** Next:** Backward residuals
** Up:** THE LSL ALGORITHM
** Previous:** The basic LSL algorithm:

The forward residuals can be expressed in terms of orthogonal
projections. Effectively, defining
*P*_{1,n,T}=*A*_{1,n,T}(*A*'_{1,n,T}*A*_{1,n,T})^{-1}*A*'_{1,n,T}, we have:
Because *P*^{2}_{1,n,T}=*P*'_{1,n,T}=*P*_{1,n,T}, the operator *P*_{1,n,T} is an
**orthogonal projector**. It projects the data vector *y*_{T} on the space
*Y*_{1,n,T} spanned by the columns of *A*_{1,n,T}. These columns are simply
the past observations (or shifted data), *Zy*_{T} to *Z*^{n}*y*_{T}, where *Z*
is the unit-delay shift operator.
The *predictable* part of *y*_{T} is *P*_{1,n,T}*y*_{T}, the **orthogonal
projection** of *y*_{T} on *Y*_{1,n,T}. The *residuals* are then
orthogonal to this space: they are called the **orthogonal complement**
of *y*_{T} on *Y*_{1,n,T}. Their geometric interpretation is illustrated in
Figure .
Defining the operator , the forward residuals
are ; is also an
orthogonal projector.

So, we need to compute *P*_{1,n,T}*y*_{T}. But this projection is easy to compute,
if we already know an orthogonal basis of the
space *Y*_{1,n,T}. Effectively, the projection of *y*_{T} is:

The backward residuals will in fact form this orthogonal basis.

** Next:** Backward residuals
** Up:** THE LSL ALGORITHM
** Previous:** The basic LSL algorithm:
Stanford Exploration Project

1/13/1998