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## Backward residuals

We define the vectors of backward residuals rk,T-1 with the recursions:
 (1)
The index appears because these residuals can all be expressed as Z applied to a known vector: r0,T=yT for example. These recursions are in fact equivalent to performing a Gram-Schmidt orthogonalization of the space . Then, for any k, the space is equal to ; also, rk,T-1 is orthogonal to .So, with these recursions, we see that rk,T-1 is the orthogonal complement of the shifted data Zk+1yT on :

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 rk,T-1.
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:

Next: Recursions: order updating Up: THE LSL ALGORITHM Previous: Forward residuals
Stanford Exploration Project
1/13/1998