# conversion of multivariate time series into functional data analysis

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?

$$f_1, \ldots, f_n$$ 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 $$\hat f_{n+1} = \sum_{j=1}^K \hat a_{n+1, j} \phi_j$$