= Linear backfitting equations
{title2=$f_j+T_j\sum_{k\ne j}f_k=T_jCy$}
For <linear smoothers>, let $C=I-\mathbf1\mathbf1^T/n$ and $T_j=CS_jC$. 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 $(I-T_1T_2)f_1=T_1(I-T_2)Cy$. A common centered vector reproduced by both smoothers yields opposite perturbations of the components without changing their sum.
// Target: probability-and-statistics.bigb
Back to article page
