A Predictive Functional Principal Component Analysis of Well-Being Data

We consider the case of the principal components of a multivariate random vector that obeys a linear mixed model. The random vector itself then lies in a lower dimensional subspace. This situation suggests that this subspace can be modeled by the probabilistic (random-effects) principal components. We employ a linear predictor adjusted by the residual part of the probabilistic principal components, that results not explained by the longitudinal time-varying linear model. A new predictor is given, employing the vector of scores that comes from that principal components, and the resulting scores approximated by spline curves as functionals. The application to the official Italian well-being data shows some of the features of the method.