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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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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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Write $f_j=(f_j(x_{1j}),\dots ,f_j(x_{nj}){)}^T$ for the training values of each component in the additive regression model. Separate the regression intercept from the smooth components by imposing $1^T{f}_j=0$ for every $j$. Without these constraints, adding a constant to a component and subtracting it from the intercept gives exactly the same mean function. A population alternative is $\mathbb{E}{f}_j(X_j)=0$ when those expectations exist. Additional identifiability of additive regression components requirements exclude redundant component directions, as discussed below.

The backfitting algorithm proceeds as follows:

Because every component remains centered, the least-squares update of the intercept remains $\overline{y}$ throughout. The essential update isFor penalized least-squares smoothers these updates are coordinate descent on the additive fitting objective. For a completely arbitrary smoother, convergence is an additional property to establish, not a consequence of merely specifying the iteration. Centering removes the constant ambiguity but does not by itself remove concurvity.

There are two distinct identifiability issues. Without centering, constants can move between the regression intercept and the additive components. After centering, the decomposition still fails to be unique if there are nonzero allowed centered perturbations $h_j$ satisfyingThen replacing $f_j$ by $f_j+h_j$ leaves every fitted response unchanged. This nonlinear analogue of multicollinearity is concurvity. For example, if two predictor columns are identical, a centered function $h$ can be added to the effect of the first and subtracted from the effect of the second. Conceptually, the data observe their combined effect but cannot tell which predictor deserves the credit. A unique fitted sum can therefore coexist with nonunique individual effects. Smoothness penalties can distinguish some such decompositions, so a representation ambiguity is not automatically an ambiguity of every penalized estimator.

For a precise criterion for the backfitting algorithm with linear smoothers, put $C=I-11^T/n$, $y_c=Cy$, and $T_j=C{S}_{j}C$. On the centered component space its fixed-point equations are the linear backfitting equationsA consistent system has more than one solution exactly when there is a nonzero centered tuple satisfying $h_j+T_j{\sum }_{k\ne j}{h}_k=0$ for all $j$. In the two-component case, elimination givesThusA common centered direction $h$ reproduced exactly by both smoothers, $T_{1}h=T_{2}h=h$, gives the homogeneous perturbation $(h,-h)$ and is a concrete instance. For example, two identical predictor columns with smoothers that reproduce the same centered linear trend have this property. Approximate concurvity yields an eigenvalue near one, causing poor conditioning and slow or unstable component estimation, but it does not constitute exact nonuniqueness. Failure of an iteration to converge is also distinct from the existence of multiple fixed points.

Use the backfitting algorithm to alternate conditional updates of the two additive functions. Initialize $\alpha =\overline{Y}$ and both centered function vectors at zero. With $S_e,S_d$ the smoothing matrices for the chosen experience and education spline penalties, one cycle isfollowed byThe centering transfers any constant component to the intercept; update $\alpha$ to the mean of $Y-f_e-f_d$ if necessary, which is $\overline{Y}$ under the stated constraints. Repeat until changes in the functions or objective are negligible. Each function is estimated by smoothing its partial residual, namely the response after subtracting all other currently fitted terms.

For fixed penalties these updates are block minimizations ofsubject to the centering constraints. Each step cannot increase the objective; with identifiable additive terms and a positive-definite constrained quadratic, backfitting converges to its unique minimizer. Highly dependent predictors can make the decomposition poorly identified or slow convergence. The Gaussian identity-link model needs these least-squares updates directly; non-Gaussian generalized additive models use weighted backfitting inside iteratively reweighted least squares. The package implementation also separates each spline's unpenalized linear component from its nonlinear component, giving the two ANOVA tables displayed in part (e).