Write for the training values of each component in the additive regression model. Separate the regression intercept from the smooth components by imposing for every . 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 when those expectations exist. Additional identifiability of additive regression components requirements exclude redundant component directions, as discussed below.
The backfitting algorithm proceeds as follows:
- Initialize and for all , or use another centered starting fit. Choose the smoother and its tuning parameters for each predictor.
- Cycle through . Form the partial residual , using the newest available component estimates.
- Apply the generic smoother in predictor : . Center its fitted values and replace the current component by . The same centering constant is subtracted from its function when predicting at a new predictor value.
- Repeat complete cycles until the components and fitted sum change by less than the chosen numerical tolerance. Return .
Because every component remains centered, the least-squares update of the intercept remains 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.
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