Because P21,n,T=P'1,n,T=P1,n,T, the operator P1,n,T is an orthogonal projector. It projects the data vector yT on the space Y1,n,T spanned by the columns of A1,n,T. These columns are simply the past observations (or shifted data), ZyT to ZnyT, where Z is the unit-delay shift operator.
The predictable part of yT is P1,n,TyT, the orthogonal projection of yT on Y1,n,T. The residuals are then orthogonal to this space: they are called the orthogonal complement of yT on Y1,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 P1,n,TyT. But this projection is easy to compute, if we already know an orthogonal basis of the space Y1,n,T. Effectively, the projection of yT is:
The backward residuals will in fact form this orthogonal basis.