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This section introduces the general class of additive models and motivates the rep- resentation of its smooth terms through spline functions. The additive model is a generalization of the usual linear regression model. Hence, we start this section with a review of the linear model and its limitations.

Consider the standard multiple linear regression problem. Given a response variable Y and pselected explanatory variables (or covariates) X1, . . . , Xp, our goal is to model the dependency of Y on X1, . . . , Xp. The standard tool is the multiple

linear regression model:

Y =β0+β1X1+· · ·+βpXp +ǫ , (2.6) where ǫ _∼ N(0, σ2_{), independently of the Xj’s. This model makes the strong as-}

sumption that the dependency of E[Y] on X1, . . . , Xp is linear in each of the pre-

dictors. If this assumption holds, even approximately, then the linear regression model in (2.6) is an extremely useful tool because not only does it provide a simple description of the data, it also summarizes the contribution of each predictor with a single coefficient, the βj’s.

Linear models such as that in (2.6) are often too simple for most practical appli- cations. There are many ways in which the linear model (2.6) can be generalized. A straightforward extension consists in adding polynomial terms like X2

j, Xj3, etc., to the model in (2.6). However, it is usually difficult to determine which such terms should be added in order to obtain a good fit to the data. Furthermore, polynomial models suffer from a series of drawbacks. As mentioned in Section 2.1, the data fits

are not local, which means that perturbations in the data may affect the polynomial fit in remote regions. Also, the fitting process for polynomials can be numerically ill-conditioned due to the presence of collinearity, a result of the spurious corre- lation among the power transformations of the predictors Xj. Surface smoothers are another possible generalization of the model (2.6). Essentially these model the dependency of Y onX1, . . . , Xp through a p-dimensional surface as follows:

Y =g(X1, . . . , Xp) +ǫ . (2.7)

However, it is not clear how one should define a local fit in p dimensions, though tensor product of splines attempt to deal with such issue (see, e.g., Wahba, 1990; Wood, 2006b). Moreover, the effect of any individual predictor on the response Y is difficult to interpret if the surface smoother in (2.7) is fitted to the data.

The aforementioned interpretation problem highlights an important feature of the linear model that has made it so popular for statistical inference: the linear model is additive in the predictor effects. This additivity property implies that one can examine the predictor effects separately, in the absence of interactions. Additive models (Hastie & Tibshirani, 1986, 1990b) retain this important feature as they are additive in the predictor effects. An additive model is defined by

Y =β0+

p X

j=1

gj(Xj) +ǫ , (2.8)

where as before the error ǫ is independent of the Xj’s withǫ_∼N(0, σ2_{). The com-}

ponents gj(Xj) are arbitrary smooth univariate functions. They define ‘transformed predictors’ gj(Xj) which act additively on the response variable. Thus we can ex-

amine the effect of each predictor in the model separately. For estimation purposes the assumption that E[gj(Xj)] = 0 is required, since otherwise the intercept β0 in

(2.8) is unidentifiable, i.e., any of the fitted gj could be shifted by some arbitrary constant, accompanied by an offsetting shift inβ0, and the resulting set of estimates

would fit the data equally well.

The model in (2.8) assumes that all predictors take values over a continuous domain and estimates all its components in a nonparametric fashion. In some situa- tions, however, one or more predictors may be categorical variables. In this case the additive model in (2.8) can be modified to include both linear and smooth terms. If there exist q such categorical predictors, V1, . . . , Vq say, a semiparametric model is

defined as Y =β0+β1V1+· · ·+βqVq+ p X j=1 gj(Xj) +ǫ . (2.9)

More complex models, involving interaction terms, like g(Xu, Xv) for example, can also be included in the model. However, in this thesis we shall focus on additive models such as those in (2.8) and (2.9).

Having decided on the regression model, one needs to choose how to represent the unknown smooth terms gj. The discussion in Section 2.1 and Section 2.2 regarding the local nature of spline functions presents them as a flexible approximation tool. Additionally, De Boor (1978), and also Green & Silverman (1994), derive important results regarding the numerical optimality of spline interpolants, and in particular of cubic spline interpolants, since these minimize an intuitive measure of roughness based upon the curvature of the function to be estimated. We shall see in the next chapter that cubic spline functions are also optimal in the sense that they are the solution to an optimization problem within the framework of curve fitting.

The discussion above suggests that spline functions ought to be a good choice for representing smooth curves in any statistical framework, and hence are the building block of all the smooths presented in this thesis.