the feature vector [ uu - ]yielding linear prediction function in similar waythe right-hand side of the polynomial prediction function in ( can be viewed as linear function of the extended feature vector ph( [ ]the function ph is called feature map the second strategy is to transform the response variable and possibly also the explanatory variable such that the transformed variables ye are related in simpler (ideally linearway for examplefor the exponential prediction function -bx we have ln ln bxwhich is linear relation between ln and [ ]example (chlorinetable lists the free chlorine concentration (in mg per literin swimming poolrecorded every hours for days simple chemistry-based model for the chlorine concentration as function of time is - where is the initial concentration and is the reaction rate table chlorine concentration (in mg/las function of time (hourshours concentration hours concentration the exponential relationship -bt suggests that log transformation of will result in linear relationship between ln and the feature vector [ ]thusif for some given data ( )(tn yn )we plot ( ln )(tn ln yn )these points should approximately lie on straight lineand hence the simple linear regression model applies the left panel of figure illustrates that the transformed data indeed lie approximately on straight line the estimated regression line is also drawn here the intercept and slope are - and - here the original (non-transformeddata is shown in the right panel of figure along with the fitted curve -bt where exp( and - feature map