In other words, rk,T-1(t) is the residual obtained from the optimal prediction of y(t-(k+1)) with the future values ;this explains the term ``backward'' residuals. The geometrical interpretations of and rk,T-1 appear together on Figure .
Geometrical interpretation of the residuals and
is the orthogonal complement of the data yT on Y1,k,T.
The predictable part of yT of order k is the orthogonal projection of yT on Y1,k,T.
rk,T-1 is the orthogonal complement of the shifted data Zk+1yT on Y1,k,T.
The new space of regressors Y1,k+1,T is .
is the orthogonal complement of yT on Y1,k+1,T.
is equal to rk,T-1 multiplied by the reflection coefficient Krk+1.
Finally, knowing that the forward residuals are the orthogonal complement of yT on , so on , we have the recursive formulas: