** Next:** Recursions: order updating
** Up:** THE LSL ALGORITHM
** Previous:** Forward residuals

We define the vectors of backward residuals *r*_{k,T-1} with the recursions:
| |
(1) |

The index appears because these residuals can all be expressed as *Z*
applied to a known vector: *r*_{0,T}=*y*_{T} 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,
*r*_{k,T-1} is orthogonal to .So, with these recursions, we see that *r*_{k,T-1} is the **orthogonal
complement** of the shifted data *Z*^{k+1}*y*_{T} on :
In other words, *r*_{k,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 *r*_{k,T-1} appear together on Figure .
Geometrical interpretation of the residuals and
*r*_{k,T-1}.

is the orthogonal complement of the data *y*_{T} on *Y*_{1,k,T}.

The predictable part of *y*_{T} of order *k* is the orthogonal projection of *y*_{T} on *Y*_{1,k,T}.

*r*_{k,T-1} is the orthogonal complement of the shifted data *Z*^{k+1}*y*_{T} on *Y*_{1,k,T}.

The new space of regressors *Y*_{1,k+1,T} is .

is the orthogonal complement of *y*_{T} on *Y*_{1,k+1,T}.

is equal to *r*_{k,T-1} multiplied by the reflection
coefficient *K*^{r}_{k+1}.

Finally, knowing that the forward residuals are the orthogonal
complement of *y*_{T} 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