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 and . The centered fixed-point equations of the backfitting algorithm have diagonal identity blocks and off-diagonal blocks . 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 . A common centered vector reproduced by both smoothers yields opposite perturbations of the components without changing their sum.
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Backfitting is an iterative algorithm used primarily in the context of fitting additive models, particularly generalized additive models (GAMs). An additive model assumes that the response variable can be expressed as a sum of smooth functions of predictor variables. The backfitting algorithm helps to estimate the smooth functions in such models.