Preface to First Edition - lib

184 SMOOTHERS AND GENERALISED ADDITIVE MODELStions. Instead of (10.1) the criterion used to determine g isn∑∫(y i − g(x i )) 2 + λ g ′′ (x) 2 dx (10.2)i=1where g ′′ (x) represents the second derivation of g(x) with respect to x. Althoughwritten formally this criterion looks a little formidable, it is reallynothing more than an effort to govern the trade-off between the goodnessof-fitof the data (as measured by ∑ (y i − g(x i )) 2 ) and the ‘wiggliness’ ordeparture of linearity of g measured by ∫ g ′′ (x) 2 dx; for a linear function, thispart of (10.2) would be zero. The parameter λ governs the smoothness of g,with larger values resulting in a smoother curve.The cubic spline which minimises (10.2) is a series of cubic polynomialsjoined at the unique observed values of the explanatory variables, x i , (formore details, see Keele, 2008).The ‘effective number of parameters’ (analogous to the number of parametersin a parametric fit) or degrees of freedom of a cubic spline smoother isgenerally used to specify its smoothness rather than λ directly. A numericalsearch is then used to determine the value of λ corresponding to the requireddegrees of freedom. Roughly, the complexity of a cubic spline is about the sameas a polynomial of degree one less than the degrees of freedom (see Keele, 2008,for details). But the cubic spline smoother ‘spreads out’ its parameters in amore even way and hence is much more flexible than is polynomial regression.The spline smoother does have a number of technical advantages over thelowess smoother such as providing the best mean square error and avoidingoverfitting that can cause smoothers to display unimportant variation betweenx and y that is of no real interest. But in practise the lowess smoother andthe cubic spline smoother will give very similar results on many examples.10.2.2 Generalised Additive ModelsThe scatterplot smoothers described above are the basis of a more general,semi-parametric approach to modelling situations where there is more than asingle explanatory variable, such as the air pollution data in Table 10.2 andthe kyphosis data in Table 10.3. These models are usually called generalisedadditive models (GAMs) and allow the investigator to model the relationshipbetween the response variable and some of the explanatory variables using thenon-parametric lowess or cubic splines smoothers, with this relationship forother explanatory variables being estimated in the usual parametric fashion.So returning for a moment to the multiple linear regression model described inChapter 6 in which there is a dependent variable, y, and a set of explanatoryvariables, x 1 , ...,x q , and the model assumed isy = β 0 +q∑β j x j + ε.j=1© 2010 by Taylor and Francis Group, LLC