conversion of multivariate time series into functional data analysis

Question

I am trying to generalize the functional auto-regressive model of order one to some order $p$. For this, I've calculated the functional principle components and choose a particular $pc$s which explain a given amount of variation. Then from these $Fpc$s I calculate its scores and using the var package to predict the given data for one a head. Now I want to convert these vectors into functional form? How do I do this?

Answer 1

If I understand correctly you have the following model:

$f_1, \ldots, f_n$ functional observations. You wish to predict $f_{n+1}$. For that you find the functional principal components $\phi_1, \ldots, \phi_K$ and scores $ (a_{i,1}, \ldots, a_{i,K})$ such that $< f_i, \phi_j> = a_{i,j}$. As in the multivariate case, the $\{ \phi_j\}$ form an orthonormal basis of the space that best aproximates (among all K dimentional subspces) the functionals observations. You have that $$f_i \approx \sum_{j=1}^K a_{i,j} \phi_j.$$ Then you are then fitting a vector arima to the multivariate scores and predict the next step $(\hat a_{n+1, 1}, \ldots, \hat a_{n+1, K})$. The natural way to convert back the prediction is $$ \hat f_{n+1} = \sum_{j=1}^K \hat a_{n+1, j} \phi_j$$

Of course the same idea holds if you are predicting any number of steps ahead.