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?
1 Answer
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.
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$\begingroup$ Thanks a lot Manuel a really understand your point of view, but as i choose the first 4 FPC and their score, which only four forecast of single day, i need to forecast each 24 hours(mean a day) and then need to generalize the result for 365 days (mean a year) please guide me how i do this in R . $\endgroup$ Oct 17, 2020 at 3:23