Generalized additive model

A generalized additive model has linear predictor $g({\mu }_i)=\alpha +{\sum }_j{s}_j(x_{ij})$. Centering the smooth functions identifies the intercept. The response family and link function remain part of the model; an unspecified software family must not be inferred from the response name.

An additive regression model represents a conditional mean as a sum of univariate functions and an intercept. It permits nonlinear effects without estimating a general high-dimensional response surface. The backfitting algorithm smooths each component's partial residual. Centering separates constants from the intercept, while concurvity can still prevent component identification.

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In an additive regression model, each component can transfer a constant to the regression intercept, so impose empirical mean-zero constraints, or population mean-zero constraints when appropriate. Even after centering, nonzero component perturbations $h_j$ with ${\sum }_j{h}_j(x_{ij})=0$ at every observed point make the additive decomposition nonidentifiable. Extra smoothing penalties may resolve the fitted decomposition, but centering alone does not remove concurvity.

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Concurvity is the nonlinear analogue of multicollinearity: a function of one predictor is exactly or approximately expressible as a sum of functions of other predictors. Exact concurvity can make an additive decomposition nonunique; approximate concurvity makes components difficult to estimate stably. A unique fitted sum need not imply unique individual components.

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The backfitting algorithm estimates each additive mean function by smoothing its partial residual, obtained after subtracting the current fits of all other terms, and cycling through terms until convergence. Centering each smooth separates its constant part from the intercept. For fixed quadratic smoothness penalties it is block coordinate minimization of a penalized least-squares objective; identifiability and positive definiteness of the constrained problem ensure a unique converged fit. Non-Gaussian generalized additive models can use weighted backfitting inside iteratively reweighted least squares.

For linear smoothers, let $C=I-11^T/n$ and $T_j=C{S}_{j}C$. The centered fixed-point equations of the backfitting algorithm have diagonal identity blocks and off-diagonal blocks $T_j$. On the centered component space, a consistent system has a unique answer exactly when its homogeneous system has only the zero solution. For two components the reduced equation is $(I-T_1{T}_2)f_1=T_1(I-T_2)Cy$. A common centered vector reproduced by both smoothers yields opposite perturbations of the components without changing their sum.

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A generalized additive mixed model supplements smooth population effects with latent random effects. Given these effects, responses follow a specified response family; integrating them out induces dependence between related observations.