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## 2 Method 

## 2.1 Background: classical functional linear regression 

The functional approach to longitudinal data analysis assumes that the observations ( x$_{1}$ , y$_{1}$ ) , . . . , ( x$_{n}$ , y$_{n}$ ) are discrete measurements of underlying continuous curves, so 

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where { X$_{i}$ ( s ) } and { Y$_{i}$ ( t ) } are the trajectories of interest, { ε$_{ij}$ } and { ε ′ ij } are random mea- surement errors, and { s$_{ij}$ } and { t$_{ij}$ } are the time points where the data is observed. The X$_{i}$ ( s ) s and the Y$_{i}$ ( t ) s are random functions that we assume independent and identically distributed realizations of a pair ( X ( s ) , Y ( t )) . 

Suppose X ( s ) and Y ( t ) are square-integrable functions on an interval [ a, b ] . Define the norm ‖ f ‖ = { ∫ b a f $^{2}$( s ) ds } 1 / 2 and the inner product 〈 f, g 〉 = ∫ b a f ( s ) g ( s ) ds . If E( ‖ X ‖ $^{2}$) and E( ‖ Y ‖ $^{2}$) are finite, then X ( s ) and Y ( t ) admit the decomposition 

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known as the Karhunen–Lo`eve decomposition (Ash and Gardne r 1975, ch. 1.4), where µ$_{X}$ ( s ) = E { X ( s ) } , µ$_{Y}$ ( t ) = E { Y ( t ) } , { φ$_{k}$ ( s ) } and { ψ$_{l}$ ( t ) } are orthonormal functions (i.e. 〈 φ$_{k}$, φ$_{k}$ ′ 〉 = δ$_{kk}$ ′ and 〈 ψ$_{l}$, ψ$_{l}$ ′ 〉 = δ$_{ll}$ ′ , where δ is Kronecker’s delta), and { U$_{k}$ } and { V$_{l}$ } are random variables with zero mean and finite variance (with out loss of generality, 

