content_image/304698.jpg content_image/304697.jpg content_image/304696.jpg content_image/304695.jpg
we can rewrite (5) as 

<img src='content_image/304698.jpg'>

Then the only parameter that remains to be estimated is the regression slope β$_{0}$ . 

Since { φ$_{k}$ } is an orthonormal basis of the X -space and { ψ$_{l}$ } is an orthonormal basis of the Y -space, without loss of generality the regression slope can be expressed as 

<img src='content_image/304697.jpg'>

In matrix form, β$_{0}$ ( s, t ) = φ ( s ) $^{T}$Θ$_{0}$ ψ ( t ) , where φ ( s ) = ( φ$_{1}$ ( s ) , . . . , φ$_{p}$ ( s )) T and ψ ( t ) = ( ψ$_{1}$ ( t ) , . . . , ψ$_{q}$ ( t )) $^{T}$. If we also collect the component scores { U$_{k}$ } and { V$_{l}$ } into vectors U ∈ R p and V ∈ R $^{q}$, from (3), (4), (6) and (7) we obtain 

<img src='content_image/304696.jpg'>

where W ∈ R q is the random vector with elements W$_{l}$ = 〈 Z, ψ$_{l}$ 〉 . This reduces the functional regression model (6) to a simpler multivariate regression model, 

and the problem now is to estimate the regression matrix Θ$_{0}$ . 

<img src='content_image/304695.jpg'>

